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  • Committer: Package Import Robot
  • Author(s): Sylvestre Ledru
  • Date: 2012-08-30 14:42:38 UTC
  • mfrom: (1.4.7)
  • Revision ID: package-import@ubuntu.com-20120830144238-c1y2og7dbm7m9nig
Tags: 5.4.0-beta-3-1~exp1
* New upstream release
* Update the scirenderer dep
* Get ride of libjhdf5-java dependency

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modules/differential_equations/help/ja_JP/bvode.xml
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<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" version="5.0-subset Scilab" xml:id="bvode" xml:lang="ja">
15
 
  <refnamediv>
16
 
    <refname>bvode</refname>
17
 
    <refpurpose>コロケーション法によるODEの境界値問題</refpurpose>
18
 
  </refnamediv>
19
 
  <refnamediv xml:id="bvodeS">
20
 
    <refname>bvodeS</refname>
21
 
    <refpurpose>bvodeの簡易的なコール</refpurpose>
22
 
  </refnamediv>
23
 
  <refsynopsisdiv>
24
 
    <title>呼び出し手順</title>
25
 
    <synopsis>
26
 
      zu=bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,fsub,dfsub,gsub,dgsub,guess)
27
 
    </synopsis>
28
 
    <synopsis>
29
 
      zu=bvodeS(xpoints,m,N,x_low,x_up,fsub,gsub,zeta, &lt;optional_args&gt;)
30
 
    </synopsis>
31
 
  </refsynopsisdiv>
32
 
  <refsection>
33
 
    <title>パラメータ</title>
34
 
    <variablelist>
35
 
      <varlistentry>
36
 
        <term>zu</term>
37
 
        <listitem>
38
 
          <para>
39
 
            大きさMの列ベクトル.
40
 
            複数の点で指定したメッシュ上で評価したODEの解.
41
 
            各点における z(u(x)) が含まれます.
42
 
          </para>
43
 
        </listitem>
44
 
      </varlistentry>
45
 
      <varlistentry>
46
 
        <term>xpoints</term>
47
 
        <listitem>
48
 
          <para>
49
 
            解を観測する点を指定する配列.
50
 
          </para>
51
 
        </listitem>
52
 
      </varlistentry>
53
 
      <varlistentry>
54
 
        <term>N</term>
55
 
        <listitem>
56
 
          <para>整数値を有するスカラー,微分方程式の数
57
 
            (N &lt;= 20).
58
 
          </para>
59
 
        </listitem>
60
 
      </varlistentry>
61
 
      <varlistentry>
62
 
        <term>m</term>
63
 
        <listitem>
64
 
          <para>
65
 
            整数要素を有する大きさ <literal>N</literal> のベクトル.
66
 
            <literal>m</literal>は各微分方程式の次数のベクトルで,
67
 
            <literal>m(i)</literal>はi番目の微分方程式の次数を指定します.
68
 
            以下に示す <literal>M</literal>は
69
 
            <literal>m</literal>の要素の合計を表します.
70
 
          </para>
71
 
        </listitem>
72
 
      </varlistentry>
73
 
      <varlistentry>
74
 
        <term>x_low</term>
75
 
        <listitem>
76
 
          <para>スカラー: 区間の左端</para>
77
 
        </listitem>
78
 
      </varlistentry>
79
 
      <varlistentry>
80
 
        <term>x_up</term>
81
 
        <listitem>
82
 
          <para>スカラー: 区間の右端</para>
83
 
        </listitem>
84
 
      </varlistentry>
85
 
      <varlistentry>
86
 
        <term>zeta</term>
87
 
        <listitem>
88
 
          <para>
89
 
            大きさ Mのベクトル,<literal>zeta(j)</literal> 
90
 
            は, j番目の端の条件点(境界値)を指定します.
91
 
            x_low&lt;=zeta(j)
92
 
            &lt;= zeta(j+1)<literal>&lt;=x_up</literal>
93
 
            となります.
94
 
          </para>
95
 
          <para>
96
 
            全ての境界条件点は,
97
 
            使用される全てのメッシュにおけるメッシュ点とする必要があります.
98
 
            以下の <literal>ipar(11)</literal> および
99
 
            <literal>fixpnt</literal>の説明を参照ください.
100
 
          </para>
101
 
        </listitem>
102
 
      </varlistentry>
103
 
      <varlistentry>
104
 
        <term>ipar</term>
105
 
        <listitem>
106
 
          <para>11個の整数要素を有する配列:</para>
107
 
          <para>
108
 
            [<literal>nonlin, collpnt, subint, ntol, ndimf, ndimi, iprint,
109
 
              iread, iguess, rstart,nfxpnt]
110
 
            </literal>
111
 
          </para>
112
 
          <variablelist>
113
 
            <varlistentry>
114
 
              <term>nonlin: ipar(1)</term>
115
 
              <listitem>
116
 
                <para>問題が線形の場合: 0 ,問題が非線形の場合: 1</para>
117
 
              </listitem>
118
 
            </varlistentry>
119
 
            <varlistentry>
120
 
              <term>collpnt: ipar(2)</term>
121
 
              <listitem>
122
 
                <para>
123
 
                  サブ区間毎のコロケーション点の数を指定します.
124
 
                  ただし, 
125
 
                  where max(m(j)) &lt;= collpnt &lt;= 7
126
 
                </para>
127
 
                <para>
128
 
                  <literal>ipar(2)=0</literal> の場合,
129
 
                  <literal>collpnt</literal> は ( max(m(j))+1 ,
130
 
                  5-max(m(j)) )に設定されます
131
 
                </para>
132
 
              </listitem>
133
 
            </varlistentry>
134
 
            <varlistentry>
135
 
              <term>subint: ipar(3)</term>
136
 
              <listitem>
137
 
                <para>初期メッシュのサブ区間の数を指定します.
138
 
                  <literal>ipar(3) = 0</literal> の場合,
139
 
                  適宜 
140
 
                  <literal>subint = 5</literal>と設定します.
141
 
                </para>
142
 
              </listitem>
143
 
            </varlistentry>
144
 
            <varlistentry>
145
 
              <term>ntol: ipar(4)</term>
146
 
              <listitem>
147
 
                <para>
148
 
                  解と微分の許容誤差の数を指定します.
149
 
                  0 &lt; ntol &lt;= M とする必要があります.
150
 
                  <literal>ipar(4)</literal> は,
151
 
                  <literal>tol</literal> 引数の次元または
152
 
                  <literal>0</literal>に設定する必要があります.
153
 
                  後者の場合, 実際の値は自動的に
154
 
                  <literal>size(tol,'*')</literal> に設定されます.
155
 
                </para>
156
 
              </listitem>
157
 
            </varlistentry>
158
 
            <varlistentry>
159
 
              <term>ndimf: ipar(5)</term>
160
 
              <listitem>
161
 
                <para>
162
 
                  <literal>fspace</literal>(実数の作業配列)の次元を指定します.
163
 
                  この値は, 
164
 
                  サブ区間の最大数 nmax への拘束条件を指定します.
165
 
                </para>
166
 
                <para>ipar(5) の値は,拘束条件
167
 
                  ipar(5)&gt;=nmax*nsizef を考慮する必要があります. ただし
168
 
                </para>
169
 
                <para>
170
 
                  <literal>nsizef=4+3*M+(5+collpnt*N)*(collpnt*N+M)+(2*M-nrec)*2*M</literal>
171
 
                  (<literal>nrec</literal> は終端境界条件の数です
172
 
                  ).
173
 
                </para>
174
 
              </listitem>
175
 
            </varlistentry>
176
 
            <varlistentry>
177
 
              <term>ndimi: ipar(6)</term>
178
 
              <listitem>
179
 
                <para>ispace (整数作業配列)の次元を指定します.
180
 
                  この値は, サブ区間の最大数 nmax への拘束条件を指定します.
181
 
                </para>
182
 
                <para>ipar(6) の値は, 拘束条件 ipar(6)&gt;=nmax*nsizei
183
 
                  を考慮する必要があります. ただし,
184
 
                </para>
185
 
                <para>
186
 
                  <literal>nsizei=3 +
187
 
                  </literal>
188
 
                  <literal>collpnt*N+M</literal>.
189
 
                </para>
190
 
              </listitem>
191
 
            </varlistentry>
192
 
            <varlistentry>
193
 
              <term>iprint: ipar(7)</term>
194
 
              <listitem>
195
 
                <para>出力制御, 以下の値とします:</para>
196
 
                <variablelist>
197
 
                  <varlistentry>
198
 
                    <term>-1</term>
199
 
                    <listitem>
200
 
                      <para>完全な診断出力の場合</para>
201
 
                    </listitem>
202
 
                  </varlistentry>
203
 
                  <varlistentry>
204
 
                    <term>0</term>
205
 
                    <listitem>
206
 
                      <para>部分的な出力の場合</para>
207
 
                    </listitem>
208
 
                  </varlistentry>
209
 
                  <varlistentry>
210
 
                    <term>1</term>
211
 
                    <listitem>
212
 
                      <para>出力しない場合</para>
213
 
                    </listitem>
214
 
                  </varlistentry>
215
 
                </variablelist>
216
 
              </listitem>
217
 
            </varlistentry>
218
 
            <varlistentry>
219
 
              <term>iread: ipar(8)</term>
220
 
              <listitem>
221
 
                <variablelist>
222
 
                  <varlistentry>
223
 
                    <term>= 0</term>
224
 
                    <listitem>
225
 
                      <para>
226
 
                        bvodeが一様な初期メッシュを生成するようにします.
227
 
                      </para>
228
 
                    </listitem>
229
 
                  </varlistentry>
230
 
                  <varlistentry>
231
 
                    <term>= xx</term>
232
 
                    <listitem>
233
 
                      <para>他の値は, Scilabでは未実装です.</para>
234
 
                      <variablelist>
235
 
                        <varlistentry>
236
 
                          <term>= 1</term>
237
 
                          <listitem>
238
 
                            <para>初期メッシュがユーザにより指定される場合.
239
 
                              fspace で以下のように定義されます: 
240
 
                            </para>
241
 
                            <para>
242
 
                              メッシュは <literal>fspace(1), ...,
243
 
                                fspace(n+1)
244
 
                              </literal>
245
 
                              となります. 
246
 
                              ユーザは内部メッシュ点<literal>fspace(j) = x(j),
247
 
                                j = 2, ..., n.
248
 
                              </literal>
249
 
                              のみを指定する必要があります.
250
 
                            </para>
251
 
                          </listitem>
252
 
                        </varlistentry>
253
 
                        <varlistentry>
254
 
                          <term>= 2 初期メッシュがユーザにより指定される場合.</term>
255
 
                          <listitem>
256
 
                            <para>
257
 
                              <literal>ipar(8)=1</literal>が指定された時と同様であり, 
258
 
                              加えて,適応型メッシュ選択は行われません.
259
 
                            </para>
260
 
                          </listitem>
261
 
                        </varlistentry>
262
 
                      </variablelist>
263
 
                    </listitem>
264
 
                  </varlistentry>
265
 
                </variablelist>
266
 
              </listitem>
267
 
            </varlistentry>
268
 
            <varlistentry>
269
 
              <term>iguess: ipar(9)</term>
270
 
              <listitem>
271
 
                <variablelist>
272
 
                  <varlistentry>
273
 
                    <term>= 0</term>
274
 
                    <listitem>
275
 
                      <para>解の初期推定値が指定されない場合.</para>
276
 
                    </listitem>
277
 
                  </varlistentry>
278
 
                  <varlistentry>
279
 
                    <term>= 1</term>
280
 
                    <listitem>
281
 
                      <para>
282
 
                        引数 <literal>guess</literal> により,
283
 
                        初期推定値がユーザにより指定される場合.
284
 
                      </para>
285
 
                    </listitem>
286
 
                  </varlistentry>
287
 
                  <varlistentry>
288
 
                    <term>= 2</term>
289
 
                    <listitem>
290
 
                      <para>
291
 
                        初期メッシュと近似解係数がユーザにより fspace で指定される場合
292
 
                        (前のメッシュと新規メッシュは同じものになります).
293
 
                      </para>
294
 
                    </listitem>
295
 
                  </varlistentry>
296
 
                  <varlistentry>
297
 
                    <term>= 3</term>
298
 
                    <listitem>
299
 
                      <para>
300
 
                        前のメッシュと近似解の係数がユーザによりfspace で指定される場合,
301
 
                        新規メッシュは 2倍荒くとられます.
302
 
                        すなわち, 前のメッシュの各2点毎に1点となります．
303
 
                      </para>
304
 
                    </listitem>
305
 
                  </varlistentry>
306
 
                  <varlistentry>
307
 
                    <term>= 4</term>
308
 
                    <listitem>
309
 
                      <para>
310
 
                        前の初期メッシュと近似解係数に加えて,同時に
311
 
                        新規メッシュが fspace に出力されます.
312
 
                        (詳細はiguess = 2, 3, および 4の出力の説明を参照ください)
313
 
                      </para>
314
 
                    </listitem>
315
 
                  </varlistentry>
316
 
                </variablelist>
317
 
              </listitem>
318
 
            </varlistentry>
319
 
            <varlistentry>
320
 
              <term>ireg: ipar(10)</term>
321
 
              <listitem>
322
 
                <variablelist>
323
 
                  <varlistentry>
324
 
                    <term>= 0</term>
325
 
                    <listitem>
326
 
                      <para>問題が標準の場合</para>
327
 
                    </listitem>
328
 
                  </varlistentry>
329
 
                  <varlistentry>
330
 
                    <term>= 1</term>
331
 
                    <listitem>
332
 
                      <para>
333
 
                        最初の緩和係数が <literal>ireg</literal> に等しい場合,
334
 
                        非線形反復は過去の収束結果に基づきません.
335
 
                        (別の敏感な非線形問題のみで使用されます)
336
 
                      </para>
337
 
                    </listitem>
338
 
                  </varlistentry>
339
 
                  <varlistentry>
340
 
                    <term>= 2</term>
341
 
                    <listitem>
342
 
                      <para>
343
 
                        以下の場合,直ちにリターンします.
344
 
                        (a) 2回連続で収束しない場合,
345
 
                        または
346
 
                        (b)最初に誤差推定値を得た後.
347
 
                      </para>
348
 
                    </listitem>
349
 
                  </varlistentry>
350
 
                </variablelist>
351
 
              </listitem>
352
 
            </varlistentry>
353
 
            <varlistentry>
354
 
              <term>nfxpnt: ipar(11)</term>
355
 
              <listitem>
356
 
                <para>
357
 
                  <literal>x_low</literal> および <literal>x_up</literal>
358
 
                  以外のメッシュの固定点の数(<literal>fixpnt</literal>の次元)を指定します.
359
 
                  <literal>ipar(11)</literal> は,引数
360
 
                  <literal>fixpnt</literal>の次元または
361
 
                  <literal>0</literal>に設定する必要があります.
362
 
                  後者の場合, 実際の値は自動的に
363
 
                  <literal>size(fixpnt,'*')</literal> に設定されます.
364
 
                </para>
365
 
              </listitem>
366
 
            </varlistentry>
367
 
          </variablelist>
368
 
        </listitem>
369
 
      </varlistentry>
370
 
      <varlistentry>
371
 
        <term>ltol</term>
372
 
        <listitem>
373
 
          <para>
374
 
            次元 <literal>ntol=ipar(4)</literal>の配列.
375
 
            <literal>ltol(j) = l</literal> は,
376
 
            <literal> tol</literal> 配列の <literal>j</literal>番目の許容誤差を指定します.これは,
377
 
            以下の<literal>l</literal>番目の要素の誤差を制御します.
378
 
            <inlinemediaobject>
379
 
              <imageobject>
380
 
                <imagedata>
381
 
                  <mml:math>
382
 
                    <mml:semantics>
383
 
                      <mml:mi>z</mml:mi>
384
 
                      <mml:mrow>
385
 
                        <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
386
 
                      </mml:mrow>
387
 
                    </mml:semantics>
388
 
                  </mml:math>
389
 
                </imagedata>
390
 
              </imageobject>
391
 
            </inlinemediaobject>
392
 
            .以下も必要です:
393
 
          </para>
394
 
          <para>
395
 
            <literal>1 &lt;= ltol(1) &lt; ltol(2) &lt; ... &lt; ltol(ntol)
396
 
              &lt;= M
397
 
            </literal>
398
 
          </para>
399
 
        </listitem>
400
 
      </varlistentry>
401
 
      <varlistentry>
402
 
        <term>tol</term>
403
 
        <listitem>
404
 
          <para>
405
 
            <literal>ntol=ipar(4)</literal>次の配列.
406
 
          </para>
407
 
          <para>
408
 
            <literal>tol(j)</literal>は,
409
 
            以下の <literal>ltol(j)</literal> 番目の要素の許容誤差です.
410
 
            <inlinemediaobject>
411
 
              <imageobject>
412
 
                <imagedata>
413
 
                  <mml:math>
414
 
                    <mml:semantics>
415
 
                      <mml:mi>z</mml:mi>
416
 
                      <mml:mrow>
417
 
                        <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
418
 
                      </mml:mrow>
419
 
                    </mml:semantics>
420
 
                  </mml:math>
421
 
                </imagedata>
422
 
              </imageobject>
423
 
            </inlinemediaobject>
424
 
            .つまり, このコードは以下の条件を
425
 
            満たそうとします.
426
 
            <inlinemediaobject>
427
 
              <imageobject>
428
 
                <imagedata>
429
 
                  <mml:math>
430
 
                    <mml:semantics>
431
 
                      <mml:mrow>
432
 
                        <mml:mrow>
433
 
                          <mml:mfenced mml:close="∣" mml:open="∣">
434
 
                            <mml:msub>
435
 
                              <mml:mrow>
436
 
                                <mml:mo mml:stretchy="false">(</mml:mo>
437
 
                                <mml:mrow>
438
 
                                  <mml:mi>z</mml:mi>
439
 
                                  <mml:mrow>
440
 
                                    <mml:mrow>
441
 
                                      <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
442
 
                                    </mml:mrow>
443
 
                                    <mml:mo mml:stretchy="false">−</mml:mo><mml:mi>z</mml:mi>
444
 
                                  </mml:mrow>
445
 
                                  <mml:mrow>
446
 
                                    <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
447
 
                                  </mml:mrow>
448
 
                                </mml:mrow>
449
 
                                <mml:mo mml:stretchy="false">)</mml:mo>
450
 
                              </mml:mrow>
451
 
                              <mml:mrow>
452
 
                                <mml:mi mml:fontstyle="italic">ltol</mml:mi>
453
 
                                <mml:mrow>
454
 
                                  <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
455
 
                                </mml:mrow>
456
 
                              </mml:mrow>
457
 
                            </mml:msub>
458
 
                          </mml:mfenced>
459
 
                          <mml:mo mml:stretchy="false">≤</mml:mo><mml:mi mml:fontstyle="italic">tol</mml:mi>
460
 
                        </mml:mrow>
461
 
                        <mml:mrow>
462
 
                          <mml:mrow>
463
 
                            <mml:mrow>
464
 
                              <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
465
 
                            </mml:mrow>
466
 
                            <mml:mo mml:stretchy="false">⋅</mml:mo>
467
 
                            <mml:mfenced mml:close="∣" mml:open="∣">
468
 
                              <mml:msub>
469
 
                                <mml:mrow>
470
 
                                  <mml:mo mml:stretchy="false">(</mml:mo>
471
 
                                  <mml:mrow>
472
 
                                    <mml:mi>z</mml:mi>
473
 
                                    <mml:mrow>
474
 
                                      <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
475
 
                                    </mml:mrow>
476
 
                                  </mml:mrow>
477
 
                                  <mml:mo mml:stretchy="false">)</mml:mo>
478
 
                                </mml:mrow>
479
 
                                <mml:mrow>
480
 
                                  <mml:mi mml:fontstyle="italic">ltol</mml:mi>
481
 
                                  <mml:mrow>
482
 
                                    <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
483
 
                                  </mml:mrow>
484
 
                                </mml:mrow>
485
 
                              </mml:msub>
486
 
                            </mml:mfenced>
487
 
                          </mml:mrow>
488
 
                          <mml:mo mml:stretchy="false">+</mml:mo><mml:mi mml:fontstyle="italic">tol</mml:mi>
489
 
                        </mml:mrow>
490
 
                        <mml:mrow>
491
 
                          <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
492
 
                        </mml:mrow>
493
 
                        <mml:mi>,</mml:mi><mml:mi mml:fontstyle="normal">for</mml:mi>
494
 
                        <mml:mrow>
495
 
                          <mml:mi>j</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn>
496
 
                        </mml:mrow>
497
 
                        <mml:mi mml:fontstyle="normal">:</mml:mi><mml:mi mml:fontstyle="normal">ntol</mml:mi>
498
 
                      </mml:mrow>
499
 
                      <mml:annotation mml:encoding="StarMath 5.0">abs(z(v)-z(u))_{ltol(j)}
500
 
                        &lt;= tol(j) cdot abs(z(u))_{ltol(j)} + tol(j), for
501
 
                        j=1:ntol
502
 
                      </mml:annotation>
503
 
                    </mml:semantics>
504
 
                  </mml:math>
505
 
                </imagedata>
506
 
              </imageobject>
507
 
            </inlinemediaobject>
508
 
            (サブ区間について)
509
 
          </para>
510
 
          <para>
511
 
            <inlinemediaobject>
512
 
              <imageobject>
513
 
                <imagedata>
514
 
                  <mml:math>
515
 
                    <mml:semantics>
516
 
                      <mml:mi>v</mml:mi>
517
 
                    </mml:semantics>
518
 
                  </mml:math>
519
 
                </imagedata>
520
 
              </imageobject>
521
 
            </inlinemediaobject>
522
 
            が近似解ベクトルの場合,
523
 
            <inlinemediaobject>
524
 
              <imageobject>
525
 
                <imagedata>
526
 
                  <mml:math>
527
 
                    <mml:semantics>
528
 
                      <mml:mi>u</mml:mi>
529
 
                    </mml:semantics>
530
 
                  </mml:math>
531
 
                </imagedata>
532
 
              </imageobject>
533
 
            </inlinemediaobject>
534
 
             は厳密解(未知)となります.
535
 
          </para>
536
 
        </listitem>
537
 
      </varlistentry>
538
 
      <varlistentry>
539
 
        <term>fixpnt</term>
540
 
        <listitem>
541
 
          <para>
542
 
            <literal>nfxpnt=ipar(11)</literal>次の配列. 
543
 
            各メッシュに含まれる<literal>x_low</literal> および
544
 
            <literal>x_up</literal>以外の点を含みます.
545
 
            このコードは,<literal>x_low</literal> および <literal>x_up</literal>
546
 
            以外の全ての周辺条件点(<literal>zeta</literal>の説明を参照)が
547
 
            <literal>fixpnt</literal>の中の固定点として
548
 
            含まれることを必要とします.
549
 
          </para>
550
 
        </listitem>
551
 
      </varlistentry>
552
 
      <varlistentry>
553
 
        <term>fsub</term>
554
 
        <listitem>
555
 
          <para>
556
 
            <link linkend="external">external</link> は,以下の
557
 
            列ベクトルの評価に使用されます. <literal>f=</literal>
558
 
            <inlinemediaobject>
559
 
              <imageobject>
560
 
                <imagedata>
561
 
                  <mml:math>
562
 
                    <mml:semantics>
563
 
                      <mml:mrow>
564
 
                        <mml:mo mml:stretchy="false">[</mml:mo>
565
 
                        <mml:mrow>
566
 
                          <mml:msub>
567
 
                            <mml:mi>f</mml:mi><mml:mn>1</mml:mn>
568
 
                          </mml:msub>
569
 
                          <mml:mrow>
570
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
571
 
                            <mml:mrow>
572
 
                              <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
573
 
                              <mml:mrow>
574
 
                                <mml:mo mml:stretchy="false">(</mml:mo>
575
 
                                <mml:mrow>
576
 
                                  <mml:mi>u</mml:mi>
577
 
                                  <mml:mrow>
578
 
                                    <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
579
 
                                  </mml:mrow>
580
 
                                </mml:mrow>
581
 
                                <mml:mo mml:stretchy="false">)</mml:mo>
582
 
                              </mml:mrow>
583
 
                            </mml:mrow>
584
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
585
 
                          </mml:mrow>
586
 
                          <mml:mi>;</mml:mi>
587
 
                          <mml:msub>
588
 
                            <mml:mi>f</mml:mi><mml:mn>2</mml:mn>
589
 
                          </mml:msub>
590
 
                          <mml:mrow>
591
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
592
 
                            <mml:mrow>
593
 
                              <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
594
 
                              <mml:mrow>
595
 
                                <mml:mo mml:stretchy="false">(</mml:mo>
596
 
                                <mml:mrow>
597
 
                                  <mml:mi>u</mml:mi>
598
 
                                  <mml:mrow>
599
 
                                    <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
600
 
                                  </mml:mrow>
601
 
                                </mml:mrow>
602
 
                                <mml:mo mml:stretchy="false">)</mml:mo>
603
 
                              </mml:mrow>
604
 
                            </mml:mrow>
605
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
606
 
                          </mml:mrow>
607
 
                          <mml:mi>,</mml:mi><mml:mo mml:stretchy="false">⋯</mml:mo><mml:mi>,</mml:mi>
608
 
                          <mml:msub>
609
 
                            <mml:mi>f</mml:mi><mml:mi>N</mml:mi>
610
 
                          </mml:msub>
611
 
                          <mml:mrow>
612
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
613
 
                            <mml:mrow>
614
 
                              <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
615
 
                              <mml:mrow>
616
 
                                <mml:mo mml:stretchy="false">(</mml:mo>
617
 
                                <mml:mrow>
618
 
                                  <mml:mi>u</mml:mi>
619
 
                                  <mml:mrow>
620
 
                                    <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
621
 
                                  </mml:mrow>
622
 
                                </mml:mrow>
623
 
                                <mml:mo mml:stretchy="false">)</mml:mo>
624
 
                              </mml:mrow>
625
 
                            </mml:mrow>
626
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
627
 
                          </mml:mrow>
628
 
                        </mml:mrow>
629
 
                        <mml:mo mml:stretchy="false">]</mml:mo>
630
 
                      </mml:mrow>
631
 
                      <mml:annotation mml:encoding="StarMath 5.0">[f_1(x,z(u(x)));
632
 
                        f_2(x,z(u(x))),dotsaxis,
633
 
                        f_N(x,z(u(x)))]
634
 
                      </mml:annotation>
635
 
                    </mml:semantics>
636
 
                  </mml:math>
637
 
                </imagedata>
638
 
              </imageobject>
639
 
            </inlinemediaobject>
640
 
            
641
 
            ただし,
642
 
            <literal>x_low</literal> &lt;= <literal>x</literal> &lt;= <literal>x_up</literal> の任意の<literal>x</literal>,
643
 
            および <literal>z=z(u(x))</literal> となる任意の<literal>z</literal> とします.
644
 
            (以下の説明を参照)
645
 
          </para>
646
 
          <para>外部関数は以下のように宣言を必要とします:</para>
647
 
          <itemizedlist>
648
 
            <listitem>
649
 
              <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
650
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
15
    <refnamediv>
 
16
        <refname>bvode</refname>
 
17
        <refpurpose>コロケーション法によるODEの境界値問題</refpurpose>
 
18
    </refnamediv>
 
19
    <refnamediv xml:id="bvodeS">
 
20
        <refname>bvodeS</refname>
 
21
        <refpurpose>bvodeの簡易的なコール</refpurpose>
 
22
    </refnamediv>
 
23
    <refsynopsisdiv>
 
24
        <title>呼び出し手順</title>
 
25
        <synopsis>
 
26
            zu=bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,fsub,dfsub,gsub,dgsub,guess)
 
27
        </synopsis>
 
28
        <synopsis>
 
29
            zu=bvodeS(xpoints,m,N,x_low,x_up,fsub,gsub,zeta, &lt;optional_args&gt;)
 
30
        </synopsis>
 
31
    </refsynopsisdiv>
 
32
    <refsection>
 
33
        <title>パラメータ</title>
 
34
        <variablelist>
 
35
            <varlistentry>
 
36
                <term>zu</term>
 
37
                <listitem>
 
38
                    <para>
 
39
                        大きさMの列ベクトル.
 
40
                        複数の点で指定したメッシュ上で評価したODEの解.
 
41
                        各点における z(u(x)) が含まれます.
 
42
                    </para>
 
43
                </listitem>
 
44
            </varlistentry>
 
45
            <varlistentry>
 
46
                <term>xpoints</term>
 
47
                <listitem>
 
48
                    <para>
 
49
                        解を観測する点を指定する配列.
 
50
                    </para>
 
51
                </listitem>
 
52
            </varlistentry>
 
53
            <varlistentry>
 
54
                <term>N</term>
 
55
                <listitem>
 
56
                    <para>整数値を有するスカラー,微分方程式の数
 
57
                        (N &lt;= 20).
 
58
                    </para>
 
59
                </listitem>
 
60
            </varlistentry>
 
61
            <varlistentry>
 
62
                <term>m</term>
 
63
                <listitem>
 
64
                    <para>
 
65
                        整数要素を有する大きさ <literal>N</literal> のベクトル.
 
66
                        <literal>m</literal>は各微分方程式の次数のベクトルで,
 
67
                        <literal>m(i)</literal>はi番目の微分方程式の次数を指定します.
 
68
                        以下に示す <literal>M</literal>は
 
69
                        <literal>m</literal>の要素の合計を表します.
 
70
                    </para>
 
71
                </listitem>
 
72
            </varlistentry>
 
73
            <varlistentry>
 
74
                <term>x_low</term>
 
75
                <listitem>
 
76
                    <para>スカラー: 区間の左端</para>
 
77
                </listitem>
 
78
            </varlistentry>
 
79
            <varlistentry>
 
80
                <term>x_up</term>
 
81
                <listitem>
 
82
                    <para>スカラー: 区間の右端</para>
 
83
                </listitem>
 
84
            </varlistentry>
 
85
            <varlistentry>
 
86
                <term>zeta</term>
 
87
                <listitem>
 
88
                    <para>
 
89
                        大きさ Mのベクトル,<literal>zeta(j)</literal> 
 
90
                        は, j番目の端の条件点(境界値)を指定します.
 
91
                        x_low&lt;=zeta(j)
 
92
                        &lt;= zeta(j+1)<literal>&lt;=x_up</literal>
 
93
                        となります.
 
94
                    </para>
 
95
                    <para>
 
96
                        全ての境界条件点は,
 
97
                        使用される全てのメッシュにおけるメッシュ点とする必要があります.
 
98
                        以下の <literal>ipar(11)</literal> および
 
99
                        <literal>fixpnt</literal>の説明を参照ください.
 
100
                    </para>
 
101
                </listitem>
 
102
            </varlistentry>
 
103
            <varlistentry>
 
104
                <term>ipar</term>
 
105
                <listitem>
 
106
                    <para>11個の整数要素を有する配列:</para>
 
107
                    <para>
 
108
                        [<literal>nonlin, collpnt, subint, ntol, ndimf, ndimi, iprint,
 
109
                            iread, iguess, rstart,nfxpnt]
 
110
                        </literal>
 
111
                    </para>
 
112
                    <variablelist>
 
113
                        <varlistentry>
 
114
                            <term>nonlin: ipar(1)</term>
 
115
                            <listitem>
 
116
                                <para>問題が線形の場合: 0 ,問題が非線形の場合: 1</para>
 
117
                            </listitem>
 
118
                        </varlistentry>
 
119
                        <varlistentry>
 
120
                            <term>collpnt: ipar(2)</term>
 
121
                            <listitem>
 
122
                                <para>
 
123
                                    サブ区間毎のコロケーション点の数を指定します.
 
124
                                    ただし, 
 
125
                                    where max(m(j)) &lt;= collpnt &lt;= 7
 
126
                                </para>
 
127
                                <para>
 
128
                                    <literal>ipar(2)=0</literal> の場合,
 
129
                                    <literal>collpnt</literal> は ( max(m(j))+1 ,
 
130
                                    5-max(m(j)) )に設定されます
 
131
                                </para>
 
132
                            </listitem>
 
133
                        </varlistentry>
 
134
                        <varlistentry>
 
135
                            <term>subint: ipar(3)</term>
 
136
                            <listitem>
 
137
                                <para>初期メッシュのサブ区間の数を指定します.
 
138
                                    <literal>ipar(3) = 0</literal> の場合,
 
139
                                    適宜 
 
140
                                    <literal>subint = 5</literal>と設定します.
 
141
                                </para>
 
142
                            </listitem>
 
143
                        </varlistentry>
 
144
                        <varlistentry>
 
145
                            <term>ntol: ipar(4)</term>
 
146
                            <listitem>
 
147
                                <para>
 
148
                                    解と微分の許容誤差の数を指定します.
 
149
                                    0 &lt; ntol &lt;= M とする必要があります.
 
150
                                    <literal>ipar(4)</literal> は,
 
151
                                    <literal>tol</literal> 引数の次元または
 
152
                                    <literal>0</literal>に設定する必要があります.
 
153
                                    後者の場合, 実際の値は自動的に
 
154
                                    <literal>size(tol,'*')</literal> に設定されます.
 
155
                                </para>
 
156
                            </listitem>
 
157
                        </varlistentry>
 
158
                        <varlistentry>
 
159
                            <term>ndimf: ipar(5)</term>
 
160
                            <listitem>
 
161
                                <para>
 
162
                                    <literal>fspace</literal>(実数の作業配列)の次元を指定します.
 
163
                                    この値は, 
 
164
                                    サブ区間の最大数 nmax への拘束条件を指定します.
 
165
                                </para>
 
166
                                <para>ipar(5) の値は,拘束条件
 
167
                                    ipar(5)&gt;=nmax*nsizef を考慮する必要があります. ただし
 
168
                                </para>
 
169
                                <para>
 
170
                                    <literal>nsizef=4+3*M+(5+collpnt*N)*(collpnt*N+M)+(2*M-nrec)*2*M</literal>
 
171
                                    (<literal>nrec</literal> は終端境界条件の数です
 
172
                                    ).
 
173
                                </para>
 
174
                            </listitem>
 
175
                        </varlistentry>
 
176
                        <varlistentry>
 
177
                            <term>ndimi: ipar(6)</term>
 
178
                            <listitem>
 
179
                                <para>ispace (整数作業配列)の次元を指定します.
 
180
                                    この値は, サブ区間の最大数 nmax への拘束条件を指定します.
 
181
                                </para>
 
182
                                <para>ipar(6) の値は, 拘束条件 ipar(6)&gt;=nmax*nsizei
 
183
                                    を考慮する必要があります. ただし,
 
184
                                </para>
 
185
                                <para>
 
186
                                    <literal>nsizei=3 +
 
187
                                    </literal>
 
188
                                    <literal>collpnt*N+M</literal>.
 
189
                                </para>
 
190
                            </listitem>
 
191
                        </varlistentry>
 
192
                        <varlistentry>
 
193
                            <term>iprint: ipar(7)</term>
 
194
                            <listitem>
 
195
                                <para>出力制御, 以下の値とします:</para>
 
196
                                <variablelist>
 
197
                                    <varlistentry>
 
198
                                        <term>-1</term>
 
199
                                        <listitem>
 
200
                                            <para>完全な診断出力の場合</para>
 
201
                                        </listitem>
 
202
                                    </varlistentry>
 
203
                                    <varlistentry>
 
204
                                        <term>0</term>
 
205
                                        <listitem>
 
206
                                            <para>部分的な出力の場合</para>
 
207
                                        </listitem>
 
208
                                    </varlistentry>
 
209
                                    <varlistentry>
 
210
                                        <term>1</term>
 
211
                                        <listitem>
 
212
                                            <para>出力しない場合</para>
 
213
                                        </listitem>
 
214
                                    </varlistentry>
 
215
                                </variablelist>
 
216
                            </listitem>
 
217
                        </varlistentry>
 
218
                        <varlistentry>
 
219
                            <term>iread: ipar(8)</term>
 
220
                            <listitem>
 
221
                                <variablelist>
 
222
                                    <varlistentry>
 
223
                                        <term>= 0</term>
 
224
                                        <listitem>
 
225
                                            <para>
 
226
                                                bvodeが一様な初期メッシュを生成するようにします.
 
227
                                            </para>
 
228
                                        </listitem>
 
229
                                    </varlistentry>
 
230
                                    <varlistentry>
 
231
                                        <term>= xx</term>
 
232
                                        <listitem>
 
233
                                            <para>他の値は, Scilabでは未実装です.</para>
 
234
                                            <variablelist>
 
235
                                                <varlistentry>
 
236
                                                    <term>= 1</term>
 
237
                                                    <listitem>
 
238
                                                        <para>初期メッシュがユーザにより指定される場合.
 
239
                                                            fspace で以下のように定義されます: 
 
240
                                                        </para>
 
241
                                                        <para>
 
242
                                                            メッシュは <literal>fspace(1), ...,
 
243
                                                                fspace(n+1)
 
244
                                                            </literal>
 
245
                                                            となります. 
 
246
                                                            ユーザは内部メッシュ点<literal>fspace(j) = x(j),
 
247
                                                                j = 2, ..., n.
 
248
                                                            </literal>
 
249
                                                            のみを指定する必要があります.
 
250
                                                        </para>
 
251
                                                    </listitem>
 
252
                                                </varlistentry>
 
253
                                                <varlistentry>
 
254
                                                    <term>= 2 初期メッシュがユーザにより指定される場合.</term>
 
255
                                                    <listitem>
 
256
                                                        <para>
 
257
                                                            <literal>ipar(8)=1</literal>が指定された時と同様であり, 
 
258
                                                            加えて,適応型メッシュ選択は行われません.
 
259
                                                        </para>
 
260
                                                    </listitem>
 
261
                                                </varlistentry>
 
262
                                            </variablelist>
 
263
                                        </listitem>
 
264
                                    </varlistentry>
 
265
                                </variablelist>
 
266
                            </listitem>
 
267
                        </varlistentry>
 
268
                        <varlistentry>
 
269
                            <term>iguess: ipar(9)</term>
 
270
                            <listitem>
 
271
                                <variablelist>
 
272
                                    <varlistentry>
 
273
                                        <term>= 0</term>
 
274
                                        <listitem>
 
275
                                            <para>解の初期推定値が指定されない場合.</para>
 
276
                                        </listitem>
 
277
                                    </varlistentry>
 
278
                                    <varlistentry>
 
279
                                        <term>= 1</term>
 
280
                                        <listitem>
 
281
                                            <para>
 
282
                                                引数 <literal>guess</literal> により,
 
283
                                                初期推定値がユーザにより指定される場合.
 
284
                                            </para>
 
285
                                        </listitem>
 
286
                                    </varlistentry>
 
287
                                    <varlistentry>
 
288
                                        <term>= 2</term>
 
289
                                        <listitem>
 
290
                                            <para>
 
291
                                                初期メッシュと近似解係数がユーザにより fspace で指定される場合
 
292
                                                (前のメッシュと新規メッシュは同じものになります).
 
293
                                            </para>
 
294
                                        </listitem>
 
295
                                    </varlistentry>
 
296
                                    <varlistentry>
 
297
                                        <term>= 3</term>
 
298
                                        <listitem>
 
299
                                            <para>
 
300
                                                前のメッシュと近似解の係数がユーザによりfspace で指定される場合,
 
301
                                                新規メッシュは 2倍荒くとられます.
 
302
                                                すなわち, 前のメッシュの各2点毎に1点となります．
 
303
                                            </para>
 
304
                                        </listitem>
 
305
                                    </varlistentry>
 
306
                                    <varlistentry>
 
307
                                        <term>= 4</term>
 
308
                                        <listitem>
 
309
                                            <para>
 
310
                                                前の初期メッシュと近似解係数に加えて,同時に
 
311
                                                新規メッシュが fspace に出力されます.
 
312
                                                (詳細はiguess = 2, 3, および 4の出力の説明を参照ください)
 
313
                                            </para>
 
314
                                        </listitem>
 
315
                                    </varlistentry>
 
316
                                </variablelist>
 
317
                            </listitem>
 
318
                        </varlistentry>
 
319
                        <varlistentry>
 
320
                            <term>ireg: ipar(10)</term>
 
321
                            <listitem>
 
322
                                <variablelist>
 
323
                                    <varlistentry>
 
324
                                        <term>= 0</term>
 
325
                                        <listitem>
 
326
                                            <para>問題が標準の場合</para>
 
327
                                        </listitem>
 
328
                                    </varlistentry>
 
329
                                    <varlistentry>
 
330
                                        <term>= 1</term>
 
331
                                        <listitem>
 
332
                                            <para>
 
333
                                                最初の緩和係数が <literal>ireg</literal> に等しい場合,
 
334
                                                非線形反復は過去の収束結果に基づきません.
 
335
                                                (別の敏感な非線形問題のみで使用されます)
 
336
                                            </para>
 
337
                                        </listitem>
 
338
                                    </varlistentry>
 
339
                                    <varlistentry>
 
340
                                        <term>= 2</term>
 
341
                                        <listitem>
 
342
                                            <para>
 
343
                                                以下の場合,直ちにリターンします.
 
344
                                                (a) 2回連続で収束しない場合,
 
345
                                                または
 
346
                                                (b)最初に誤差推定値を得た後.
 
347
                                            </para>
 
348
                                        </listitem>
 
349
                                    </varlistentry>
 
350
                                </variablelist>
 
351
                            </listitem>
 
352
                        </varlistentry>
 
353
                        <varlistentry>
 
354
                            <term>nfxpnt: ipar(11)</term>
 
355
                            <listitem>
 
356
                                <para>
 
357
                                    <literal>x_low</literal> および <literal>x_up</literal>
 
358
                                    以外のメッシュの固定点の数(<literal>fixpnt</literal>の次元)を指定します.
 
359
                                    <literal>ipar(11)</literal> は,引数
 
360
                                    <literal>fixpnt</literal>の次元または
 
361
                                    <literal>0</literal>に設定する必要があります.
 
362
                                    後者の場合, 実際の値は自動的に
 
363
                                    <literal>size(fixpnt,'*')</literal> に設定されます.
 
364
                                </para>
 
365
                            </listitem>
 
366
                        </varlistentry>
 
367
                    </variablelist>
 
368
                </listitem>
 
369
            </varlistentry>
 
370
            <varlistentry>
 
371
                <term>ltol</term>
 
372
                <listitem>
 
373
                    <para>
 
374
                        次元 <literal>ntol=ipar(4)</literal>の配列.
 
375
                        <literal>ltol(j) = l</literal> は,
 
376
                        <literal> tol</literal> 配列の <literal>j</literal>番目の許容誤差を指定します.これは,
 
377
                        以下の<literal>l</literal>番目の要素の誤差を制御します.
 
378
                        <inlinemediaobject>
 
379
                            <imageobject>
 
380
                                <imagedata>
 
381
                                    <mml:math>
 
382
                                        <mml:semantics>
 
383
                                            <mml:mi>z</mml:mi>
 
384
                                            <mml:mrow>
 
385
                                                <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
386
                                            </mml:mrow>
 
387
                                        </mml:semantics>
 
388
                                    </mml:math>
 
389
                                </imagedata>
 
390
                            </imageobject>
 
391
                        </inlinemediaobject>
 
392
                        .以下も必要です:
 
393
                    </para>
 
394
                    <para>
 
395
                        <literal>1 &lt;= ltol(1) &lt; ltol(2) &lt; ... &lt; ltol(ntol)
 
396
                            &lt;= M
 
397
                        </literal>
 
398
                    </para>
 
399
                </listitem>
 
400
            </varlistentry>
 
401
            <varlistentry>
 
402
                <term>tol</term>
 
403
                <listitem>
 
404
                    <para>
 
405
                        <literal>ntol=ipar(4)</literal>次の配列.
 
406
                    </para>
 
407
                    <para>
 
408
                        <literal>tol(j)</literal>は,
 
409
                        以下の <literal>ltol(j)</literal> 番目の要素の許容誤差です.
 
410
                        <inlinemediaobject>
 
411
                            <imageobject>
 
412
                                <imagedata>
 
413
                                    <mml:math>
 
414
                                        <mml:semantics>
 
415
                                            <mml:mi>z</mml:mi>
 
416
                                            <mml:mrow>
 
417
                                                <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
418
                                            </mml:mrow>
 
419
                                        </mml:semantics>
 
420
                                    </mml:math>
 
421
                                </imagedata>
 
422
                            </imageobject>
 
423
                        </inlinemediaobject>
 
424
                        .つまり, このコードは以下の条件を
 
425
                        満たそうとします.
 
426
                        <inlinemediaobject>
 
427
                            <imageobject>
 
428
                                <imagedata>
 
429
                                    <mml:math>
 
430
                                        <mml:semantics>
 
431
                                            <mml:mrow>
 
432
                                                <mml:mrow>
 
433
                                                    <mml:mfenced mml:close="∣" mml:open="∣">
 
434
                                                        <mml:msub>
 
435
                                                            <mml:mrow>
 
436
                                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
437
                                                                <mml:mrow>
 
438
                                                                    <mml:mi>z</mml:mi>
 
439
                                                                    <mml:mrow>
 
440
                                                                        <mml:mrow>
 
441
                                                                            <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
442
                                                                        </mml:mrow>
 
443
                                                                        <mml:mo mml:stretchy="false">−</mml:mo><mml:mi>z</mml:mi>
 
444
                                                                    </mml:mrow>
 
445
                                                                    <mml:mrow>
 
446
                                                                        <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
447
                                                                    </mml:mrow>
 
448
                                                                </mml:mrow>
 
449
                                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
450
                                                            </mml:mrow>
 
451
                                                            <mml:mrow>
 
452
                                                                <mml:mi mml:fontstyle="italic">ltol</mml:mi>
 
453
                                                                <mml:mrow>
 
454
                                                                    <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
455
                                                                </mml:mrow>
 
456
                                                            </mml:mrow>
 
457
                                                        </mml:msub>
 
458
                                                    </mml:mfenced>
 
459
                                                    <mml:mo mml:stretchy="false">≤</mml:mo><mml:mi mml:fontstyle="italic">tol</mml:mi>
 
460
                                                </mml:mrow>
 
461
                                                <mml:mrow>
 
462
                                                    <mml:mrow>
 
463
                                                        <mml:mrow>
 
464
                                                            <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
465
                                                        </mml:mrow>
 
466
                                                        <mml:mo mml:stretchy="false">⋅</mml:mo>
 
467
                                                        <mml:mfenced mml:close="∣" mml:open="∣">
 
468
                                                            <mml:msub>
 
469
                                                                <mml:mrow>
 
470
                                                                    <mml:mo mml:stretchy="false">(</mml:mo>
 
471
                                                                    <mml:mrow>
 
472
                                                                        <mml:mi>z</mml:mi>
 
473
                                                                        <mml:mrow>
 
474
                                                                            <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
475
                                                                        </mml:mrow>
 
476
                                                                    </mml:mrow>
 
477
                                                                    <mml:mo mml:stretchy="false">)</mml:mo>
 
478
                                                                </mml:mrow>
 
479
                                                                <mml:mrow>
 
480
                                                                    <mml:mi mml:fontstyle="italic">ltol</mml:mi>
 
481
                                                                    <mml:mrow>
 
482
                                                                        <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
483
                                                                    </mml:mrow>
 
484
                                                                </mml:mrow>
 
485
                                                            </mml:msub>
 
486
                                                        </mml:mfenced>
 
487
                                                    </mml:mrow>
 
488
                                                    <mml:mo mml:stretchy="false">+</mml:mo><mml:mi mml:fontstyle="italic">tol</mml:mi>
 
489
                                                </mml:mrow>
 
490
                                                <mml:mrow>
 
491
                                                    <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
492
                                                </mml:mrow>
 
493
                                                <mml:mi>,</mml:mi><mml:mi mml:fontstyle="normal">for</mml:mi>
 
494
                                                <mml:mrow>
 
495
                                                    <mml:mi>j</mml:mi><mml:mo mml:stretchy="false">=</mml:mo><mml:mn>1</mml:mn>
 
496
                                                </mml:mrow>
 
497
                                                <mml:mi mml:fontstyle="normal">:</mml:mi><mml:mi mml:fontstyle="normal">ntol</mml:mi>
 
498
                                            </mml:mrow>
 
499
                                            <mml:annotation mml:encoding="StarMath 5.0">abs(z(v)-z(u))_{ltol(j)}
 
500
                                                &lt;= tol(j) cdot abs(z(u))_{ltol(j)} + tol(j), for
 
501
                                                j=1:ntol
 
502
                                            </mml:annotation>
 
503
                                        </mml:semantics>
 
504
                                    </mml:math>
 
505
                                </imagedata>
 
506
                            </imageobject>
 
507
                        </inlinemediaobject>
 
508
                        (サブ区間について)
 
509
                    </para>
 
510
                    <para>
 
511
                        <inlinemediaobject>
 
512
                            <imageobject>
 
513
                                <imagedata>
 
514
                                    <mml:math>
 
515
                                        <mml:semantics>
 
516
                                            <mml:mi>v</mml:mi>
 
517
                                        </mml:semantics>
 
518
                                    </mml:math>
 
519
                                </imagedata>
 
520
                            </imageobject>
 
521
                        </inlinemediaobject>
 
522
                        が近似解ベクトルの場合,
 
523
                        <inlinemediaobject>
 
524
                            <imageobject>
 
525
                                <imagedata>
 
526
                                    <mml:math>
 
527
                                        <mml:semantics>
 
528
                                            <mml:mi>u</mml:mi>
 
529
                                        </mml:semantics>
 
530
                                    </mml:math>
 
531
                                </imagedata>
 
532
                            </imageobject>
 
533
                        </inlinemediaobject>
 
534
                        は厳密解(未知)となります.
 
535
                    </para>
 
536
                </listitem>
 
537
            </varlistentry>
 
538
            <varlistentry>
 
539
                <term>fixpnt</term>
 
540
                <listitem>
 
541
                    <para>
 
542
                        <literal>nfxpnt=ipar(11)</literal>次の配列. 
 
543
                        各メッシュに含まれる<literal>x_low</literal> および
 
544
                        <literal>x_up</literal>以外の点を含みます.
 
545
                        このコードは,<literal>x_low</literal> および <literal>x_up</literal>
 
546
                        以外の全ての周辺条件点(<literal>zeta</literal>の説明を参照)が
 
547
                        <literal>fixpnt</literal>の中の固定点として
 
548
                        含まれることを必要とします.
 
549
                    </para>
 
550
                </listitem>
 
551
            </varlistentry>
 
552
            <varlistentry>
 
553
                <term>fsub</term>
 
554
                <listitem>
 
555
                    <para>
 
556
                        <link linkend="external">external</link> は,以下の
 
557
                        列ベクトルの評価に使用されます. <literal>f=</literal>
 
558
                        <inlinemediaobject>
 
559
                            <imageobject>
 
560
                                <imagedata>
 
561
                                    <mml:math>
 
562
                                        <mml:semantics>
 
563
                                            <mml:mrow>
 
564
                                                <mml:mo mml:stretchy="false">[</mml:mo>
 
565
                                                <mml:mrow>
 
566
                                                    <mml:msub>
 
567
                                                        <mml:mi>f</mml:mi><mml:mn>1</mml:mn>
 
568
                                                    </mml:msub>
 
569
                                                    <mml:mrow>
 
570
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
571
                                                        <mml:mrow>
 
572
                                                            <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
 
573
                                                            <mml:mrow>
 
574
                                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
575
                                                                <mml:mrow>
 
576
                                                                    <mml:mi>u</mml:mi>
 
577
                                                                    <mml:mrow>
 
578
                                                                        <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
579
                                                                    </mml:mrow>
 
580
                                                                </mml:mrow>
 
581
                                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
582
                                                            </mml:mrow>
 
583
                                                        </mml:mrow>
 
584
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
585
                                                    </mml:mrow>
 
586
                                                    <mml:mi>;</mml:mi>
 
587
                                                    <mml:msub>
 
588
                                                        <mml:mi>f</mml:mi><mml:mn>2</mml:mn>
 
589
                                                    </mml:msub>
 
590
                                                    <mml:mrow>
 
591
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
592
                                                        <mml:mrow>
 
593
                                                            <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
 
594
                                                            <mml:mrow>
 
595
                                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
596
                                                                <mml:mrow>
 
597
                                                                    <mml:mi>u</mml:mi>
 
598
                                                                    <mml:mrow>
 
599
                                                                        <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
600
                                                                    </mml:mrow>
 
601
                                                                </mml:mrow>
 
602
                                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
603
                                                            </mml:mrow>
 
604
                                                        </mml:mrow>
 
605
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
606
                                                    </mml:mrow>
 
607
                                                    <mml:mi>,</mml:mi><mml:mo mml:stretchy="false">⋯</mml:mo><mml:mi>,</mml:mi>
 
608
                                                    <mml:msub>
 
609
                                                        <mml:mi>f</mml:mi><mml:mi>N</mml:mi>
 
610
                                                    </mml:msub>
 
611
                                                    <mml:mrow>
 
612
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
613
                                                        <mml:mrow>
 
614
                                                            <mml:mi>x</mml:mi><mml:mi>,</mml:mi><mml:mi>z</mml:mi>
 
615
                                                            <mml:mrow>
 
616
                                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
617
                                                                <mml:mrow>
 
618
                                                                    <mml:mi>u</mml:mi>
 
619
                                                                    <mml:mrow>
 
620
                                                                        <mml:mo mml:stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo mml:stretchy="false">)</mml:mo>
 
621
                                                                    </mml:mrow>
 
622
                                                                </mml:mrow>
 
623
                                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
624
                                                            </mml:mrow>
 
625
                                                        </mml:mrow>
 
626
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
627
                                                    </mml:mrow>
 
628
                                                </mml:mrow>
 
629
                                                <mml:mo mml:stretchy="false">]</mml:mo>
 
630
                                            </mml:mrow>
 
631
                                            <mml:annotation mml:encoding="StarMath 5.0">[f_1(x,z(u(x)));
 
632
                                                f_2(x,z(u(x))),dotsaxis,
 
633
                                                f_N(x,z(u(x)))]
 
634
                                            </mml:annotation>
 
635
                                        </mml:semantics>
 
636
                                    </mml:math>
 
637
                                </imagedata>
 
638
                            </imageobject>
 
639
                        </inlinemediaobject>
 
640
                        
 
641
                        ただし,
 
642
                        <literal>x_low</literal> &lt;= <literal>x</literal> &lt;= <literal>x_up</literal> の任意の<literal>x</literal>,
 
643
                        および <literal>z=z(u(x))</literal> となる任意の<literal>z</literal> とします.
 
644
                        (以下の説明を参照)
 
645
                    </para>
 
646
                    <para>外部関数は以下のように宣言を必要とします:</para>
 
647
                    <itemizedlist>
 
648
                        <listitem>
 
649
                            <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
 
650
                            <programlisting role="no-scilab-exec"><![CDATA[  
651
651
subroutine fsub(x,zu,f)
652
652
double precision zu(*), f(*),x 
653
653
 ]]></programlisting>
654
 
            </listitem>
655
 
            <listitem>
656
 
              <para>Cの場合,関数プロトタイプを以下のようにします:</para>
657
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
654
                        </listitem>
 
655
                        <listitem>
 
656
                            <para>Cの場合,関数プロトタイプを以下のようにします:</para>
 
657
                            <programlisting role="no-scilab-exec"><![CDATA[  
658
658
void fsub(double *x, double *zu, double *f)
659
659
 ]]></programlisting>
660
 
            </listitem>
661
 
            <listitem>
662
 
              <para>そしてScilab上では:</para>
663
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
660
                        </listitem>
 
661
                        <listitem>
 
662
                            <para>そしてScilab上では:</para>
 
663
                            <programlisting role="no-scilab-exec"><![CDATA[  
664
664
function f=fsub(x,zu,parameters)
665
665
 ]]></programlisting>
666
 
            </listitem>
667
 
          </itemizedlist>
668
 
        </listitem>
669
 
      </varlistentry>
670
 
      <varlistentry>
671
 
        <term>dfsub</term>
672
 
        <listitem>
673
 
          <para>
674
 
            <link linkend="external">external</link>は,
675
 
            点<literal>x</literal>におけるヤコビアン<literal>f(x,z(u))</literal>の
676
 
            評価に使用されます.
677
 
            ただし,<literal>z(u(x))</literal>は,
678
 
            <literal>fsub</literal>と同様に定義され,
679
 
            <literal>N</literal> 行 <literal>M</literal>列配列
680
 
            dfには,<literal>f</literal>の偏微分が代入されます:
681
 
          </para>
682
 
          <informalequation>
683
 
            <mediaobject>
684
 
              <imageobject>
685
 
                <imagedata align="center">
686
 
                  <mml:math>
687
 
                    <mml:semantics>
688
 
                      <mml:mrow>
689
 
                        <mml:mi mml:fontstyle="italic">df</mml:mi>
690
 
                        <mml:mrow>
691
 
                          <mml:mrow>
692
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
693
 
                            <mml:mrow>
694
 
                              <mml:mi>i</mml:mi>
695
 
                              <mml:mi>,</mml:mi>
696
 
                              <mml:mi>j</mml:mi>
697
 
                            </mml:mrow>
698
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
699
 
                          </mml:mrow>
700
 
                          <mml:mo mml:stretchy="false">=</mml:mo>
701
 
                          <mml:mfrac>
702
 
                            <mml:mrow>
703
 
                              <mml:mo mml:stretchy="false">∂</mml:mo>
704
 
                              <mml:msub>
705
 
                                <mml:mi>f</mml:mi>
706
 
                                <mml:mi>i</mml:mi>
707
 
                              </mml:msub>
708
 
                            </mml:mrow>
709
 
                            <mml:mrow>
710
 
                              <mml:mo mml:stretchy="false">∂</mml:mo>
711
 
                              <mml:msub>
712
 
                                <mml:mi>z</mml:mi>
713
 
                                <mml:mi>j</mml:mi>
714
 
                              </mml:msub>
715
 
                            </mml:mrow>
716
 
                          </mml:mfrac>
717
 
                        </mml:mrow>
718
 
                        <mml:mtext> for
719
 
                        </mml:mtext>
720
 
                        <mml:mrow>
721
 
                          <mml:mo mml:stretchy="true">{</mml:mo>
722
 
                          <mml:mtable>
723
 
                            <mml:mtr>
724
 
                              <mml:mrow>
725
 
                                <mml:mrow>
726
 
                                  <mml:mi>i</mml:mi>
727
 
                                  <mml:mo mml:stretchy="false">=</mml:mo>
728
 
                                  <mml:mn>1</mml:mn>
729
 
                                </mml:mrow>
730
 
                                <mml:mi mml:fontstyle="normal">:</mml:mi>
731
 
                                <mml:mi mml:fontstyle="italic">N</mml:mi>
732
 
                              </mml:mrow>
733
 
                            </mml:mtr>
734
 
                            <mml:mtr>
735
 
                              <mml:mrow>
736
 
                                <mml:mrow>
737
 
                                  <mml:mi>j</mml:mi>
738
 
                                  <mml:mo mml:stretchy="false">=</mml:mo>
739
 
                                  <mml:mn>1</mml:mn>
740
 
                                </mml:mrow>
741
 
                                <mml:mi mml:fontstyle="normal">:</mml:mi>
742
 
                                <mml:mi mml:fontstyle="italic">M</mml:mi>
743
 
                              </mml:mrow>
744
 
                            </mml:mtr>
745
 
                          </mml:mtable>
746
 
                        </mml:mrow>
747
 
                      </mml:mrow>
748
 
                      <mml:annotation mml:encoding="StarMath 5.0">df(i,j)= {partial{f_i}} over
749
 
                        {partial{z_j}} ~ for ~ left lbrace binom{i=1:N}{j=1:M} right
750
 
                        none
751
 
                      </mml:annotation>
752
 
                    </mml:semantics>
753
 
                  </mml:math>
754
 
                </imagedata>
755
 
              </imageobject>
756
 
            </mediaobject>
757
 
          </informalequation>
758
 
          <para>外部関数は以下のように宣言を必要とします:</para>
759
 
          <itemizedlist>
760
 
            <listitem>
761
 
              <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
762
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
666
                        </listitem>
 
667
                    </itemizedlist>
 
668
                </listitem>
 
669
            </varlistentry>
 
670
            <varlistentry>
 
671
                <term>dfsub</term>
 
672
                <listitem>
 
673
                    <para>
 
674
                        <link linkend="external">external</link>は,
 
675
                        点<literal>x</literal>におけるヤコビアン<literal>f(x,z(u))</literal>の
 
676
                        評価に使用されます.
 
677
                        ただし,<literal>z(u(x))</literal>は,
 
678
                        <literal>fsub</literal>と同様に定義され,
 
679
                        <literal>N</literal> 行 <literal>M</literal>列配列
 
680
                        dfには,<literal>f</literal>の偏微分が代入されます:
 
681
                    </para>
 
682
                    <informalequation>
 
683
                        <mediaobject>
 
684
                            <imageobject>
 
685
                                <imagedata align="center">
 
686
                                    <mml:math>
 
687
                                        <mml:semantics>
 
688
                                            <mml:mrow>
 
689
                                                <mml:mi mml:fontstyle="italic">df</mml:mi>
 
690
                                                <mml:mrow>
 
691
                                                    <mml:mrow>
 
692
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
693
                                                        <mml:mrow>
 
694
                                                            <mml:mi>i</mml:mi>
 
695
                                                            <mml:mi>,</mml:mi>
 
696
                                                            <mml:mi>j</mml:mi>
 
697
                                                        </mml:mrow>
 
698
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
699
                                                    </mml:mrow>
 
700
                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
701
                                                    <mml:mfrac>
 
702
                                                        <mml:mrow>
 
703
                                                            <mml:mo mml:stretchy="false">∂</mml:mo>
 
704
                                                            <mml:msub>
 
705
                                                                <mml:mi>f</mml:mi>
 
706
                                                                <mml:mi>i</mml:mi>
 
707
                                                            </mml:msub>
 
708
                                                        </mml:mrow>
 
709
                                                        <mml:mrow>
 
710
                                                            <mml:mo mml:stretchy="false">∂</mml:mo>
 
711
                                                            <mml:msub>
 
712
                                                                <mml:mi>z</mml:mi>
 
713
                                                                <mml:mi>j</mml:mi>
 
714
                                                            </mml:msub>
 
715
                                                        </mml:mrow>
 
716
                                                    </mml:mfrac>
 
717
                                                </mml:mrow>
 
718
                                                <mml:mtext> for
 
719
                                                </mml:mtext>
 
720
                                                <mml:mrow>
 
721
                                                    <mml:mo mml:stretchy="true">{</mml:mo>
 
722
                                                    <mml:mtable>
 
723
                                                        <mml:mtr>
 
724
                                                            <mml:mrow>
 
725
                                                                <mml:mrow>
 
726
                                                                    <mml:mi>i</mml:mi>
 
727
                                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
728
                                                                    <mml:mn>1</mml:mn>
 
729
                                                                </mml:mrow>
 
730
                                                                <mml:mi mml:fontstyle="normal">:</mml:mi>
 
731
                                                                <mml:mi mml:fontstyle="italic">N</mml:mi>
 
732
                                                            </mml:mrow>
 
733
                                                        </mml:mtr>
 
734
                                                        <mml:mtr>
 
735
                                                            <mml:mrow>
 
736
                                                                <mml:mrow>
 
737
                                                                    <mml:mi>j</mml:mi>
 
738
                                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
739
                                                                    <mml:mn>1</mml:mn>
 
740
                                                                </mml:mrow>
 
741
                                                                <mml:mi mml:fontstyle="normal">:</mml:mi>
 
742
                                                                <mml:mi mml:fontstyle="italic">M</mml:mi>
 
743
                                                            </mml:mrow>
 
744
                                                        </mml:mtr>
 
745
                                                    </mml:mtable>
 
746
                                                </mml:mrow>
 
747
                                            </mml:mrow>
 
748
                                            <mml:annotation mml:encoding="StarMath 5.0">df(i,j)= {partial{f_i}} over
 
749
                                                {partial{z_j}} ~ for ~ left lbrace binom{i=1:N}{j=1:M} right
 
750
                                                none
 
751
                                            </mml:annotation>
 
752
                                        </mml:semantics>
 
753
                                    </mml:math>
 
754
                                </imagedata>
 
755
                            </imageobject>
 
756
                        </mediaobject>
 
757
                    </informalequation>
 
758
                    <para>外部関数は以下のように宣言を必要とします:</para>
 
759
                    <itemizedlist>
 
760
                        <listitem>
 
761
                            <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
 
762
                            <programlisting role="no-scilab-exec"><![CDATA[  
763
763
subroutine dfsub(x,zu,df)
764
764
double precision zu(*), df(*),x
765
765
 ]]></programlisting>
766
 
            </listitem>
767
 
            <listitem>
768
 
              <para>Cの場合, 関数プロトタイプを以下のようにします:</para>
769
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
766
                        </listitem>
 
767
                        <listitem>
 
768
                            <para>Cの場合, 関数プロトタイプを以下のようにします:</para>
 
769
                            <programlisting role="no-scilab-exec"><![CDATA[  
770
770
void dfsub(double *x, double *zu, double *df)
771
771
 ]]></programlisting>
772
 
            </listitem>
773
 
            <listitem>
774
 
              <para>そして,Scilabにおいては:</para>
775
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
772
                        </listitem>
 
773
                        <listitem>
 
774
                            <para>そして,Scilabにおいては:</para>
 
775
                            <programlisting role="no-scilab-exec"><![CDATA[  
776
776
function df=dfsub(x,zu,parameters)
777
777
 ]]></programlisting>
778
 
            </listitem>
779
 
          </itemizedlist>
780
 
        </listitem>
781
 
      </varlistentry>
782
 
      <varlistentry>
783
 
        <term>gsub</term>
784
 
        <listitem>
785
 
          <para>
786
 
            以下の評価に使用される<link linkend="external">external</link>
787
 
            <inlinemediaobject>
788
 
              <imageobject>
789
 
                <imagedata>
790
 
                  <mml:math>
791
 
                    <mml:semantics>
792
 
                      <mml:mrow>
793
 
                        <mml:msub>
794
 
                          <mml:mi>g</mml:mi><mml:mi>i</mml:mi>
795
 
                        </mml:msub>
796
 
                        <mml:mrow>
797
 
                          <mml:mo mml:stretchy="false">(</mml:mo>
798
 
                          <mml:mrow>
799
 
                            <mml:msub>
800
 
                              <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
801
 
                            </mml:msub>
802
 
                            <mml:mi>,</mml:mi><mml:mi>z</mml:mi>
803
 
                            <mml:mrow>
804
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
805
 
                              <mml:mrow>
806
 
                                <mml:mi>u</mml:mi>
807
 
                                <mml:mrow>
808
 
                                  <mml:mo mml:stretchy="false">(</mml:mo>
809
 
                                  <mml:msub>
810
 
                                    <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
811
 
                                  </mml:msub>
812
 
                                  <mml:mo mml:stretchy="false">)</mml:mo>
813
 
                                </mml:mrow>
814
 
                              </mml:mrow>
815
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
816
 
                            </mml:mrow>
817
 
                          </mml:mrow>
818
 
                          <mml:mo mml:stretchy="false">)</mml:mo>
819
 
                        </mml:mrow>
820
 
                      </mml:mrow>
821
 
                      <mml:annotation mml:encoding="StarMath 5.0">g_i(%zeta_i,z(u(%zeta_i)))</mml:annotation>
822
 
                    </mml:semantics>
823
 
                  </mml:math>
824
 
                </imagedata>
825
 
              </imageobject>
826
 
            </inlinemediaobject>
827
 
            given z=
828
 
            <inlinemediaobject>
829
 
              <imageobject>
830
 
                <imagedata>
831
 
                  <mml:math>
832
 
                    <mml:semantics>
833
 
                      <mml:mrow>
834
 
                        <mml:mi>z</mml:mi>
835
 
                        <mml:mrow>
836
 
                          <mml:mo mml:stretchy="false">(</mml:mo>
837
 
                          <mml:mrow>
838
 
                            <mml:mi>u</mml:mi>
839
 
                            <mml:mrow>
840
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
841
 
                              <mml:msub>
842
 
                                <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
843
 
                              </mml:msub>
844
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
845
 
                            </mml:mrow>
846
 
                          </mml:mrow>
847
 
                          <mml:mo mml:stretchy="false">)</mml:mo>
848
 
                        </mml:mrow>
849
 
                      </mml:mrow>
850
 
                      <mml:annotation mml:encoding="StarMath 5.0">z(u(%zeta_i))</mml:annotation>
851
 
                    </mml:semantics>
852
 
                  </mml:math>
853
 
                </imagedata>
854
 
              </imageobject>
855
 
            </inlinemediaobject>
856
 
            
857
 
            <literal>z = zeta(i)</literal> ただし,
858
 
            <literal>1&lt;=i&lt;=M.</literal>
859
 
          </para>
860
 
          <para>外部関数は以下のように宣言を必要とします:</para>
861
 
          <itemizedlist>
862
 
            <listitem>
863
 
              <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
864
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
778
                        </listitem>
 
779
                    </itemizedlist>
 
780
                </listitem>
 
781
            </varlistentry>
 
782
            <varlistentry>
 
783
                <term>gsub</term>
 
784
                <listitem>
 
785
                    <para>
 
786
                        以下の評価に使用される<link linkend="external">external</link>
 
787
                        <inlinemediaobject>
 
788
                            <imageobject>
 
789
                                <imagedata>
 
790
                                    <mml:math>
 
791
                                        <mml:semantics>
 
792
                                            <mml:mrow>
 
793
                                                <mml:msub>
 
794
                                                    <mml:mi>g</mml:mi><mml:mi>i</mml:mi>
 
795
                                                </mml:msub>
 
796
                                                <mml:mrow>
 
797
                                                    <mml:mo mml:stretchy="false">(</mml:mo>
 
798
                                                    <mml:mrow>
 
799
                                                        <mml:msub>
 
800
                                                            <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
 
801
                                                        </mml:msub>
 
802
                                                        <mml:mi>,</mml:mi><mml:mi>z</mml:mi>
 
803
                                                        <mml:mrow>
 
804
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
805
                                                            <mml:mrow>
 
806
                                                                <mml:mi>u</mml:mi>
 
807
                                                                <mml:mrow>
 
808
                                                                    <mml:mo mml:stretchy="false">(</mml:mo>
 
809
                                                                    <mml:msub>
 
810
                                                                        <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
 
811
                                                                    </mml:msub>
 
812
                                                                    <mml:mo mml:stretchy="false">)</mml:mo>
 
813
                                                                </mml:mrow>
 
814
                                                            </mml:mrow>
 
815
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
816
                                                        </mml:mrow>
 
817
                                                    </mml:mrow>
 
818
                                                    <mml:mo mml:stretchy="false">)</mml:mo>
 
819
                                                </mml:mrow>
 
820
                                            </mml:mrow>
 
821
                                            <mml:annotation mml:encoding="StarMath 5.0">g_i(%zeta_i,z(u(%zeta_i)))</mml:annotation>
 
822
                                        </mml:semantics>
 
823
                                    </mml:math>
 
824
                                </imagedata>
 
825
                            </imageobject>
 
826
                        </inlinemediaobject>
 
827
                        given z=
 
828
                        <inlinemediaobject>
 
829
                            <imageobject>
 
830
                                <imagedata>
 
831
                                    <mml:math>
 
832
                                        <mml:semantics>
 
833
                                            <mml:mrow>
 
834
                                                <mml:mi>z</mml:mi>
 
835
                                                <mml:mrow>
 
836
                                                    <mml:mo mml:stretchy="false">(</mml:mo>
 
837
                                                    <mml:mrow>
 
838
                                                        <mml:mi>u</mml:mi>
 
839
                                                        <mml:mrow>
 
840
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
841
                                                            <mml:msub>
 
842
                                                                <mml:mo mml:stretchy="false">ζ</mml:mo><mml:mi>i</mml:mi>
 
843
                                                            </mml:msub>
 
844
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
845
                                                        </mml:mrow>
 
846
                                                    </mml:mrow>
 
847
                                                    <mml:mo mml:stretchy="false">)</mml:mo>
 
848
                                                </mml:mrow>
 
849
                                            </mml:mrow>
 
850
                                            <mml:annotation mml:encoding="StarMath 5.0">z(u(%zeta_i))</mml:annotation>
 
851
                                        </mml:semantics>
 
852
                                    </mml:math>
 
853
                                </imagedata>
 
854
                            </imageobject>
 
855
                        </inlinemediaobject>
 
856
                        
 
857
                        <literal>z = zeta(i)</literal> ただし,
 
858
                        <literal>1&lt;=i&lt;=M.</literal>
 
859
                    </para>
 
860
                    <para>外部関数は以下のように宣言を必要とします:</para>
 
861
                    <itemizedlist>
 
862
                        <listitem>
 
863
                            <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
 
864
                            <programlisting role="no-scilab-exec"><![CDATA[  
865
865
subroutine gsub(i,zu,g)
866
866
double precision zu(*), g(*)
867
867
integer i
868
868
 ]]></programlisting>
869
 
            </listitem>
870
 
            <listitem>
871
 
              <para>Cの場合, 関数プロトタイプを以下のようにします:</para>
872
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
869
                        </listitem>
 
870
                        <listitem>
 
871
                            <para>Cの場合, 関数プロトタイプを以下のようにします:</para>
 
872
                            <programlisting role="no-scilab-exec"><![CDATA[  
873
873
void gsub(int *i, double *zu, double *g)
874
874
 ]]></programlisting>
875
 
            </listitem>
876
 
            <listitem>
877
 
              <para>そして,Scilabでは:</para>
878
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
875
                        </listitem>
 
876
                        <listitem>
 
877
                            <para>そして,Scilabでは:</para>
 
878
                            <programlisting role="no-scilab-exec"><![CDATA[  
879
879
function g=gsub(i,zu,parameters)
880
880
 ]]></programlisting>
881
 
              <para>
882
 
                <literal>fsub</literal>の<literal>f</literal>とは逆に
883
 
                <literal>g</literal>では
884
 
                コール毎に一つの値のみが返されることに注意してください.
885
 
              </para>
886
 
            </listitem>
887
 
          </itemizedlist>
888
 
        </listitem>
889
 
      </varlistentry>
890
 
      <varlistentry>
891
 
        <term>dgsub</term>
892
 
        <listitem>
893
 
          <para>
894
 
            an <link linkend="external">external</link> used to evaluate
895
 
            the <literal>i</literal>-th row of the Jacobian of <literal>g(x,u(x))</literal>. Where
896
 
            <literal>z(u)</literal> is as for <literal>fsub</literal>, <literal>i</literal> as for 
897
 
            <literal>gsub</literal> and the <literal>M</literal>-vector <literal>dg</literal> should 
898
 
            be filled with the partial derivatives of <literal>g</literal>, viz, for a particular call one 
899
 
            calculates
900
 
          </para>
901
 
          <informalequation>
902
 
            <mediaobject>
903
 
              <imageobject>
904
 
                <imagedata align="center">
905
 
                  <mml:math>
906
 
                    <mml:semantics>
907
 
                      <mml:mrow>
908
 
                        <mml:mi mml:fontstyle="italic">dg</mml:mi>
909
 
                        <mml:mrow>
910
 
                          <mml:mrow>
911
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
912
 
                            <mml:mrow>
913
 
                              <mml:mi>i</mml:mi>
914
 
                              <mml:mi>,</mml:mi>
915
 
                              <mml:mi>j</mml:mi>
916
 
                            </mml:mrow>
917
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
918
 
                          </mml:mrow>
919
 
                          <mml:mo mml:stretchy="false">=</mml:mo>
920
 
                          <mml:mfrac>
921
 
                            <mml:mrow>
922
 
                              <mml:mo mml:stretchy="false">∂</mml:mo>
923
 
                              <mml:msub>
924
 
                                <mml:mi>g</mml:mi>
925
 
                                <mml:mi>i</mml:mi>
926
 
                              </mml:msub>
927
 
                            </mml:mrow>
928
 
                            <mml:mrow>
929
 
                              <mml:mo mml:stretchy="false">∂</mml:mo>
930
 
                              <mml:msub>
931
 
                                <mml:mi>z</mml:mi>
932
 
                                <mml:mi>j</mml:mi>
933
 
                              </mml:msub>
934
 
                            </mml:mrow>
935
 
                          </mml:mfrac>
936
 
                        </mml:mrow>
937
 
                        <mml:mtext> for
938
 
                        </mml:mtext>
939
 
                        <mml:mrow>
940
 
                          <mml:mo mml:stretchy="true">{</mml:mo>
941
 
                          <mml:mtable>
942
 
                            <mml:mtr>
943
 
                              <mml:mrow>
944
 
                                <mml:mrow>
945
 
                                  <mml:mi>i</mml:mi>
946
 
                                  <mml:mo mml:stretchy="false">=</mml:mo>
947
 
                                  <mml:mn>1</mml:mn>
948
 
                                </mml:mrow>
949
 
                                <mml:mi mml:fontstyle="normal">:</mml:mi>
950
 
                                <mml:mi mml:fontstyle="italic">M</mml:mi>
951
 
                              </mml:mrow>
952
 
                            </mml:mtr>
953
 
                            <mml:mtr>
954
 
                              <mml:mrow>
955
 
                                <mml:mrow>
956
 
                                  <mml:mi>j</mml:mi>
957
 
                                  <mml:mo mml:stretchy="false">=</mml:mo>
958
 
                                  <mml:mn>1</mml:mn>
959
 
                                </mml:mrow>
960
 
                                <mml:mi mml:fontstyle="normal">:</mml:mi>
961
 
                                <mml:mi mml:fontstyle="italic">M</mml:mi>
962
 
                              </mml:mrow>
963
 
                            </mml:mtr>
964
 
                          </mml:mtable>
965
 
                        </mml:mrow>
966
 
                      </mml:mrow>
967
 
                      <mml:annotation mml:encoding="StarMath 5.0">dg(i,j)= {partial{g_i}} over
968
 
                        {partial{z_j}} ~ for ~ left lbrace binom{i=1:M}{j=1:M} right
969
 
                        none
970
 
                      </mml:annotation>
971
 
                    </mml:semantics>
972
 
                  </mml:math>
973
 
                </imagedata>
974
 
              </imageobject>
975
 
            </mediaobject>
976
 
          </informalequation>
977
 
          <para>外部関数は以下のような宣言を必要とします:</para>
978
 
          <itemizedlist>
979
 
            <listitem>
980
 
              <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
981
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
881
                            <para>
 
882
                                <literal>fsub</literal>の<literal>f</literal>とは逆に
 
883
                                <literal>g</literal>では
 
884
                                コール毎に一つの値のみが返されることに注意してください.
 
885
                            </para>
 
886
                        </listitem>
 
887
                    </itemizedlist>
 
888
                </listitem>
 
889
            </varlistentry>
 
890
            <varlistentry>
 
891
                <term>dgsub</term>
 
892
                <listitem>
 
893
                    <para>
 
894
                        an <link linkend="external">external</link> used to evaluate
 
895
                        the <literal>i</literal>-th row of the Jacobian of <literal>g(x,u(x))</literal>. Where
 
896
                        <literal>z(u)</literal> is as for <literal>fsub</literal>, <literal>i</literal> as for 
 
897
                        <literal>gsub</literal> and the <literal>M</literal>-vector <literal>dg</literal> should 
 
898
                        be filled with the partial derivatives of <literal>g</literal>, viz, for a particular call one 
 
899
                        calculates
 
900
                    </para>
 
901
                    <informalequation>
 
902
                        <mediaobject>
 
903
                            <imageobject>
 
904
                                <imagedata align="center">
 
905
                                    <mml:math>
 
906
                                        <mml:semantics>
 
907
                                            <mml:mrow>
 
908
                                                <mml:mi mml:fontstyle="italic">dg</mml:mi>
 
909
                                                <mml:mrow>
 
910
                                                    <mml:mrow>
 
911
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
912
                                                        <mml:mrow>
 
913
                                                            <mml:mi>i</mml:mi>
 
914
                                                            <mml:mi>,</mml:mi>
 
915
                                                            <mml:mi>j</mml:mi>
 
916
                                                        </mml:mrow>
 
917
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
918
                                                    </mml:mrow>
 
919
                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
920
                                                    <mml:mfrac>
 
921
                                                        <mml:mrow>
 
922
                                                            <mml:mo mml:stretchy="false">∂</mml:mo>
 
923
                                                            <mml:msub>
 
924
                                                                <mml:mi>g</mml:mi>
 
925
                                                                <mml:mi>i</mml:mi>
 
926
                                                            </mml:msub>
 
927
                                                        </mml:mrow>
 
928
                                                        <mml:mrow>
 
929
                                                            <mml:mo mml:stretchy="false">∂</mml:mo>
 
930
                                                            <mml:msub>
 
931
                                                                <mml:mi>z</mml:mi>
 
932
                                                                <mml:mi>j</mml:mi>
 
933
                                                            </mml:msub>
 
934
                                                        </mml:mrow>
 
935
                                                    </mml:mfrac>
 
936
                                                </mml:mrow>
 
937
                                                <mml:mtext> for
 
938
                                                </mml:mtext>
 
939
                                                <mml:mrow>
 
940
                                                    <mml:mo mml:stretchy="true">{</mml:mo>
 
941
                                                    <mml:mtable>
 
942
                                                        <mml:mtr>
 
943
                                                            <mml:mrow>
 
944
                                                                <mml:mrow>
 
945
                                                                    <mml:mi>i</mml:mi>
 
946
                                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
947
                                                                    <mml:mn>1</mml:mn>
 
948
                                                                </mml:mrow>
 
949
                                                                <mml:mi mml:fontstyle="normal">:</mml:mi>
 
950
                                                                <mml:mi mml:fontstyle="italic">M</mml:mi>
 
951
                                                            </mml:mrow>
 
952
                                                        </mml:mtr>
 
953
                                                        <mml:mtr>
 
954
                                                            <mml:mrow>
 
955
                                                                <mml:mrow>
 
956
                                                                    <mml:mi>j</mml:mi>
 
957
                                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
958
                                                                    <mml:mn>1</mml:mn>
 
959
                                                                </mml:mrow>
 
960
                                                                <mml:mi mml:fontstyle="normal">:</mml:mi>
 
961
                                                                <mml:mi mml:fontstyle="italic">M</mml:mi>
 
962
                                                            </mml:mrow>
 
963
                                                        </mml:mtr>
 
964
                                                    </mml:mtable>
 
965
                                                </mml:mrow>
 
966
                                            </mml:mrow>
 
967
                                            <mml:annotation mml:encoding="StarMath 5.0">dg(i,j)= {partial{g_i}} over
 
968
                                                {partial{z_j}} ~ for ~ left lbrace binom{i=1:M}{j=1:M} right
 
969
                                                none
 
970
                                            </mml:annotation>
 
971
                                        </mml:semantics>
 
972
                                    </mml:math>
 
973
                                </imagedata>
 
974
                            </imageobject>
 
975
                        </mediaobject>
 
976
                    </informalequation>
 
977
                    <para>外部関数は以下のような宣言を必要とします:</para>
 
978
                    <itemizedlist>
 
979
                        <listitem>
 
980
                            <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
 
981
                            <programlisting role="no-scilab-exec"><![CDATA[  
982
982
subroutine dgsub(i,zu,dg)
983
983
double precision zu(*), dg(*)
984
984
 ]]></programlisting>
985
 
            </listitem>
986
 
            <listitem>
987
 
              <para>Cの場合, 関数プロトタイプを以下のようにします:</para>
988
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
985
                        </listitem>
 
986
                        <listitem>
 
987
                            <para>Cの場合, 関数プロトタイプを以下のようにします:</para>
 
988
                            <programlisting role="no-scilab-exec"><![CDATA[  
989
989
void dgsub(int *i, double *zu, double *dg)
990
990
 ]]></programlisting>
991
 
            </listitem>
992
 
            <listitem>
993
 
              <para>そして,Scilabでは:</para>
994
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
991
                        </listitem>
 
992
                        <listitem>
 
993
                            <para>そして,Scilabでは:</para>
 
994
                            <programlisting role="no-scilab-exec"><![CDATA[  
995
995
function dg=dgsub(i,zu,parameters)
996
996
 ]]></programlisting>
997
 
            </listitem>
998
 
          </itemizedlist>
999
 
        </listitem>
1000
 
      </varlistentry>
1001
 
      <varlistentry>
1002
 
        <term>guess</term>
1003
 
        <listitem>
1004
 
          <para>
1005
 
            <literal>z(u(x))</literal>および
1006
 
            <literal>u(x)</literal>の<literal>mj</literal>階微分<literal>dmval(u(x))</literal>
1007
 
            の初期近似値を見積もるために
1008
 
            <link linkend="external">external</link>が使用されます.
1009
 
            <literal>ipar(9) = 1</literal>の場合のみ, このサブルーチンが
1010
 
            使用されることに注意してください.
1011
 
            このため，<literal>zu</literal>の<literal>M</literal>個の要素および <literal>dmval</literal> の <literal>N</literal>個の
1012
 
            要素が<literal>x_low</literal> &lt;= <literal>x</literal> &lt;= <literal>x_up</literal>の範囲の任意の <literal>x</literal> について計算される
1013
 
            必要があります.
1014
 
          </para>
1015
 
          <para>外部関数は以下のように宣言を必要とします:</para>
1016
 
          <itemizedlist>
1017
 
            <listitem>
1018
 
              <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
1019
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
997
                        </listitem>
 
998
                    </itemizedlist>
 
999
                </listitem>
 
1000
            </varlistentry>
 
1001
            <varlistentry>
 
1002
                <term>guess</term>
 
1003
                <listitem>
 
1004
                    <para>
 
1005
                        <literal>z(u(x))</literal>および
 
1006
                        <literal>u(x)</literal>の<literal>mj</literal>階微分<literal>dmval(u(x))</literal>
 
1007
                        の初期近似値を見積もるために
 
1008
                        <link linkend="external">external</link>が使用されます.
 
1009
                        <literal>ipar(9) = 1</literal>の場合のみ, このサブルーチンが
 
1010
                        使用されることに注意してください.
 
1011
                        このため，<literal>zu</literal>の<literal>M</literal>個の要素および <literal>dmval</literal> の <literal>N</literal>個の
 
1012
                        要素が<literal>x_low</literal> &lt;= <literal>x</literal> &lt;= <literal>x_up</literal>の範囲の任意の <literal>x</literal> について計算される
 
1013
                        必要があります.
 
1014
                    </para>
 
1015
                    <para>外部関数は以下のように宣言を必要とします:</para>
 
1016
                    <itemizedlist>
 
1017
                        <listitem>
 
1018
                            <para>Fortranの場合, 呼び出し手順を以下のようにします:</para>
 
1019
                            <programlisting role="no-scilab-exec"><![CDATA[  
1020
1020
subroutine guess(x,zu,dmval)
1021
1021
double precision x,z(*), dmval(*)
1022
1022
 ]]></programlisting>
1023
 
            </listitem>
1024
 
            <listitem>
1025
 
              <para>Cの場合, 関数プロトタイプを以下のようにします</para>
1026
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
1023
                        </listitem>
 
1024
                        <listitem>
 
1025
                            <para>Cの場合, 関数プロトタイプを以下のようにします</para>
 
1026
                            <programlisting role="no-scilab-exec"><![CDATA[  
1027
1027
void fsub(double *x, double *zu, double *dmval)
1028
1028
 ]]></programlisting>
1029
 
            </listitem>
1030
 
            <listitem>
1031
 
              <para>そしてScilab上では, </para>
1032
 
              <programlisting role="no-scilab-exec"><![CDATA[  
 
1029
                        </listitem>
 
1030
                        <listitem>
 
1031
                            <para>そしてScilab上では, </para>
 
1032
                            <programlisting role="no-scilab-exec"><![CDATA[  
1033
1033
function [dmval,zu]=fsub(x,parameters)
1034
1034
 ]]></programlisting>
1035
 
            </listitem>
1036
 
          </itemizedlist>
1037
 
        </listitem>
1038
 
      </varlistentry>
1039
 
      <varlistentry>
1040
 
        <term>&lt;optional_args&gt;</term>
1041
 
        <listitem>
1042
 
          <para>以下のどれかとする必要があります:</para>
1043
 
          <itemizedlist>
1044
 
            <listitem>
1045
 
              <para>
1046
 
                以下の一連の値:
1047
 
                <literal>guess, dfsub, dgsub, fixpnt, ndimf, ndimi, ltol, tol,
1048
 
                  ntol,nonlin, collpnt, subint, iprint, ireg, ifail
1049
 
                </literal>
1050
 
              </para>
1051
 
            </listitem>
1052
 
            <listitem>
1053
 
              <para>
1054
 
                もしくは,一連の<literal>arg_name=argvalue</literal>
1055
 
                で指定します.
1056
 
                ただし <literal>arg_name</literal> は以下のどれかとなります: 
1057
 
                <literal>guess</literal>,
1058
 
                <literal>dfsub</literal>, <literal>dgsub</literal>,
1059
 
                <literal>fixpnt</literal>, <literal>ndimf</literal>,
1060
 
                <literal>ndimi</literal>, <literal>ltol</literal>,
1061
 
                <literal>tol</literal>, <literal>ntol</literal>,
1062
 
                <literal>nonlin</literal>, <literal>collpnt</literal>,
1063
 
                <literal>subint</literal>, <literal>iprint</literal>,
1064
 
                <literal>ireg</literal>, <literal>ifail</literal>
1065
 
              </para>
1066
 
            </listitem>
1067
 
          </itemizedlist>
1068
 
          <para>
1069
 
            これらの引数が指定される場合は,常に
1070
 
            <literal>ifail</literal>は上記のように説明されます.
1071
 
            <literal>ifail</literal>は選択されたオプション引数
1072
 
            に対応する bvode コールを表示するために使用できます.
1073
 
            もし，<literal>guess</literal> が指定された場合,
1074
 
            <literal>iguess</literal>が 1 に設定されます.
1075
 
          </para>
1076
 
        </listitem>
1077
 
      </varlistentry>
1078
 
    </variablelist>
1079
 
  </refsection>
1080
 
  <refsection>
1081
 
    <title>説明</title>
1082
 
    <para>
1083
 
      これらの関数は, 以下のように定義された ode-s の
1084
 
      混合次数に関連するシステム多点境界値問題を解きます.
1085
 
    </para>
1086
 
    <informalequation>
1087
 
      <mediaobject>
1088
 
        <imageobject>
1089
 
          <imagedata align="left">
1090
 
            <mml:math>
1091
 
              <mml:semantics>
1092
 
                <mml:mrow>
1093
 
                  <mml:mo mml:stretchy="true">{</mml:mo>
1094
 
                  <mml:mtable>
1095
 
                    <mml:mtr>
1096
 
                      <mml:mrow>
1097
 
                        <mml:mrow>
1098
 
                          <mml:msubsup>
1099
 
                            <mml:mi>u</mml:mi>
1100
 
                            <mml:mi>i</mml:mi>
1101
 
                            <mml:mrow>
1102
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1103
 
                              <mml:msub>
1104
 
                                <mml:mi>m</mml:mi>
1105
 
                                <mml:mi>i</mml:mi>
1106
 
                              </mml:msub>
1107
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1108
 
                            </mml:mrow>
1109
 
                          </mml:msubsup>
1110
 
                          <mml:mo mml:stretchy="false">=</mml:mo>
1111
 
                          <mml:msub>
1112
 
                            <mml:mi>f</mml:mi>
1113
 
                            <mml:mi>i</mml:mi>
1114
 
                          </mml:msub>
1115
 
                        </mml:mrow>
1116
 
                        <mml:mrow>
1117
 
                          <mml:mo mml:stretchy="false">(</mml:mo>
1118
 
                          <mml:mrow>
1119
 
                            <mml:mi>x</mml:mi>
1120
 
                            <mml:mi>,</mml:mi>
1121
 
                            <mml:mi>z</mml:mi>
1122
 
                            <mml:mrow>
1123
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1124
 
                              <mml:mrow>
1125
 
                                <mml:mi>u</mml:mi>
1126
 
                                <mml:mrow>
1127
 
                                  <mml:mo mml:stretchy="false">(</mml:mo>
1128
 
                                  <mml:mi>x</mml:mi>
1129
 
                                  <mml:mo mml:stretchy="false">)</mml:mo>
1130
 
                                </mml:mrow>
1131
 
                              </mml:mrow>
1132
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1133
 
                            </mml:mrow>
1134
 
                          </mml:mrow>
1135
 
                          <mml:mo mml:stretchy="false">)</mml:mo>
1136
 
                        </mml:mrow>
1137
 
                        <mml:mi/>
1138
 
                        <mml:mrow>
1139
 
                          <mml:mo mml:stretchy="true">{</mml:mo>
1140
 
                          <mml:mtable>
1141
 
                            <mml:mtr>
1142
 
                              <mml:mrow>
1143
 
                                <mml:mrow>
1144
 
                                  <mml:mi>i</mml:mi>
1145
 
                                  <mml:mo mml:stretchy="false">=</mml:mo>
1146
 
                                  <mml:mn>1</mml:mn>
1147
 
                                </mml:mrow>
1148
 
                                <mml:mi mml:fontstyle="normal">:</mml:mi>
1149
 
                                <mml:mi>N</mml:mi>
1150
 
                              </mml:mrow>
1151
 
                            </mml:mtr>
1152
 
                            <mml:mtr>
1153
 
                              <mml:mrow>
1154
 
                                <mml:mi>x</mml:mi>
1155
 
                                <mml:mo mml:stretchy="false">∈</mml:mo>
1156
 
                                <mml:mfenced mml:close="[" mml:open="]">
1157
 
                                  <mml:mrow>
1158
 
                                    <mml:msub>
1159
 
                                      <mml:mi>x</mml:mi>
1160
 
                                      <mml:mi>l</mml:mi>
1161
 
                                    </mml:msub>
1162
 
                                    <mml:mi>,</mml:mi>
1163
 
                                    <mml:msub>
1164
 
                                      <mml:mi>x</mml:mi>
1165
 
                                      <mml:mi>u</mml:mi>
1166
 
                                    </mml:msub>
1167
 
                                  </mml:mrow>
1168
 
                                </mml:mfenced>
1169
 
                              </mml:mrow>
1170
 
                            </mml:mtr>
1171
 
                          </mml:mtable>
1172
 
                        </mml:mrow>
1173
 
                      </mml:mrow>
1174
 
                    </mml:mtr>
1175
 
                    <mml:mtr>
1176
 
                      <mml:mi/>
1177
 
                    </mml:mtr>
1178
 
                    <mml:mtr>
1179
 
                      <mml:mrow>
1180
 
                        <mml:msub>
1181
 
                          <mml:mi>g</mml:mi>
1182
 
                          <mml:mi>j</mml:mi>
1183
 
                        </mml:msub>
1184
 
                        <mml:mrow>
1185
 
                          <mml:mrow>
1186
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
1187
 
                            <mml:mrow>
1188
 
                              <mml:msub>
1189
 
                                <mml:mo mml:stretchy="false">ζ</mml:mo>
1190
 
                                <mml:mi>j</mml:mi>
1191
 
                              </mml:msub>
1192
 
                              <mml:mi>,</mml:mi>
1193
 
                              <mml:mi>z</mml:mi>
1194
 
                              <mml:mrow>
1195
 
                                <mml:mo mml:stretchy="false">(</mml:mo>
1196
 
                                <mml:mrow>
1197
 
                                  <mml:mi>u</mml:mi>
1198
 
                                  <mml:mrow>
1199
 
                                    <mml:mo mml:stretchy="false">(</mml:mo>
1200
 
                                    <mml:msub>
1201
 
                                      <mml:mo mml:stretchy="false">ζ</mml:mo>
1202
 
                                      <mml:mi>j</mml:mi>
1203
 
                                    </mml:msub>
1204
 
                                    <mml:mo mml:stretchy="false">)</mml:mo>
1205
 
                                  </mml:mrow>
1206
 
                                </mml:mrow>
1207
 
                                <mml:mo mml:stretchy="false">)</mml:mo>
1208
 
                              </mml:mrow>
1209
 
                            </mml:mrow>
1210
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
1211
 
                          </mml:mrow>
1212
 
                          <mml:mo mml:stretchy="false">=</mml:mo>
1213
 
                          <mml:mn>0</mml:mn>
1214
 
                        </mml:mrow>
1215
 
                        <mml:mi/>
1216
 
                        <mml:mrow>
1217
 
                          <mml:mtext>,</mml:mtext>
1218
 
                          <mml:mi>j</mml:mi>
1219
 
                          <mml:mo mml:stretchy="false">=</mml:mo>
1220
 
                          <mml:mn>1</mml:mn>
1221
 
                        </mml:mrow>
1222
 
                        <mml:mi mml:fontstyle="normal">:</mml:mi>
1223
 
                        <mml:mi>M</mml:mi>
1224
 
                      </mml:mrow>
1225
 
                    </mml:mtr>
1226
 
                  </mml:mtable>
1227
 
                </mml:mrow>
1228
 
                <mml:annotation mml:encoding="StarMath 5.0">left lbrace
1229
 
                  stack{u_i^(m_i)=f_i(x,z(u(x))) ~~left lbrace { binom{i=1:N}{x
1230
 
                  in left ] a_l, a_u right [} } right none# `#
1231
 
                  g_j(%zeta_j,z(u(%zeta_j))) = 0~~j=1:M} right
1232
 
                  none
1233
 
                </mml:annotation>
1234
 
              </mml:semantics>
1235
 
            </mml:math>
1236
 
          </imagedata>
1237
 
        </imageobject>
1238
 
      </mediaobject>
1239
 
    </informalequation>
1240
 
    <para>ただし,e</para>
1241
 
    <mediaobject>
1242
 
      <imageobject>
1243
 
        <imagedata align="left">
1244
 
          <mml:math>
1245
 
            <mml:semantics>
1246
 
              <mml:mrow>
1247
 
                <mml:mo mml:stretchy="true">{</mml:mo>
1248
 
                <mml:mtable>
1249
 
                  <mml:mtr>
1250
 
                    <mml:mrow>
1251
 
                      <mml:mi>M</mml:mi>
1252
 
                      <mml:mo mml:stretchy="false">=</mml:mo>
1253
 
                      <mml:mrow>
1254
 
                        <mml:munderover>
1255
 
                          <mml:mo mml:stretchy="false">∑</mml:mo>
1256
 
                          <mml:mrow>
1257
 
                            <mml:mi>i</mml:mi>
1258
 
                            <mml:mo mml:stretchy="false">=</mml:mo>
1259
 
                            <mml:mn>1</mml:mn>
1260
 
                          </mml:mrow>
1261
 
                          <mml:mi>N</mml:mi>
1262
 
                        </mml:munderover>
1263
 
                        <mml:msub>
1264
 
                          <mml:mi>m</mml:mi>
1265
 
                          <mml:mi>i</mml:mi>
1266
 
                        </mml:msub>
1267
 
                      </mml:mrow>
1268
 
                    </mml:mrow>
1269
 
                  </mml:mtr>
1270
 
                  <mml:mtr>
1271
 
                    <mml:mi/>
1272
 
                  </mml:mtr>
1273
 
                  <mml:mtr>
1274
 
                    <mml:mrow>
1275
 
                      <mml:mi>z</mml:mi>
1276
 
                      <mml:mrow>
1277
 
                        <mml:mrow>
1278
 
                          <mml:mo mml:stretchy="false">(</mml:mo>
1279
 
                          <mml:mrow>
1280
 
                            <mml:mi>u</mml:mi>
1281
 
                            <mml:mrow>
1282
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1283
 
                              <mml:mi>x</mml:mi>
1284
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1285
 
                            </mml:mrow>
1286
 
                          </mml:mrow>
1287
 
                          <mml:mo mml:stretchy="false">)</mml:mo>
1288
 
                        </mml:mrow>
1289
 
                        <mml:mo mml:stretchy="false">=</mml:mo>
1290
 
                        <mml:mrow>
1291
 
                          <mml:mo mml:stretchy="false">[</mml:mo>
1292
 
                          <mml:mrow>
1293
 
                            <mml:msub>
1294
 
                              <mml:mi>u</mml:mi>
1295
 
                              <mml:mn>1</mml:mn>
1296
 
                            </mml:msub>
1297
 
                            <mml:mrow>
1298
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1299
 
                              <mml:mi>x</mml:mi>
1300
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1301
 
                            </mml:mrow>
1302
 
                            <mml:mi>;</mml:mi>
1303
 
                            <mml:msubsup>
1304
 
                              <mml:mi>u</mml:mi>
1305
 
                              <mml:mn>1</mml:mn>
1306
 
                              <mml:mn>1</mml:mn>
1307
 
                            </mml:msubsup>
1308
 
                            <mml:mrow>
1309
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1310
 
                              <mml:mi>x</mml:mi>
1311
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1312
 
                            </mml:mrow>
1313
 
                            <mml:mi>;</mml:mi>
1314
 
                            <mml:mo mml:stretchy="false">⋯</mml:mo>
1315
 
                            <mml:msubsup>
1316
 
                              <mml:mi>u</mml:mi>
1317
 
                              <mml:mn>1</mml:mn>
1318
 
                              <mml:mrow>
1319
 
                                <mml:mo mml:stretchy="false">(</mml:mo>
1320
 
                                <mml:mrow>
1321
 
                                  <mml:msub>
1322
 
                                    <mml:mi>m</mml:mi>
1323
 
                                    <mml:mn>1</mml:mn>
1324
 
                                  </mml:msub>
1325
 
                                  <mml:mo mml:stretchy="false">−</mml:mo>
1326
 
                                  <mml:mn>1</mml:mn>
1327
 
                                </mml:mrow>
1328
 
                                <mml:mo mml:stretchy="false">)</mml:mo>
1329
 
                              </mml:mrow>
1330
 
                            </mml:msubsup>
1331
 
                            <mml:mrow>
1332
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1333
 
                              <mml:mi>x</mml:mi>
1334
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1335
 
                            </mml:mrow>
1336
 
                            <mml:mi>;</mml:mi>
1337
 
                            <mml:mi/>
1338
 
                            <mml:mo mml:stretchy="false">⋯</mml:mo>
1339
 
                            <mml:mi>;</mml:mi>
1340
 
                            <mml:mi/>
1341
 
                            <mml:msub>
1342
 
                              <mml:mi>u</mml:mi>
1343
 
                              <mml:mi>N</mml:mi>
1344
 
                            </mml:msub>
1345
 
                            <mml:mrow>
1346
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1347
 
                              <mml:mi>x</mml:mi>
1348
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1349
 
                            </mml:mrow>
1350
 
                            <mml:mi>;</mml:mi>
1351
 
                            <mml:msubsup>
1352
 
                              <mml:mi>u</mml:mi>
1353
 
                              <mml:mi>N</mml:mi>
1354
 
                              <mml:mn>1</mml:mn>
1355
 
                            </mml:msubsup>
1356
 
                            <mml:mrow>
1357
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1358
 
                              <mml:mi>x</mml:mi>
1359
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1360
 
                            </mml:mrow>
1361
 
                            <mml:mi>;</mml:mi>
1362
 
                            <mml:mo mml:stretchy="false">⋯</mml:mo>
1363
 
                            <mml:mi>;</mml:mi>
1364
 
                            <mml:msubsup>
1365
 
                              <mml:mi>u</mml:mi>
1366
 
                              <mml:mi>N</mml:mi>
1367
 
                              <mml:mrow>
1368
 
                                <mml:mo mml:stretchy="false">(</mml:mo>
1369
 
                                <mml:mrow>
1370
 
                                  <mml:msub>
1371
 
                                    <mml:mi>m</mml:mi>
1372
 
                                    <mml:mi>N</mml:mi>
1373
 
                                  </mml:msub>
1374
 
                                  <mml:mo mml:stretchy="false">−</mml:mo>
1375
 
                                  <mml:mn>1</mml:mn>
1376
 
                                </mml:mrow>
1377
 
                                <mml:mo mml:stretchy="false">)</mml:mo>
1378
 
                              </mml:mrow>
1379
 
                            </mml:msubsup>
1380
 
                            <mml:mrow>
1381
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1382
 
                              <mml:mi>x</mml:mi>
1383
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1384
 
                            </mml:mrow>
1385
 
                          </mml:mrow>
1386
 
                          <mml:mo mml:stretchy="false">]</mml:mo>
1387
 
                        </mml:mrow>
1388
 
                      </mml:mrow>
1389
 
                    </mml:mrow>
1390
 
                  </mml:mtr>
1391
 
                  <mml:mtr>
1392
 
                    <mml:mi/>
1393
 
                  </mml:mtr>
1394
 
                  <mml:mtr>
1395
 
                    <mml:mrow>
1396
 
                      <mml:mrow>
1397
 
                        <mml:mrow>
1398
 
                          <mml:mrow>
1399
 
                            <mml:msub>
1400
 
                              <mml:mi>x</mml:mi>
1401
 
                              <mml:mi>l</mml:mi>
1402
 
                            </mml:msub>
1403
 
                            <mml:mo mml:stretchy="false">≤</mml:mo>
1404
 
                            <mml:msub>
1405
 
                              <mml:mo mml:stretchy="false">ζ</mml:mo>
1406
 
                              <mml:mn>1</mml:mn>
1407
 
                            </mml:msub>
1408
 
                          </mml:mrow>
1409
 
                          <mml:mo mml:stretchy="false">≤</mml:mo>
1410
 
                          <mml:mo mml:stretchy="false">⋯</mml:mo>
1411
 
                        </mml:mrow>
1412
 
                        <mml:mo mml:stretchy="false">≤</mml:mo>
1413
 
                        <mml:msub>
1414
 
                          <mml:mo mml:stretchy="false">ζ</mml:mo>
1415
 
                          <mml:mi>M</mml:mi>
1416
 
                        </mml:msub>
1417
 
                      </mml:mrow>
1418
 
                      <mml:mo mml:stretchy="false">≤</mml:mo>
1419
 
                      <mml:msub>
1420
 
                        <mml:mi>x</mml:mi>
1421
 
                        <mml:mi>u</mml:mi>
1422
 
                      </mml:msub>
1423
 
                    </mml:mrow>
1424
 
                  </mml:mtr>
1425
 
                </mml:mtable>
1426
 
              </mml:mrow>
1427
 
              <mml:annotation mml:encoding="StarMath 5.0">left lbrace stack
1428
 
                { M= sum from {i=1} to {N} m_i # ~# z(u(x)) =
1429
 
                [u_1(x);u_1^{1}(x);dotsaxis u_1^(m_1-1)(x);~ dotsaxis;
1430
 
                ~u_N(x);u_N^{1}(x);dotsaxis; u_N^(m_N-1)(x)]# ~# x_l &lt;=
1431
 
                %zeta_1 &lt;= dotsaxis &lt;= %zeta_M &lt;=x_u}right
1432
 
                none
1433
 
              </mml:annotation>
1434
 
            </mml:semantics>
1435
 
          </mml:math>
1436
 
        </imagedata>
1437
 
      </imageobject>
1438
 
    </mediaobject>
1439
 
    <para>
1440
 
      外部関数で使用され, <literal>bvode</literal>  により返される
1441
 
      引数 <literal>zu</literal> は, 
1442
 
      指定した <literal>x</literal> に関する <literal>z(u(x))</literal>の要素により
1443
 
      構成される列ベクトルです.
1444
 
    </para>
1445
 
    <para>
1446
 
      解 u を近似するために使用される手法は,
1447
 
      ガウス点におけるコロケーションです.
1448
 
      この方法は,i番目(i = 1:N)の要素についてm(i)-1階の連続微分が可であることを
1449
 
      必要とします.
1450
 
      ここで, k はサブ区間毎のコロケーション点(ステージ)の数で,
1451
 
      k .ge. max m(i) となるように選択されます.
1452
 
      ルンゲクッタのノミナル解表現が使用されます.
1453
 
    </para>
1454
 
  </refsection>
1455
 
  <refsection>
1456
 
    <title>例</title>
1457
 
    <para>
1458
 
      最初の2つの問題は参考文献の論文 [1] からの引用です.
1459
 
    </para>
1460
 
    <itemizedlist>
1461
 
      <listitem>
1462
 
        <para>
1463
 
          <emphasis role="bold">問題 1</emphasis> は
1464
 
          両端で単純支持され,均等に負荷がかかった剛性が可変の梁について
1465
 
          記述します.
1466
 
        </para>
1467
 
        <para>以下のように定義することができます :</para>
1468
 
        <para>次の4次の微分方程式を解きます:</para>
1469
 
        <mediaobject>
1470
 
          <imageobject>
1471
 
            <imagedata align="left">
1472
 
              <mml:math>
1473
 
                <mml:semantics>
1474
 
                  <mml:mrow>
1475
 
                    <mml:mfrac>
1476
 
                      <mml:msup>
1477
 
                        <mml:mi>d</mml:mi>
1478
 
                        <mml:mn>4</mml:mn>
1479
 
                      </mml:msup>
1480
 
                      <mml:mrow>
1481
 
                        <mml:mi>d</mml:mi>
1482
 
                        <mml:msup>
1483
 
                          <mml:mi>x</mml:mi>
1484
 
                          <mml:mn>4</mml:mn>
1485
 
                        </mml:msup>
1486
 
                      </mml:mrow>
1487
 
                    </mml:mfrac>
1488
 
                    <mml:mi>u</mml:mi>
1489
 
                    <mml:mrow>
1490
 
                      <mml:mrow>
1491
 
                        <mml:mo mml:stretchy="false">(</mml:mo>
1492
 
                        <mml:mi>x</mml:mi>
1493
 
                        <mml:mo mml:stretchy="false">)</mml:mo>
1494
 
                      </mml:mrow>
1495
 
                      <mml:mo mml:stretchy="false">=</mml:mo>
1496
 
                      <mml:mfrac>
1497
 
                        <mml:mrow>
1498
 
                          <mml:mrow>
1499
 
                            <mml:mn>1</mml:mn>
1500
 
                            <mml:mo mml:stretchy="false">−</mml:mo>
1501
 
                            <mml:mrow>
1502
 
                              <mml:mrow>
1503
 
                                <mml:mn>6</mml:mn>
1504
 
                                <mml:mo mml:stretchy="false">.</mml:mo>
1505
 
                                <mml:msup>
1506
 
                                  <mml:mi>x</mml:mi>
1507
 
                                  <mml:mn>2</mml:mn>
1508
 
                                </mml:msup>
1509
 
                              </mml:mrow>
1510
 
                              <mml:mo mml:stretchy="false">.</mml:mo>
1511
 
                              <mml:mfrac>
1512
 
                                <mml:msup>
1513
 
                                  <mml:mi>d</mml:mi>
1514
 
                                  <mml:mn>3</mml:mn>
1515
 
                                </mml:msup>
1516
 
                                <mml:mrow>
1517
 
                                  <mml:mi>d</mml:mi>
1518
 
                                  <mml:msup>
1519
 
                                    <mml:mi>x</mml:mi>
1520
 
                                    <mml:mn>3</mml:mn>
1521
 
                                  </mml:msup>
1522
 
                                </mml:mrow>
1523
 
                              </mml:mfrac>
1524
 
                            </mml:mrow>
1525
 
                          </mml:mrow>
1526
 
                          <mml:mi>u</mml:mi>
1527
 
                          <mml:mrow>
1528
 
                            <mml:mrow>
1529
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1530
 
                              <mml:mi>x</mml:mi>
1531
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1532
 
                            </mml:mrow>
1533
 
                            <mml:mo mml:stretchy="false">−</mml:mo>
1534
 
                            <mml:mrow>
1535
 
                              <mml:mrow>
1536
 
                                <mml:mn>6</mml:mn>
1537
 
                                <mml:mo mml:stretchy="false">.</mml:mo>
1538
 
                                <mml:mi>x</mml:mi>
1539
 
                              </mml:mrow>
1540
 
                              <mml:mo mml:stretchy="false">.</mml:mo>
1541
 
                              <mml:mfrac>
1542
 
                                <mml:msup>
1543
 
                                  <mml:mi>d</mml:mi>
1544
 
                                  <mml:mn>2</mml:mn>
1545
 
                                </mml:msup>
1546
 
                                <mml:mrow>
1547
 
                                  <mml:mi>d</mml:mi>
1548
 
                                  <mml:msup>
1549
 
                                    <mml:mi>x</mml:mi>
1550
 
                                    <mml:mn>2</mml:mn>
1551
 
                                  </mml:msup>
1552
 
                                </mml:mrow>
1553
 
                              </mml:mfrac>
1554
 
                            </mml:mrow>
1555
 
                          </mml:mrow>
1556
 
                          <mml:mi>u</mml:mi>
1557
 
                          <mml:mrow>
1558
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
1559
 
                            <mml:mi>x</mml:mi>
1560
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
1561
 
                          </mml:mrow>
1562
 
                        </mml:mrow>
1563
 
                        <mml:msup>
1564
 
                          <mml:mi>x</mml:mi>
1565
 
                          <mml:mn>3</mml:mn>
1566
 
                        </mml:msup>
1567
 
                      </mml:mfrac>
1568
 
                    </mml:mrow>
1569
 
                  </mml:mrow>
1570
 
                  <mml:annotation mml:encoding="StarMath 5.0">d^4 over {d
1571
 
                    x^4} u(x)={1 -6*x^2*{{d^3} over {d x^3}} u(x)-6*x*{{d^2}
1572
 
                    over {d x^2}} u(x)} over x^3
1573
 
                  </mml:annotation>
1574
 
                </mml:semantics>
1575
 
              </mml:math>
1576
 
            </imagedata>
1577
 
          </imageobject>
1578
 
        </mediaobject>
1579
 
        <para>設定する境界条件:</para>
1580
 
        <mediaobject>
1581
 
          <imageobject>
1582
 
            <imagedata align="left">
1583
 
              <mml:math>
1584
 
                <mml:semantics>
1585
 
                  <mml:mrow>
1586
 
                    <mml:mo mml:stretchy="true">{</mml:mo>
1587
 
                    <mml:mtable>
1588
 
                      <mml:mtr>
1589
 
                        <mml:mrow>
1590
 
                          <mml:mi>u</mml:mi>
1591
 
                          <mml:mrow>
1592
 
                            <mml:mrow>
1593
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1594
 
                              <mml:mn>1</mml:mn>
1595
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1596
 
                            </mml:mrow>
1597
 
                            <mml:mo mml:stretchy="false">=</mml:mo>
1598
 
                            <mml:mn>0</mml:mn>
1599
 
                          </mml:mrow>
1600
 
                        </mml:mrow>
1601
 
                      </mml:mtr>
1602
 
                      <mml:mtr>
1603
 
                        <mml:mrow>
1604
 
                          <mml:mfrac>
1605
 
                            <mml:msup>
1606
 
                              <mml:mi>d</mml:mi>
1607
 
                              <mml:mn>2</mml:mn>
1608
 
                            </mml:msup>
1609
 
                            <mml:mrow>
1610
 
                              <mml:mi>d</mml:mi>
1611
 
                              <mml:msup>
1612
 
                                <mml:mi>x</mml:mi>
1613
 
                                <mml:mn>2</mml:mn>
1614
 
                              </mml:msup>
1615
 
                            </mml:mrow>
1616
 
                          </mml:mfrac>
1617
 
                          <mml:mi>u</mml:mi>
1618
 
                          <mml:mrow>
1619
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
1620
 
                            <mml:mi>x</mml:mi>
1621
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
1622
 
                          </mml:mrow>
1623
 
                          <mml:mrow>
1624
 
                            <mml:mrow>
1625
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1626
 
                              <mml:mn>1</mml:mn>
1627
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1628
 
                            </mml:mrow>
1629
 
                            <mml:mo mml:stretchy="false">=</mml:mo>
1630
 
                            <mml:mn>0</mml:mn>
1631
 
                          </mml:mrow>
1632
 
                        </mml:mrow>
1633
 
                      </mml:mtr>
1634
 
                      <mml:mtr>
1635
 
                        <mml:mrow>
1636
 
                          <mml:mi>u</mml:mi>
1637
 
                          <mml:mrow>
1638
 
                            <mml:mrow>
1639
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1640
 
                              <mml:mn>2</mml:mn>
1641
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1642
 
                            </mml:mrow>
1643
 
                            <mml:mo mml:stretchy="false">=</mml:mo>
1644
 
                            <mml:mn>0</mml:mn>
1645
 
                          </mml:mrow>
1646
 
                        </mml:mrow>
1647
 
                      </mml:mtr>
1648
 
                      <mml:mtr>
1649
 
                        <mml:mrow>
1650
 
                          <mml:mfrac>
1651
 
                            <mml:msup>
1652
 
                              <mml:mi>d</mml:mi>
1653
 
                              <mml:mn>2</mml:mn>
1654
 
                            </mml:msup>
1655
 
                            <mml:mrow>
1656
 
                              <mml:mi>d</mml:mi>
1657
 
                              <mml:msup>
1658
 
                                <mml:mi>x</mml:mi>
1659
 
                                <mml:mn>2</mml:mn>
1660
 
                              </mml:msup>
1661
 
                            </mml:mrow>
1662
 
                          </mml:mfrac>
1663
 
                          <mml:mi>u</mml:mi>
1664
 
                          <mml:mrow>
1665
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
1666
 
                            <mml:mi>x</mml:mi>
1667
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
1668
 
                          </mml:mrow>
1669
 
                          <mml:mrow>
1670
 
                            <mml:mrow>
1671
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1672
 
                              <mml:mn>2</mml:mn>
1673
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1674
 
                            </mml:mrow>
1675
 
                            <mml:mo mml:stretchy="false">=</mml:mo>
1676
 
                            <mml:mn>0</mml:mn>
1677
 
                          </mml:mrow>
1678
 
                        </mml:mrow>
1679
 
                      </mml:mtr>
1680
 
                    </mml:mtable>
1681
 
                  </mml:mrow>
1682
 
                  <mml:annotation mml:encoding="StarMath 5.0">left lbrace
1683
 
                    stack{ u(1)=0# {{d^2} over {d x^2}} u(x)(1)=0# u(2)=0#
1684
 
                    {{d^2} over {d x^2}} u(x)(2)=0} right
1685
 
                    none
1686
 
                  </mml:annotation>
1687
 
                </mml:semantics>
1688
 
              </mml:math>
1689
 
            </imagedata>
1690
 
          </imageobject>
1691
 
        </mediaobject>
1692
 
        <para>この問題の厳密解は以下のようになることが知られています:</para>
1693
 
        <mediaobject>
1694
 
          <imageobject>
1695
 
            <imagedata align="left">
1696
 
              <mml:math>
1697
 
                <mml:semantics>
1698
 
                  <mml:mrow>
1699
 
                    <mml:mi>u</mml:mi>
1700
 
                    <mml:mrow>
1701
 
                      <mml:mrow>
1702
 
                        <mml:mo mml:stretchy="false">(</mml:mo>
1703
 
                        <mml:mi>x</mml:mi>
1704
 
                        <mml:mo mml:stretchy="false">)</mml:mo>
1705
 
                      </mml:mrow>
1706
 
                      <mml:mo mml:stretchy="false">=</mml:mo>
1707
 
                      <mml:mfrac>
1708
 
                        <mml:mn>1</mml:mn>
1709
 
                        <mml:mn>4</mml:mn>
1710
 
                      </mml:mfrac>
1711
 
                    </mml:mrow>
1712
 
                    <mml:mrow>
1713
 
                      <mml:mo mml:stretchy="false">(</mml:mo>
1714
 
                      <mml:mrow>
1715
 
                        <mml:mn>10</mml:mn>
1716
 
                        <mml:mi/>
1717
 
                        <mml:mi>log</mml:mi>
1718
 
                        <mml:mrow>
1719
 
                          <mml:mrow>
1720
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
1721
 
                            <mml:mn>2</mml:mn>
1722
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
1723
 
                          </mml:mrow>
1724
 
                          <mml:mo mml:stretchy="false">−</mml:mo>
1725
 
                          <mml:mn>3</mml:mn>
1726
 
                        </mml:mrow>
1727
 
                      </mml:mrow>
1728
 
                      <mml:mo mml:stretchy="false">)</mml:mo>
1729
 
                    </mml:mrow>
1730
 
                    <mml:mi/>
1731
 
                    <mml:mrow>
1732
 
                      <mml:mrow>
1733
 
                        <mml:mo mml:stretchy="false">(</mml:mo>
1734
 
                        <mml:mrow>
1735
 
                          <mml:mn>1</mml:mn>
1736
 
                          <mml:mo mml:stretchy="false">−</mml:mo>
1737
 
                          <mml:mi>x</mml:mi>
1738
 
                        </mml:mrow>
1739
 
                        <mml:mo mml:stretchy="false">)</mml:mo>
1740
 
                      </mml:mrow>
1741
 
                      <mml:mo mml:stretchy="false">+</mml:mo>
1742
 
                      <mml:mfrac>
1743
 
                        <mml:mn>1</mml:mn>
1744
 
                        <mml:mn>2</mml:mn>
1745
 
                      </mml:mfrac>
1746
 
                    </mml:mrow>
1747
 
                    <mml:mrow>
1748
 
                      <mml:mo mml:stretchy="false">[</mml:mo>
1749
 
                      <mml:mrow>
1750
 
                        <mml:mfrac>
1751
 
                          <mml:mn>1</mml:mn>
1752
 
                          <mml:mi>x</mml:mi>
1753
 
                        </mml:mfrac>
1754
 
                        <mml:mo mml:stretchy="false">+</mml:mo>
1755
 
                        <mml:mrow>
1756
 
                          <mml:mrow>
1757
 
                            <mml:mo mml:stretchy="false">(</mml:mo>
1758
 
                            <mml:mrow>
1759
 
                              <mml:mn>3</mml:mn>
1760
 
                              <mml:mo mml:stretchy="false">+</mml:mo>
1761
 
                              <mml:mi>x</mml:mi>
1762
 
                            </mml:mrow>
1763
 
                            <mml:mo mml:stretchy="false">)</mml:mo>
1764
 
                          </mml:mrow>
1765
 
                          <mml:mi/>
1766
 
                          <mml:mi>log</mml:mi>
1767
 
                          <mml:mrow>
1768
 
                            <mml:mrow>
1769
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1770
 
                              <mml:mi>x</mml:mi>
1771
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1772
 
                            </mml:mrow>
1773
 
                            <mml:mo mml:stretchy="false">−</mml:mo>
1774
 
                            <mml:mi>x</mml:mi>
1775
 
                          </mml:mrow>
1776
 
                        </mml:mrow>
1777
 
                      </mml:mrow>
1778
 
                      <mml:mo mml:stretchy="false">]</mml:mo>
1779
 
                    </mml:mrow>
1780
 
                  </mml:mrow>
1781
 
                  <mml:annotation mml:encoding="StarMath 5.0">u(x)=1 over 4
1782
 
                    (10`log(2)-3)`(1-x)+ {1 over 2} [ {1 over x} + {(3+x) `
1783
 
                    log(x)-x}]
1784
 
                  </mml:annotation>
1785
 
                </mml:semantics>
1786
 
              </mml:math>
1787
 
            </imagedata>
1788
 
          </imageobject>
1789
 
        </mediaobject>
1790
 
        <programlisting role="example"><![CDATA[ 
 
1035
                        </listitem>
 
1036
                    </itemizedlist>
 
1037
                </listitem>
 
1038
            </varlistentry>
 
1039
            <varlistentry>
 
1040
                <term>&lt;optional_args&gt;</term>
 
1041
                <listitem>
 
1042
                    <para>以下のどれかとする必要があります:</para>
 
1043
                    <itemizedlist>
 
1044
                        <listitem>
 
1045
                            <para>
 
1046
                                以下の一連の値:
 
1047
                                <literal>guess, dfsub, dgsub, fixpnt, ndimf, ndimi, ltol, tol,
 
1048
                                    ntol,nonlin, collpnt, subint, iprint, ireg, ifail
 
1049
                                </literal>
 
1050
                            </para>
 
1051
                        </listitem>
 
1052
                        <listitem>
 
1053
                            <para>
 
1054
                                もしくは,一連の<literal>arg_name=argvalue</literal>
 
1055
                                で指定します.
 
1056
                                ただし <literal>arg_name</literal> は以下のどれかとなります: 
 
1057
                                <literal>guess</literal>,
 
1058
                                <literal>dfsub</literal>, <literal>dgsub</literal>,
 
1059
                                <literal>fixpnt</literal>, <literal>ndimf</literal>,
 
1060
                                <literal>ndimi</literal>, <literal>ltol</literal>,
 
1061
                                <literal>tol</literal>, <literal>ntol</literal>,
 
1062
                                <literal>nonlin</literal>, <literal>collpnt</literal>,
 
1063
                                <literal>subint</literal>, <literal>iprint</literal>,
 
1064
                                <literal>ireg</literal>, <literal>ifail</literal>
 
1065
                            </para>
 
1066
                        </listitem>
 
1067
                    </itemizedlist>
 
1068
                    <para>
 
1069
                        これらの引数が指定される場合は,常に
 
1070
                        <literal>ifail</literal>は上記のように説明されます.
 
1071
                        <literal>ifail</literal>は選択されたオプション引数
 
1072
                        に対応する bvode コールを表示するために使用できます.
 
1073
                        もし，<literal>guess</literal> が指定された場合,
 
1074
                        <literal>iguess</literal>が 1 に設定されます.
 
1075
                    </para>
 
1076
                </listitem>
 
1077
            </varlistentry>
 
1078
        </variablelist>
 
1079
    </refsection>
 
1080
    <refsection>
 
1081
        <title>説明</title>
 
1082
        <para>
 
1083
            これらの関数は, 以下のように定義された ode-s の
 
1084
            混合次数に関連するシステム多点境界値問題を解きます.
 
1085
        </para>
 
1086
        <informalequation>
 
1087
            <mediaobject>
 
1088
                <imageobject>
 
1089
                    <imagedata align="left">
 
1090
                        <mml:math>
 
1091
                            <mml:semantics>
 
1092
                                <mml:mrow>
 
1093
                                    <mml:mo mml:stretchy="true">{</mml:mo>
 
1094
                                    <mml:mtable>
 
1095
                                        <mml:mtr>
 
1096
                                            <mml:mrow>
 
1097
                                                <mml:mrow>
 
1098
                                                    <mml:msubsup>
 
1099
                                                        <mml:mi>u</mml:mi>
 
1100
                                                        <mml:mi>i</mml:mi>
 
1101
                                                        <mml:mrow>
 
1102
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1103
                                                            <mml:msub>
 
1104
                                                                <mml:mi>m</mml:mi>
 
1105
                                                                <mml:mi>i</mml:mi>
 
1106
                                                            </mml:msub>
 
1107
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1108
                                                        </mml:mrow>
 
1109
                                                    </mml:msubsup>
 
1110
                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
1111
                                                    <mml:msub>
 
1112
                                                        <mml:mi>f</mml:mi>
 
1113
                                                        <mml:mi>i</mml:mi>
 
1114
                                                    </mml:msub>
 
1115
                                                </mml:mrow>
 
1116
                                                <mml:mrow>
 
1117
                                                    <mml:mo mml:stretchy="false">(</mml:mo>
 
1118
                                                    <mml:mrow>
 
1119
                                                        <mml:mi>x</mml:mi>
 
1120
                                                        <mml:mi>,</mml:mi>
 
1121
                                                        <mml:mi>z</mml:mi>
 
1122
                                                        <mml:mrow>
 
1123
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1124
                                                            <mml:mrow>
 
1125
                                                                <mml:mi>u</mml:mi>
 
1126
                                                                <mml:mrow>
 
1127
                                                                    <mml:mo mml:stretchy="false">(</mml:mo>
 
1128
                                                                    <mml:mi>x</mml:mi>
 
1129
                                                                    <mml:mo mml:stretchy="false">)</mml:mo>
 
1130
                                                                </mml:mrow>
 
1131
                                                            </mml:mrow>
 
1132
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1133
                                                        </mml:mrow>
 
1134
                                                    </mml:mrow>
 
1135
                                                    <mml:mo mml:stretchy="false">)</mml:mo>
 
1136
                                                </mml:mrow>
 
1137
                                                <mml:mi/>
 
1138
                                                <mml:mrow>
 
1139
                                                    <mml:mo mml:stretchy="true">{</mml:mo>
 
1140
                                                    <mml:mtable>
 
1141
                                                        <mml:mtr>
 
1142
                                                            <mml:mrow>
 
1143
                                                                <mml:mrow>
 
1144
                                                                    <mml:mi>i</mml:mi>
 
1145
                                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
1146
                                                                    <mml:mn>1</mml:mn>
 
1147
                                                                </mml:mrow>
 
1148
                                                                <mml:mi mml:fontstyle="normal">:</mml:mi>
 
1149
                                                                <mml:mi>N</mml:mi>
 
1150
                                                            </mml:mrow>
 
1151
                                                        </mml:mtr>
 
1152
                                                        <mml:mtr>
 
1153
                                                            <mml:mrow>
 
1154
                                                                <mml:mi>x</mml:mi>
 
1155
                                                                <mml:mo mml:stretchy="false">∈</mml:mo>
 
1156
                                                                <mml:mfenced mml:close="[" mml:open="]">
 
1157
                                                                    <mml:mrow>
 
1158
                                                                        <mml:msub>
 
1159
                                                                            <mml:mi>x</mml:mi>
 
1160
                                                                            <mml:mi>l</mml:mi>
 
1161
                                                                        </mml:msub>
 
1162
                                                                        <mml:mi>,</mml:mi>
 
1163
                                                                        <mml:msub>
 
1164
                                                                            <mml:mi>x</mml:mi>
 
1165
                                                                            <mml:mi>u</mml:mi>
 
1166
                                                                        </mml:msub>
 
1167
                                                                    </mml:mrow>
 
1168
                                                                </mml:mfenced>
 
1169
                                                            </mml:mrow>
 
1170
                                                        </mml:mtr>
 
1171
                                                    </mml:mtable>
 
1172
                                                </mml:mrow>
 
1173
                                            </mml:mrow>
 
1174
                                        </mml:mtr>
 
1175
                                        <mml:mtr>
 
1176
                                            <mml:mi/>
 
1177
                                        </mml:mtr>
 
1178
                                        <mml:mtr>
 
1179
                                            <mml:mrow>
 
1180
                                                <mml:msub>
 
1181
                                                    <mml:mi>g</mml:mi>
 
1182
                                                    <mml:mi>j</mml:mi>
 
1183
                                                </mml:msub>
 
1184
                                                <mml:mrow>
 
1185
                                                    <mml:mrow>
 
1186
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
1187
                                                        <mml:mrow>
 
1188
                                                            <mml:msub>
 
1189
                                                                <mml:mo mml:stretchy="false">ζ</mml:mo>
 
1190
                                                                <mml:mi>j</mml:mi>
 
1191
                                                            </mml:msub>
 
1192
                                                            <mml:mi>,</mml:mi>
 
1193
                                                            <mml:mi>z</mml:mi>
 
1194
                                                            <mml:mrow>
 
1195
                                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
1196
                                                                <mml:mrow>
 
1197
                                                                    <mml:mi>u</mml:mi>
 
1198
                                                                    <mml:mrow>
 
1199
                                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
1200
                                                                        <mml:msub>
 
1201
                                                                            <mml:mo mml:stretchy="false">ζ</mml:mo>
 
1202
                                                                            <mml:mi>j</mml:mi>
 
1203
                                                                        </mml:msub>
 
1204
                                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
1205
                                                                    </mml:mrow>
 
1206
                                                                </mml:mrow>
 
1207
                                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
1208
                                                            </mml:mrow>
 
1209
                                                        </mml:mrow>
 
1210
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
1211
                                                    </mml:mrow>
 
1212
                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
1213
                                                    <mml:mn>0</mml:mn>
 
1214
                                                </mml:mrow>
 
1215
                                                <mml:mi/>
 
1216
                                                <mml:mrow>
 
1217
                                                    <mml:mtext>,</mml:mtext>
 
1218
                                                    <mml:mi>j</mml:mi>
 
1219
                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
1220
                                                    <mml:mn>1</mml:mn>
 
1221
                                                </mml:mrow>
 
1222
                                                <mml:mi mml:fontstyle="normal">:</mml:mi>
 
1223
                                                <mml:mi>M</mml:mi>
 
1224
                                            </mml:mrow>
 
1225
                                        </mml:mtr>
 
1226
                                    </mml:mtable>
 
1227
                                </mml:mrow>
 
1228
                                <mml:annotation mml:encoding="StarMath 5.0">left lbrace
 
1229
                                    stack{u_i^(m_i)=f_i(x,z(u(x))) ~~left lbrace { binom{i=1:N}{x
 
1230
                                    in left ] a_l, a_u right [} } right none# `#
 
1231
                                    g_j(%zeta_j,z(u(%zeta_j))) = 0~~j=1:M} right
 
1232
                                    none
 
1233
                                </mml:annotation>
 
1234
                            </mml:semantics>
 
1235
                        </mml:math>
 
1236
                    </imagedata>
 
1237
                </imageobject>
 
1238
            </mediaobject>
 
1239
        </informalequation>
 
1240
        <para>ただし,e</para>
 
1241
        <mediaobject>
 
1242
            <imageobject>
 
1243
                <imagedata align="left">
 
1244
                    <mml:math>
 
1245
                        <mml:semantics>
 
1246
                            <mml:mrow>
 
1247
                                <mml:mo mml:stretchy="true">{</mml:mo>
 
1248
                                <mml:mtable>
 
1249
                                    <mml:mtr>
 
1250
                                        <mml:mrow>
 
1251
                                            <mml:mi>M</mml:mi>
 
1252
                                            <mml:mo mml:stretchy="false">=</mml:mo>
 
1253
                                            <mml:mrow>
 
1254
                                                <mml:munderover>
 
1255
                                                    <mml:mo mml:stretchy="false">∑</mml:mo>
 
1256
                                                    <mml:mrow>
 
1257
                                                        <mml:mi>i</mml:mi>
 
1258
                                                        <mml:mo mml:stretchy="false">=</mml:mo>
 
1259
                                                        <mml:mn>1</mml:mn>
 
1260
                                                    </mml:mrow>
 
1261
                                                    <mml:mi>N</mml:mi>
 
1262
                                                </mml:munderover>
 
1263
                                                <mml:msub>
 
1264
                                                    <mml:mi>m</mml:mi>
 
1265
                                                    <mml:mi>i</mml:mi>
 
1266
                                                </mml:msub>
 
1267
                                            </mml:mrow>
 
1268
                                        </mml:mrow>
 
1269
                                    </mml:mtr>
 
1270
                                    <mml:mtr>
 
1271
                                        <mml:mi/>
 
1272
                                    </mml:mtr>
 
1273
                                    <mml:mtr>
 
1274
                                        <mml:mrow>
 
1275
                                            <mml:mi>z</mml:mi>
 
1276
                                            <mml:mrow>
 
1277
                                                <mml:mrow>
 
1278
                                                    <mml:mo mml:stretchy="false">(</mml:mo>
 
1279
                                                    <mml:mrow>
 
1280
                                                        <mml:mi>u</mml:mi>
 
1281
                                                        <mml:mrow>
 
1282
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1283
                                                            <mml:mi>x</mml:mi>
 
1284
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1285
                                                        </mml:mrow>
 
1286
                                                    </mml:mrow>
 
1287
                                                    <mml:mo mml:stretchy="false">)</mml:mo>
 
1288
                                                </mml:mrow>
 
1289
                                                <mml:mo mml:stretchy="false">=</mml:mo>
 
1290
                                                <mml:mrow>
 
1291
                                                    <mml:mo mml:stretchy="false">[</mml:mo>
 
1292
                                                    <mml:mrow>
 
1293
                                                        <mml:msub>
 
1294
                                                            <mml:mi>u</mml:mi>
 
1295
                                                            <mml:mn>1</mml:mn>
 
1296
                                                        </mml:msub>
 
1297
                                                        <mml:mrow>
 
1298
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1299
                                                            <mml:mi>x</mml:mi>
 
1300
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1301
                                                        </mml:mrow>
 
1302
                                                        <mml:mi>;</mml:mi>
 
1303
                                                        <mml:msubsup>
 
1304
                                                            <mml:mi>u</mml:mi>
 
1305
                                                            <mml:mn>1</mml:mn>
 
1306
                                                            <mml:mn>1</mml:mn>
 
1307
                                                        </mml:msubsup>
 
1308
                                                        <mml:mrow>
 
1309
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1310
                                                            <mml:mi>x</mml:mi>
 
1311
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1312
                                                        </mml:mrow>
 
1313
                                                        <mml:mi>;</mml:mi>
 
1314
                                                        <mml:mo mml:stretchy="false">⋯</mml:mo>
 
1315
                                                        <mml:msubsup>
 
1316
                                                            <mml:mi>u</mml:mi>
 
1317
                                                            <mml:mn>1</mml:mn>
 
1318
                                                            <mml:mrow>
 
1319
                                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
1320
                                                                <mml:mrow>
 
1321
                                                                    <mml:msub>
 
1322
                                                                        <mml:mi>m</mml:mi>
 
1323
                                                                        <mml:mn>1</mml:mn>
 
1324
                                                                    </mml:msub>
 
1325
                                                                    <mml:mo mml:stretchy="false">−</mml:mo>
 
1326
                                                                    <mml:mn>1</mml:mn>
 
1327
                                                                </mml:mrow>
 
1328
                                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
1329
                                                            </mml:mrow>
 
1330
                                                        </mml:msubsup>
 
1331
                                                        <mml:mrow>
 
1332
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1333
                                                            <mml:mi>x</mml:mi>
 
1334
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1335
                                                        </mml:mrow>
 
1336
                                                        <mml:mi>;</mml:mi>
 
1337
                                                        <mml:mi/>
 
1338
                                                        <mml:mo mml:stretchy="false">⋯</mml:mo>
 
1339
                                                        <mml:mi>;</mml:mi>
 
1340
                                                        <mml:mi/>
 
1341
                                                        <mml:msub>
 
1342
                                                            <mml:mi>u</mml:mi>
 
1343
                                                            <mml:mi>N</mml:mi>
 
1344
                                                        </mml:msub>
 
1345
                                                        <mml:mrow>
 
1346
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1347
                                                            <mml:mi>x</mml:mi>
 
1348
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1349
                                                        </mml:mrow>
 
1350
                                                        <mml:mi>;</mml:mi>
 
1351
                                                        <mml:msubsup>
 
1352
                                                            <mml:mi>u</mml:mi>
 
1353
                                                            <mml:mi>N</mml:mi>
 
1354
                                                            <mml:mn>1</mml:mn>
 
1355
                                                        </mml:msubsup>
 
1356
                                                        <mml:mrow>
 
1357
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1358
                                                            <mml:mi>x</mml:mi>
 
1359
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1360
                                                        </mml:mrow>
 
1361
                                                        <mml:mi>;</mml:mi>
 
1362
                                                        <mml:mo mml:stretchy="false">⋯</mml:mo>
 
1363
                                                        <mml:mi>;</mml:mi>
 
1364
                                                        <mml:msubsup>
 
1365
                                                            <mml:mi>u</mml:mi>
 
1366
                                                            <mml:mi>N</mml:mi>
 
1367
                                                            <mml:mrow>
 
1368
                                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
1369
                                                                <mml:mrow>
 
1370
                                                                    <mml:msub>
 
1371
                                                                        <mml:mi>m</mml:mi>
 
1372
                                                                        <mml:mi>N</mml:mi>
 
1373
                                                                    </mml:msub>
 
1374
                                                                    <mml:mo mml:stretchy="false">−</mml:mo>
 
1375
                                                                    <mml:mn>1</mml:mn>
 
1376
                                                                </mml:mrow>
 
1377
                                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
1378
                                                            </mml:mrow>
 
1379
                                                        </mml:msubsup>
 
1380
                                                        <mml:mrow>
 
1381
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1382
                                                            <mml:mi>x</mml:mi>
 
1383
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1384
                                                        </mml:mrow>
 
1385
                                                    </mml:mrow>
 
1386
                                                    <mml:mo mml:stretchy="false">]</mml:mo>
 
1387
                                                </mml:mrow>
 
1388
                                            </mml:mrow>
 
1389
                                        </mml:mrow>
 
1390
                                    </mml:mtr>
 
1391
                                    <mml:mtr>
 
1392
                                        <mml:mi/>
 
1393
                                    </mml:mtr>
 
1394
                                    <mml:mtr>
 
1395
                                        <mml:mrow>
 
1396
                                            <mml:mrow>
 
1397
                                                <mml:mrow>
 
1398
                                                    <mml:mrow>
 
1399
                                                        <mml:msub>
 
1400
                                                            <mml:mi>x</mml:mi>
 
1401
                                                            <mml:mi>l</mml:mi>
 
1402
                                                        </mml:msub>
 
1403
                                                        <mml:mo mml:stretchy="false">≤</mml:mo>
 
1404
                                                        <mml:msub>
 
1405
                                                            <mml:mo mml:stretchy="false">ζ</mml:mo>
 
1406
                                                            <mml:mn>1</mml:mn>
 
1407
                                                        </mml:msub>
 
1408
                                                    </mml:mrow>
 
1409
                                                    <mml:mo mml:stretchy="false">≤</mml:mo>
 
1410
                                                    <mml:mo mml:stretchy="false">⋯</mml:mo>
 
1411
                                                </mml:mrow>
 
1412
                                                <mml:mo mml:stretchy="false">≤</mml:mo>
 
1413
                                                <mml:msub>
 
1414
                                                    <mml:mo mml:stretchy="false">ζ</mml:mo>
 
1415
                                                    <mml:mi>M</mml:mi>
 
1416
                                                </mml:msub>
 
1417
                                            </mml:mrow>
 
1418
                                            <mml:mo mml:stretchy="false">≤</mml:mo>
 
1419
                                            <mml:msub>
 
1420
                                                <mml:mi>x</mml:mi>
 
1421
                                                <mml:mi>u</mml:mi>
 
1422
                                            </mml:msub>
 
1423
                                        </mml:mrow>
 
1424
                                    </mml:mtr>
 
1425
                                </mml:mtable>
 
1426
                            </mml:mrow>
 
1427
                            <mml:annotation mml:encoding="StarMath 5.0">left lbrace stack
 
1428
                                { M= sum from {i=1} to {N} m_i # ~# z(u(x)) =
 
1429
                                [u_1(x);u_1^{1}(x);dotsaxis u_1^(m_1-1)(x);~ dotsaxis;
 
1430
                                ~u_N(x);u_N^{1}(x);dotsaxis; u_N^(m_N-1)(x)]# ~# x_l &lt;=
 
1431
                                %zeta_1 &lt;= dotsaxis &lt;= %zeta_M &lt;=x_u}right
 
1432
                                none
 
1433
                            </mml:annotation>
 
1434
                        </mml:semantics>
 
1435
                    </mml:math>
 
1436
                </imagedata>
 
1437
            </imageobject>
 
1438
        </mediaobject>
 
1439
        <para>
 
1440
            外部関数で使用され, <literal>bvode</literal>  により返される
 
1441
            引数 <literal>zu</literal> は, 
 
1442
            指定した <literal>x</literal> に関する <literal>z(u(x))</literal>の要素により
 
1443
            構成される列ベクトルです.
 
1444
        </para>
 
1445
        <para>
 
1446
            解 u を近似するために使用される手法は,
 
1447
            ガウス点におけるコロケーションです.
 
1448
            この方法は,i番目(i = 1:N)の要素についてm(i)-1階の連続微分が可であることを
 
1449
            必要とします.
 
1450
            ここで, k はサブ区間毎のコロケーション点(ステージ)の数で,
 
1451
            k .ge. max m(i) となるように選択されます.
 
1452
            ルンゲクッタのノミナル解表現が使用されます.
 
1453
        </para>
 
1454
    </refsection>
 
1455
    <refsection>
 
1456
        <title>例</title>
 
1457
        <para>
 
1458
            最初の2つの問題は参考文献の論文 [1] からの引用です.
 
1459
        </para>
 
1460
        <itemizedlist>
 
1461
            <listitem>
 
1462
                <para>
 
1463
                    <emphasis role="bold">問題 1</emphasis> は
 
1464
                    両端で単純支持され,均等に負荷がかかった剛性が可変の梁について
 
1465
                    記述します.
 
1466
                </para>
 
1467
                <para>以下のように定義することができます :</para>
 
1468
                <para>次の4次の微分方程式を解きます:</para>
 
1469
                <mediaobject>
 
1470
                    <imageobject>
 
1471
                        <imagedata align="left">
 
1472
                            <mml:math>
 
1473
                                <mml:semantics>
 
1474
                                    <mml:mrow>
 
1475
                                        <mml:mfrac>
 
1476
                                            <mml:msup>
 
1477
                                                <mml:mi>d</mml:mi>
 
1478
                                                <mml:mn>4</mml:mn>
 
1479
                                            </mml:msup>
 
1480
                                            <mml:mrow>
 
1481
                                                <mml:mi>d</mml:mi>
 
1482
                                                <mml:msup>
 
1483
                                                    <mml:mi>x</mml:mi>
 
1484
                                                    <mml:mn>4</mml:mn>
 
1485
                                                </mml:msup>
 
1486
                                            </mml:mrow>
 
1487
                                        </mml:mfrac>
 
1488
                                        <mml:mi>u</mml:mi>
 
1489
                                        <mml:mrow>
 
1490
                                            <mml:mrow>
 
1491
                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
1492
                                                <mml:mi>x</mml:mi>
 
1493
                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
1494
                                            </mml:mrow>
 
1495
                                            <mml:mo mml:stretchy="false">=</mml:mo>
 
1496
                                            <mml:mfrac>
 
1497
                                                <mml:mrow>
 
1498
                                                    <mml:mrow>
 
1499
                                                        <mml:mn>1</mml:mn>
 
1500
                                                        <mml:mo mml:stretchy="false">−</mml:mo>
 
1501
                                                        <mml:mrow>
 
1502
                                                            <mml:mrow>
 
1503
                                                                <mml:mn>6</mml:mn>
 
1504
                                                                <mml:mo mml:stretchy="false">.</mml:mo>
 
1505
                                                                <mml:msup>
 
1506
                                                                    <mml:mi>x</mml:mi>
 
1507
                                                                    <mml:mn>2</mml:mn>
 
1508
                                                                </mml:msup>
 
1509
                                                            </mml:mrow>
 
1510
                                                            <mml:mo mml:stretchy="false">.</mml:mo>
 
1511
                                                            <mml:mfrac>
 
1512
                                                                <mml:msup>
 
1513
                                                                    <mml:mi>d</mml:mi>
 
1514
                                                                    <mml:mn>3</mml:mn>
 
1515
                                                                </mml:msup>
 
1516
                                                                <mml:mrow>
 
1517
                                                                    <mml:mi>d</mml:mi>
 
1518
                                                                    <mml:msup>
 
1519
                                                                        <mml:mi>x</mml:mi>
 
1520
                                                                        <mml:mn>3</mml:mn>
 
1521
                                                                    </mml:msup>
 
1522
                                                                </mml:mrow>
 
1523
                                                            </mml:mfrac>
 
1524
                                                        </mml:mrow>
 
1525
                                                    </mml:mrow>
 
1526
                                                    <mml:mi>u</mml:mi>
 
1527
                                                    <mml:mrow>
 
1528
                                                        <mml:mrow>
 
1529
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1530
                                                            <mml:mi>x</mml:mi>
 
1531
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1532
                                                        </mml:mrow>
 
1533
                                                        <mml:mo mml:stretchy="false">−</mml:mo>
 
1534
                                                        <mml:mrow>
 
1535
                                                            <mml:mrow>
 
1536
                                                                <mml:mn>6</mml:mn>
 
1537
                                                                <mml:mo mml:stretchy="false">.</mml:mo>
 
1538
                                                                <mml:mi>x</mml:mi>
 
1539
                                                            </mml:mrow>
 
1540
                                                            <mml:mo mml:stretchy="false">.</mml:mo>
 
1541
                                                            <mml:mfrac>
 
1542
                                                                <mml:msup>
 
1543
                                                                    <mml:mi>d</mml:mi>
 
1544
                                                                    <mml:mn>2</mml:mn>
 
1545
                                                                </mml:msup>
 
1546
                                                                <mml:mrow>
 
1547
                                                                    <mml:mi>d</mml:mi>
 
1548
                                                                    <mml:msup>
 
1549
                                                                        <mml:mi>x</mml:mi>
 
1550
                                                                        <mml:mn>2</mml:mn>
 
1551
                                                                    </mml:msup>
 
1552
                                                                </mml:mrow>
 
1553
                                                            </mml:mfrac>
 
1554
                                                        </mml:mrow>
 
1555
                                                    </mml:mrow>
 
1556
                                                    <mml:mi>u</mml:mi>
 
1557
                                                    <mml:mrow>
 
1558
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
1559
                                                        <mml:mi>x</mml:mi>
 
1560
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
1561
                                                    </mml:mrow>
 
1562
                                                </mml:mrow>
 
1563
                                                <mml:msup>
 
1564
                                                    <mml:mi>x</mml:mi>
 
1565
                                                    <mml:mn>3</mml:mn>
 
1566
                                                </mml:msup>
 
1567
                                            </mml:mfrac>
 
1568
                                        </mml:mrow>
 
1569
                                    </mml:mrow>
 
1570
                                    <mml:annotation mml:encoding="StarMath 5.0">d^4 over {d
 
1571
                                        x^4} u(x)={1 -6*x^2*{{d^3} over {d x^3}} u(x)-6*x*{{d^2}
 
1572
                                        over {d x^2}} u(x)} over x^3
 
1573
                                    </mml:annotation>
 
1574
                                </mml:semantics>
 
1575
                            </mml:math>
 
1576
                        </imagedata>
 
1577
                    </imageobject>
 
1578
                </mediaobject>
 
1579
                <para>設定する境界条件:</para>
 
1580
                <mediaobject>
 
1581
                    <imageobject>
 
1582
                        <imagedata align="left">
 
1583
                            <mml:math>
 
1584
                                <mml:semantics>
 
1585
                                    <mml:mrow>
 
1586
                                        <mml:mo mml:stretchy="true">{</mml:mo>
 
1587
                                        <mml:mtable>
 
1588
                                            <mml:mtr>
 
1589
                                                <mml:mrow>
 
1590
                                                    <mml:mi>u</mml:mi>
 
1591
                                                    <mml:mrow>
 
1592
                                                        <mml:mrow>
 
1593
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1594
                                                            <mml:mn>1</mml:mn>
 
1595
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1596
                                                        </mml:mrow>
 
1597
                                                        <mml:mo mml:stretchy="false">=</mml:mo>
 
1598
                                                        <mml:mn>0</mml:mn>
 
1599
                                                    </mml:mrow>
 
1600
                                                </mml:mrow>
 
1601
                                            </mml:mtr>
 
1602
                                            <mml:mtr>
 
1603
                                                <mml:mrow>
 
1604
                                                    <mml:mfrac>
 
1605
                                                        <mml:msup>
 
1606
                                                            <mml:mi>d</mml:mi>
 
1607
                                                            <mml:mn>2</mml:mn>
 
1608
                                                        </mml:msup>
 
1609
                                                        <mml:mrow>
 
1610
                                                            <mml:mi>d</mml:mi>
 
1611
                                                            <mml:msup>
 
1612
                                                                <mml:mi>x</mml:mi>
 
1613
                                                                <mml:mn>2</mml:mn>
 
1614
                                                            </mml:msup>
 
1615
                                                        </mml:mrow>
 
1616
                                                    </mml:mfrac>
 
1617
                                                    <mml:mi>u</mml:mi>
 
1618
                                                    <mml:mrow>
 
1619
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
1620
                                                        <mml:mi>x</mml:mi>
 
1621
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
1622
                                                    </mml:mrow>
 
1623
                                                    <mml:mrow>
 
1624
                                                        <mml:mrow>
 
1625
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1626
                                                            <mml:mn>1</mml:mn>
 
1627
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1628
                                                        </mml:mrow>
 
1629
                                                        <mml:mo mml:stretchy="false">=</mml:mo>
 
1630
                                                        <mml:mn>0</mml:mn>
 
1631
                                                    </mml:mrow>
 
1632
                                                </mml:mrow>
 
1633
                                            </mml:mtr>
 
1634
                                            <mml:mtr>
 
1635
                                                <mml:mrow>
 
1636
                                                    <mml:mi>u</mml:mi>
 
1637
                                                    <mml:mrow>
 
1638
                                                        <mml:mrow>
 
1639
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1640
                                                            <mml:mn>2</mml:mn>
 
1641
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1642
                                                        </mml:mrow>
 
1643
                                                        <mml:mo mml:stretchy="false">=</mml:mo>
 
1644
                                                        <mml:mn>0</mml:mn>
 
1645
                                                    </mml:mrow>
 
1646
                                                </mml:mrow>
 
1647
                                            </mml:mtr>
 
1648
                                            <mml:mtr>
 
1649
                                                <mml:mrow>
 
1650
                                                    <mml:mfrac>
 
1651
                                                        <mml:msup>
 
1652
                                                            <mml:mi>d</mml:mi>
 
1653
                                                            <mml:mn>2</mml:mn>
 
1654
                                                        </mml:msup>
 
1655
                                                        <mml:mrow>
 
1656
                                                            <mml:mi>d</mml:mi>
 
1657
                                                            <mml:msup>
 
1658
                                                                <mml:mi>x</mml:mi>
 
1659
                                                                <mml:mn>2</mml:mn>
 
1660
                                                            </mml:msup>
 
1661
                                                        </mml:mrow>
 
1662
                                                    </mml:mfrac>
 
1663
                                                    <mml:mi>u</mml:mi>
 
1664
                                                    <mml:mrow>
 
1665
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
1666
                                                        <mml:mi>x</mml:mi>
 
1667
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
1668
                                                    </mml:mrow>
 
1669
                                                    <mml:mrow>
 
1670
                                                        <mml:mrow>
 
1671
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1672
                                                            <mml:mn>2</mml:mn>
 
1673
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1674
                                                        </mml:mrow>
 
1675
                                                        <mml:mo mml:stretchy="false">=</mml:mo>
 
1676
                                                        <mml:mn>0</mml:mn>
 
1677
                                                    </mml:mrow>
 
1678
                                                </mml:mrow>
 
1679
                                            </mml:mtr>
 
1680
                                        </mml:mtable>
 
1681
                                    </mml:mrow>
 
1682
                                    <mml:annotation mml:encoding="StarMath 5.0">left lbrace
 
1683
                                        stack{ u(1)=0# {{d^2} over {d x^2}} u(x)(1)=0# u(2)=0#
 
1684
                                        {{d^2} over {d x^2}} u(x)(2)=0} right
 
1685
                                        none
 
1686
                                    </mml:annotation>
 
1687
                                </mml:semantics>
 
1688
                            </mml:math>
 
1689
                        </imagedata>
 
1690
                    </imageobject>
 
1691
                </mediaobject>
 
1692
                <para>この問題の厳密解は以下のようになることが知られています:</para>
 
1693
                <mediaobject>
 
1694
                    <imageobject>
 
1695
                        <imagedata align="left">
 
1696
                            <mml:math>
 
1697
                                <mml:semantics>
 
1698
                                    <mml:mrow>
 
1699
                                        <mml:mi>u</mml:mi>
 
1700
                                        <mml:mrow>
 
1701
                                            <mml:mrow>
 
1702
                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
1703
                                                <mml:mi>x</mml:mi>
 
1704
                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
1705
                                            </mml:mrow>
 
1706
                                            <mml:mo mml:stretchy="false">=</mml:mo>
 
1707
                                            <mml:mfrac>
 
1708
                                                <mml:mn>1</mml:mn>
 
1709
                                                <mml:mn>4</mml:mn>
 
1710
                                            </mml:mfrac>
 
1711
                                        </mml:mrow>
 
1712
                                        <mml:mrow>
 
1713
                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1714
                                            <mml:mrow>
 
1715
                                                <mml:mn>10</mml:mn>
 
1716
                                                <mml:mi/>
 
1717
                                                <mml:mi>log</mml:mi>
 
1718
                                                <mml:mrow>
 
1719
                                                    <mml:mrow>
 
1720
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
1721
                                                        <mml:mn>2</mml:mn>
 
1722
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
1723
                                                    </mml:mrow>
 
1724
                                                    <mml:mo mml:stretchy="false">−</mml:mo>
 
1725
                                                    <mml:mn>3</mml:mn>
 
1726
                                                </mml:mrow>
 
1727
                                            </mml:mrow>
 
1728
                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1729
                                        </mml:mrow>
 
1730
                                        <mml:mi/>
 
1731
                                        <mml:mrow>
 
1732
                                            <mml:mrow>
 
1733
                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
1734
                                                <mml:mrow>
 
1735
                                                    <mml:mn>1</mml:mn>
 
1736
                                                    <mml:mo mml:stretchy="false">−</mml:mo>
 
1737
                                                    <mml:mi>x</mml:mi>
 
1738
                                                </mml:mrow>
 
1739
                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
1740
                                            </mml:mrow>
 
1741
                                            <mml:mo mml:stretchy="false">+</mml:mo>
 
1742
                                            <mml:mfrac>
 
1743
                                                <mml:mn>1</mml:mn>
 
1744
                                                <mml:mn>2</mml:mn>
 
1745
                                            </mml:mfrac>
 
1746
                                        </mml:mrow>
 
1747
                                        <mml:mrow>
 
1748
                                            <mml:mo mml:stretchy="false">[</mml:mo>
 
1749
                                            <mml:mrow>
 
1750
                                                <mml:mfrac>
 
1751
                                                    <mml:mn>1</mml:mn>
 
1752
                                                    <mml:mi>x</mml:mi>
 
1753
                                                </mml:mfrac>
 
1754
                                                <mml:mo mml:stretchy="false">+</mml:mo>
 
1755
                                                <mml:mrow>
 
1756
                                                    <mml:mrow>
 
1757
                                                        <mml:mo mml:stretchy="false">(</mml:mo>
 
1758
                                                        <mml:mrow>
 
1759
                                                            <mml:mn>3</mml:mn>
 
1760
                                                            <mml:mo mml:stretchy="false">+</mml:mo>
 
1761
                                                            <mml:mi>x</mml:mi>
 
1762
                                                        </mml:mrow>
 
1763
                                                        <mml:mo mml:stretchy="false">)</mml:mo>
 
1764
                                                    </mml:mrow>
 
1765
                                                    <mml:mi/>
 
1766
                                                    <mml:mi>log</mml:mi>
 
1767
                                                    <mml:mrow>
 
1768
                                                        <mml:mrow>
 
1769
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1770
                                                            <mml:mi>x</mml:mi>
 
1771
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1772
                                                        </mml:mrow>
 
1773
                                                        <mml:mo mml:stretchy="false">−</mml:mo>
 
1774
                                                        <mml:mi>x</mml:mi>
 
1775
                                                    </mml:mrow>
 
1776
                                                </mml:mrow>
 
1777
                                            </mml:mrow>
 
1778
                                            <mml:mo mml:stretchy="false">]</mml:mo>
 
1779
                                        </mml:mrow>
 
1780
                                    </mml:mrow>
 
1781
                                    <mml:annotation mml:encoding="StarMath 5.0">u(x)=1 over 4
 
1782
                                        (10`log(2)-3)`(1-x)+ {1 over 2} [ {1 over x} + {(3+x) `
 
1783
                                        log(x)-x}]
 
1784
                                    </mml:annotation>
 
1785
                                </mml:semantics>
 
1786
                            </mml:math>
 
1787
                        </imagedata>
 
1788
                    </imageobject>
 
1789
                </mediaobject>
 
1790
                <programlisting role="example"><![CDATA[ 
1791
1791
N=1;// just one differential equation
1792
1792
m=4;//a fourth order  differential equation
1793
1793
M=sum(m);
1861
1861
zu1=[];for x=xpoints,zu1=[zu1,trusol(x)]; end;  
1862
1862
norm(zu-zu1)
1863
1863
 ]]></programlisting>
1864
 
      </listitem>
1865
 
      <listitem>
1866
 
        <para>
1867
 
          同じ問題で <literal>bvodeS</literal> と初期推定を用います 
1868
 
        </para>
1869
 
        <programlisting role="example"><![CDATA[ 
 
1864
            </listitem>
 
1865
            <listitem>
 
1866
                <para>
 
1867
                    同じ問題で <literal>bvodeS</literal> と初期推定を用います 
 
1868
                </para>
 
1869
                <programlisting role="example"><![CDATA[ 
1870
1870
function [z,lhS]=zstart(x)
1871
1871
  z=zeros(5,1);z(5)=1;
1872
1872
  lhS=[0;1];
1873
1873
endfunction
1874
1874
zu=bvode(xpoints,N,m,x_low,x_up,zeta,ltol=ltol,tol=tol,guess=zstart)
1875
1875
 ]]></programlisting>
1876
 
      </listitem>
1877
 
      <listitem>
1878
 
        <para>
1879
 
          <emphasis role="bold">問題 2</emphasis>は,
1880
 
          二次関数として変化する軸対象の外部圧力分布のもとでの
1881
 
          一定の厚みの薄く浅い球状のキャップの
1882
 
          微小な有限変形を記述するものです.
1883
 
          ここで, <latex>$\varphi$</latex> は変形したシェルの経線角度変化, <latex>$\psi$</latex> は応力関数です.
1884
 
          <latex>$\varepsilon=\mu=10^{-3}$</latex>の場合,開始点に依存して2つの異なる解が得られる可能性が
1885
 
          あります.
1886
 
        </para>
1887
 
        <mediaobject>
1888
 
          <imageobject>
1889
 
            <imagedata align="left">
1890
 
              <mml:math>
1891
 
                <mml:semantics>
1892
 
                  <mml:mrow>
1893
 
                    <mml:mo mml:stretchy="true">{</mml:mo>
1894
 
                    <mml:mtable>
1895
 
                      <mml:mtr>
1896
 
                        <mml:mrow>
1897
 
                          <mml:mfrac>
1898
 
                            <mml:msup>
1899
 
                              <mml:mo mml:stretchy="false">ε</mml:mo>
1900
 
                              <mml:mn>4</mml:mn>
1901
 
                            </mml:msup>
1902
 
                            <mml:mo mml:stretchy="false">μ</mml:mo>
1903
 
                          </mml:mfrac>
1904
 
                          <mml:mrow>
1905
 
                            <mml:mrow>
1906
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1907
 
                              <mml:mrow>
1908
 
                                <mml:mo mml:stretchy="false">φ</mml:mo>
1909
 
                                <mml:mi>'</mml:mi>
1910
 
                                <mml:mrow>
1911
 
                                  <mml:mrow>
1912
 
                                    <mml:mi>'</mml:mi>
1913
 
                                    <mml:mo mml:stretchy="false">+</mml:mo>
1914
 
                                    <mml:mfrac>
1915
 
                                      <mml:mrow>
1916
 
                                        <mml:mo mml:stretchy="false">φ</mml:mo>
1917
 
                                        <mml:mi>'</mml:mi>
1918
 
                                      </mml:mrow>
1919
 
                                      <mml:mi>x</mml:mi>
1920
 
                                    </mml:mfrac>
1921
 
                                  </mml:mrow>
1922
 
                                  <mml:mo mml:stretchy="false">−</mml:mo>
1923
 
                                  <mml:mfrac>
1924
 
                                    <mml:mo mml:stretchy="false">φ</mml:mo>
1925
 
                                    <mml:msup>
1926
 
                                      <mml:mi>x</mml:mi>
1927
 
                                      <mml:mn>2</mml:mn>
1928
 
                                    </mml:msup>
1929
 
                                  </mml:mfrac>
1930
 
                                </mml:mrow>
1931
 
                              </mml:mrow>
1932
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1933
 
                            </mml:mrow>
1934
 
                            <mml:mo mml:stretchy="false">+</mml:mo>
1935
 
                            <mml:mo mml:stretchy="false">ψ</mml:mo>
1936
 
                          </mml:mrow>
1937
 
                          <mml:mrow>
1938
 
                            <mml:mrow>
1939
 
                              <mml:mrow>
1940
 
                                <mml:mo mml:stretchy="false">(</mml:mo>
1941
 
                                <mml:mrow>
1942
 
                                  <mml:mn>1</mml:mn>
1943
 
                                  <mml:mo mml:stretchy="false">−</mml:mo>
1944
 
                                  <mml:mfrac>
1945
 
                                    <mml:mo mml:stretchy="false">φ</mml:mo>
1946
 
                                    <mml:mi>x</mml:mi>
1947
 
                                  </mml:mfrac>
1948
 
                                </mml:mrow>
1949
 
                                <mml:mo mml:stretchy="false">)</mml:mo>
1950
 
                              </mml:mrow>
1951
 
                              <mml:mo mml:stretchy="false">−</mml:mo>
1952
 
                              <mml:mo mml:stretchy="false">φ</mml:mo>
1953
 
                            </mml:mrow>
1954
 
                            <mml:mo mml:stretchy="false">+</mml:mo>
1955
 
                            <mml:mo mml:stretchy="false">γ</mml:mo>
1956
 
                          </mml:mrow>
1957
 
                          <mml:mi>x</mml:mi>
1958
 
                          <mml:mrow>
1959
 
                            <mml:mrow>
1960
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1961
 
                              <mml:mrow>
1962
 
                                <mml:mn>1</mml:mn>
1963
 
                                <mml:mo mml:stretchy="false">−</mml:mo>
1964
 
                                <mml:mfrac>
1965
 
                                  <mml:msup>
1966
 
                                    <mml:mi>x</mml:mi>
1967
 
                                    <mml:mn>2</mml:mn>
1968
 
                                  </mml:msup>
1969
 
                                  <mml:mn>2</mml:mn>
1970
 
                                </mml:mfrac>
1971
 
                              </mml:mrow>
1972
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
1973
 
                            </mml:mrow>
1974
 
                            <mml:mo mml:stretchy="false">=</mml:mo>
1975
 
                            <mml:mn>0</mml:mn>
1976
 
                          </mml:mrow>
1977
 
                        </mml:mrow>
1978
 
                      </mml:mtr>
1979
 
                      <mml:mtr>
1980
 
                        <mml:mrow>
1981
 
                          <mml:mo mml:stretchy="false">μ</mml:mo>
1982
 
                          <mml:mrow>
1983
 
                            <mml:mrow>
1984
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
1985
 
                              <mml:mrow>
1986
 
                                <mml:mo mml:stretchy="false">ψ</mml:mo>
1987
 
                                <mml:mi>'</mml:mi>
1988
 
                                <mml:mrow>
1989
 
                                  <mml:mrow>
1990
 
                                    <mml:mi>'</mml:mi>
1991
 
                                    <mml:mo mml:stretchy="false">+</mml:mo>
1992
 
                                    <mml:mfrac>
1993
 
                                      <mml:mrow>
1994
 
                                        <mml:mo mml:stretchy="false">ψ</mml:mo>
1995
 
                                        <mml:mi>'</mml:mi>
1996
 
                                      </mml:mrow>
1997
 
                                      <mml:mi>x</mml:mi>
1998
 
                                    </mml:mfrac>
1999
 
                                  </mml:mrow>
2000
 
                                  <mml:mo mml:stretchy="false">−</mml:mo>
2001
 
                                  <mml:mfrac>
2002
 
                                    <mml:mo mml:stretchy="false">ψ</mml:mo>
2003
 
                                    <mml:msup>
2004
 
                                      <mml:mi>x</mml:mi>
2005
 
                                      <mml:mn>2</mml:mn>
2006
 
                                    </mml:msup>
2007
 
                                  </mml:mfrac>
2008
 
                                </mml:mrow>
2009
 
                              </mml:mrow>
2010
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
2011
 
                            </mml:mrow>
2012
 
                            <mml:mo mml:stretchy="false">−</mml:mo>
2013
 
                            <mml:mo mml:stretchy="false">φ</mml:mo>
2014
 
                          </mml:mrow>
2015
 
                          <mml:mrow>
2016
 
                            <mml:mrow>
2017
 
                              <mml:mo mml:stretchy="false">(</mml:mo>
2018
 
                              <mml:mrow>
2019
 
                                <mml:mn>1</mml:mn>
2020
 
                                <mml:mo mml:stretchy="false">−</mml:mo>
2021
 
                                <mml:mfrac>
2022
 
                                  <mml:mo mml:stretchy="false">φ</mml:mo>
2023
 
                                  <mml:mn>2x</mml:mn>
2024
 
                                </mml:mfrac>
2025
 
                              </mml:mrow>
2026
 
                              <mml:mo mml:stretchy="false">)</mml:mo>
2027
 
                            </mml:mrow>
2028
 
                            <mml:mo mml:stretchy="false">=</mml:mo>
2029
 
                            <mml:mn>0</mml:mn>
2030
 
                          </mml:mrow>
2031
 
                        </mml:mrow>
2032
 
                      </mml:mtr>
2033
 
                      <mml:mtr>
2034
 
                        <mml:mrow>
2035
 
                          <mml:mrow>
2036
 
                            <mml:mn>0</mml:mn>
2037
 
                            <mml:mo mml:stretchy="false">&lt;</mml:mo>
2038
 
                            <mml:mi>x</mml:mi>
2039
 
                          </mml:mrow>
2040
 
                          <mml:mo mml:stretchy="false">&lt;</mml:mo>
2041
 
                          <mml:mn>1</mml:mn>
2042
 
                        </mml:mrow>
2043
 
                      </mml:mtr>
2044
 
                    </mml:mtable>
2045
 
                  </mml:mrow>
2046
 
                  <mml:annotation mml:encoding="StarMath 5.0">left lbrace
2047
 
                    stack{{%epsilon^4} over %mu (%phi''+ {%phi'} over x
2048
 
                    -{%phi} over {x^2} ) +%psi (1- %phi over x ) -%phi+%gamma
2049
 
                    x(1-{x^2} over 2)=0# %mu(%psi''+{%psi'} over x-%psi over
2050
 
                    {x^2})-%phi(1-%phi over{2x})=0 # 0&lt;x&lt;1} right
2051
 
                    none
2052
 
                  </mml:annotation>
2053
 
                </mml:semantics>
2054
 
              </mml:math>
2055
 
            </imagedata>
2056
 
          </imageobject>
2057
 
        </mediaobject>
2058
 
        <para>境界条件のもとで</para>
2059
 
        <mediaobject>
2060
 
          <imageobject>
2061
 
            <imagedata align="left">
2062
 
              <mml:math>
2063
 
                <mml:semantics>
2064
 
                  <mml:mrow>
2065
 
                    <mml:mo mml:stretchy="true">{</mml:mo>
2066
 
                    <mml:mtable>
2067
 
                      <mml:mtr>
2068
 
                        <mml:mrow>
2069
 
                          <mml:mo mml:stretchy="false">φ</mml:mo>
2070
 
                          <mml:mo mml:stretchy="false">=</mml:mo>
2071
 
                          <mml:mn>0</mml:mn>
2072
 
                        </mml:mrow>
2073
 
                      </mml:mtr>
2074
 
                      <mml:mtr>
2075
 
                        <mml:mrow>
2076
 
                          <mml:mi>x</mml:mi>
2077
 
                          <mml:mi/>
2078
 
                          <mml:mo mml:stretchy="false">ψ</mml:mo>
2079
 
                          <mml:mrow>
2080
 
                            <mml:mi>'</mml:mi>
2081
 
                            <mml:mo mml:stretchy="false">−</mml:mo>
2082
 
                            <mml:mn>0.3</mml:mn>
2083
 
                          </mml:mrow>
2084
 
                          <mml:mrow>
2085
 
                            <mml:mrow>
2086
 
                              <mml:mo mml:stretchy="false">ψ</mml:mo>
2087
 
                              <mml:mo mml:stretchy="false">+</mml:mo>
2088
 
                              <mml:mn>0.7x</mml:mn>
2089
 
                            </mml:mrow>
2090
 
                            <mml:mo mml:stretchy="false">=</mml:mo>
2091
 
                            <mml:mn>0</mml:mn>
2092
 
                          </mml:mrow>
2093
 
                        </mml:mrow>
2094
 
                      </mml:mtr>
2095
 
                    </mml:mtable>
2096
 
                  </mml:mrow>
2097
 
                  <mml:annotation mml:encoding="StarMath 5.0">left lbrace
2098
 
                    stack{ %phi=0# x`%psi'-0.3%psi+0.7x=0} right
2099
 
                    none
2100
 
                  </mml:annotation>
2101
 
                </mml:semantics>
2102
 
              </mml:math>
2103
 
            </imagedata>
2104
 
          </imageobject>
2105
 
        </mediaobject>
2106
 
        <para>
2107
 
          <literal>x=0</literal> かつ <literal>x=1</literal>の場合
2108
 
        </para>
2109
 
        <programlisting role="example"><![CDATA[ 
 
1876
            </listitem>
 
1877
            <listitem>
 
1878
                <para>
 
1879
                    <emphasis role="bold">問題 2</emphasis>は,
 
1880
                    二次関数として変化する軸対象の外部圧力分布のもとでの
 
1881
                    一定の厚みの薄く浅い球状のキャップの
 
1882
                    微小な有限変形を記述するものです.
 
1883
                    ここで, <latex>$\varphi$</latex> は変形したシェルの経線角度変化, <latex>$\psi$</latex> は応力関数です.
 
1884
                    <latex>$\varepsilon=\mu=10^{-3}$</latex>の場合,開始点に依存して2つの異なる解が得られる可能性が
 
1885
                    あります.
 
1886
                </para>
 
1887
                <mediaobject>
 
1888
                    <imageobject>
 
1889
                        <imagedata align="left">
 
1890
                            <mml:math>
 
1891
                                <mml:semantics>
 
1892
                                    <mml:mrow>
 
1893
                                        <mml:mo mml:stretchy="true">{</mml:mo>
 
1894
                                        <mml:mtable>
 
1895
                                            <mml:mtr>
 
1896
                                                <mml:mrow>
 
1897
                                                    <mml:mfrac>
 
1898
                                                        <mml:msup>
 
1899
                                                            <mml:mo mml:stretchy="false">ε</mml:mo>
 
1900
                                                            <mml:mn>4</mml:mn>
 
1901
                                                        </mml:msup>
 
1902
                                                        <mml:mo mml:stretchy="false">μ</mml:mo>
 
1903
                                                    </mml:mfrac>
 
1904
                                                    <mml:mrow>
 
1905
                                                        <mml:mrow>
 
1906
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1907
                                                            <mml:mrow>
 
1908
                                                                <mml:mo mml:stretchy="false">φ</mml:mo>
 
1909
                                                                <mml:mi>'</mml:mi>
 
1910
                                                                <mml:mrow>
 
1911
                                                                    <mml:mrow>
 
1912
                                                                        <mml:mi>'</mml:mi>
 
1913
                                                                        <mml:mo mml:stretchy="false">+</mml:mo>
 
1914
                                                                        <mml:mfrac>
 
1915
                                                                            <mml:mrow>
 
1916
                                                                                <mml:mo mml:stretchy="false">φ</mml:mo>
 
1917
                                                                                <mml:mi>'</mml:mi>
 
1918
                                                                            </mml:mrow>
 
1919
                                                                            <mml:mi>x</mml:mi>
 
1920
                                                                        </mml:mfrac>
 
1921
                                                                    </mml:mrow>
 
1922
                                                                    <mml:mo mml:stretchy="false">−</mml:mo>
 
1923
                                                                    <mml:mfrac>
 
1924
                                                                        <mml:mo mml:stretchy="false">φ</mml:mo>
 
1925
                                                                        <mml:msup>
 
1926
                                                                            <mml:mi>x</mml:mi>
 
1927
                                                                            <mml:mn>2</mml:mn>
 
1928
                                                                        </mml:msup>
 
1929
                                                                    </mml:mfrac>
 
1930
                                                                </mml:mrow>
 
1931
                                                            </mml:mrow>
 
1932
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1933
                                                        </mml:mrow>
 
1934
                                                        <mml:mo mml:stretchy="false">+</mml:mo>
 
1935
                                                        <mml:mo mml:stretchy="false">ψ</mml:mo>
 
1936
                                                    </mml:mrow>
 
1937
                                                    <mml:mrow>
 
1938
                                                        <mml:mrow>
 
1939
                                                            <mml:mrow>
 
1940
                                                                <mml:mo mml:stretchy="false">(</mml:mo>
 
1941
                                                                <mml:mrow>
 
1942
                                                                    <mml:mn>1</mml:mn>
 
1943
                                                                    <mml:mo mml:stretchy="false">−</mml:mo>
 
1944
                                                                    <mml:mfrac>
 
1945
                                                                        <mml:mo mml:stretchy="false">φ</mml:mo>
 
1946
                                                                        <mml:mi>x</mml:mi>
 
1947
                                                                    </mml:mfrac>
 
1948
                                                                </mml:mrow>
 
1949
                                                                <mml:mo mml:stretchy="false">)</mml:mo>
 
1950
                                                            </mml:mrow>
 
1951
                                                            <mml:mo mml:stretchy="false">−</mml:mo>
 
1952
                                                            <mml:mo mml:stretchy="false">φ</mml:mo>
 
1953
                                                        </mml:mrow>
 
1954
                                                        <mml:mo mml:stretchy="false">+</mml:mo>
 
1955
                                                        <mml:mo mml:stretchy="false">γ</mml:mo>
 
1956
                                                    </mml:mrow>
 
1957
                                                    <mml:mi>x</mml:mi>
 
1958
                                                    <mml:mrow>
 
1959
                                                        <mml:mrow>
 
1960
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1961
                                                            <mml:mrow>
 
1962
                                                                <mml:mn>1</mml:mn>
 
1963
                                                                <mml:mo mml:stretchy="false">−</mml:mo>
 
1964
                                                                <mml:mfrac>
 
1965
                                                                    <mml:msup>
 
1966
                                                                        <mml:mi>x</mml:mi>
 
1967
                                                                        <mml:mn>2</mml:mn>
 
1968
                                                                    </mml:msup>
 
1969
                                                                    <mml:mn>2</mml:mn>
 
1970
                                                                </mml:mfrac>
 
1971
                                                            </mml:mrow>
 
1972
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
1973
                                                        </mml:mrow>
 
1974
                                                        <mml:mo mml:stretchy="false">=</mml:mo>
 
1975
                                                        <mml:mn>0</mml:mn>
 
1976
                                                    </mml:mrow>
 
1977
                                                </mml:mrow>
 
1978
                                            </mml:mtr>
 
1979
                                            <mml:mtr>
 
1980
                                                <mml:mrow>
 
1981
                                                    <mml:mo mml:stretchy="false">μ</mml:mo>
 
1982
                                                    <mml:mrow>
 
1983
                                                        <mml:mrow>
 
1984
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
1985
                                                            <mml:mrow>
 
1986
                                                                <mml:mo mml:stretchy="false">ψ</mml:mo>
 
1987
                                                                <mml:mi>'</mml:mi>
 
1988
                                                                <mml:mrow>
 
1989
                                                                    <mml:mrow>
 
1990
                                                                        <mml:mi>'</mml:mi>
 
1991
                                                                        <mml:mo mml:stretchy="false">+</mml:mo>
 
1992
                                                                        <mml:mfrac>
 
1993
                                                                            <mml:mrow>
 
1994
                                                                                <mml:mo mml:stretchy="false">ψ</mml:mo>
 
1995
                                                                                <mml:mi>'</mml:mi>
 
1996
                                                                            </mml:mrow>
 
1997
                                                                            <mml:mi>x</mml:mi>
 
1998
                                                                        </mml:mfrac>
 
1999
                                                                    </mml:mrow>
 
2000
                                                                    <mml:mo mml:stretchy="false">−</mml:mo>
 
2001
                                                                    <mml:mfrac>
 
2002
                                                                        <mml:mo mml:stretchy="false">ψ</mml:mo>
 
2003
                                                                        <mml:msup>
 
2004
                                                                            <mml:mi>x</mml:mi>
 
2005
                                                                            <mml:mn>2</mml:mn>
 
2006
                                                                        </mml:msup>
 
2007
                                                                    </mml:mfrac>
 
2008
                                                                </mml:mrow>
 
2009
                                                            </mml:mrow>
 
2010
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
2011
                                                        </mml:mrow>
 
2012
                                                        <mml:mo mml:stretchy="false">−</mml:mo>
 
2013
                                                        <mml:mo mml:stretchy="false">φ</mml:mo>
 
2014
                                                    </mml:mrow>
 
2015
                                                    <mml:mrow>
 
2016
                                                        <mml:mrow>
 
2017
                                                            <mml:mo mml:stretchy="false">(</mml:mo>
 
2018
                                                            <mml:mrow>
 
2019
                                                                <mml:mn>1</mml:mn>
 
2020
                                                                <mml:mo mml:stretchy="false">−</mml:mo>
 
2021
                                                                <mml:mfrac>
 
2022
                                                                    <mml:mo mml:stretchy="false">φ</mml:mo>
 
2023
                                                                    <mml:mn>2x</mml:mn>
 
2024
                                                                </mml:mfrac>
 
2025
                                                            </mml:mrow>
 
2026
                                                            <mml:mo mml:stretchy="false">)</mml:mo>
 
2027
                                                        </mml:mrow>
 
2028
                                                        <mml:mo mml:stretchy="false">=</mml:mo>
 
2029
                                                        <mml:mn>0</mml:mn>
 
2030
                                                    </mml:mrow>
 
2031
                                                </mml:mrow>
 
2032
                                            </mml:mtr>
 
2033
                                            <mml:mtr>
 
2034
                                                <mml:mrow>
 
2035
                                                    <mml:mrow>
 
2036
                                                        <mml:mn>0</mml:mn>
 
2037
                                                        <mml:mo mml:stretchy="false">&lt;</mml:mo>
 
2038
                                                        <mml:mi>x</mml:mi>
 
2039
                                                    </mml:mrow>
 
2040
                                                    <mml:mo mml:stretchy="false">&lt;</mml:mo>
 
2041
                                                    <mml:mn>1</mml:mn>
 
2042
                                                </mml:mrow>
 
2043
                                            </mml:mtr>
 
2044
                                        </mml:mtable>
 
2045
                                    </mml:mrow>
 
2046
                                    <mml:annotation mml:encoding="StarMath 5.0">left lbrace
 
2047
                                        stack{{%epsilon^4} over %mu (%phi''+ {%phi'} over x
 
2048
                                        -{%phi} over {x^2} ) +%psi (1- %phi over x ) -%phi+%gamma
 
2049
                                        x(1-{x^2} over 2)=0# %mu(%psi''+{%psi'} over x-%psi over
 
2050
                                        {x^2})-%phi(1-%phi over{2x})=0 # 0&lt;x&lt;1} right
 
2051
                                        none
 
2052
                                    </mml:annotation>
 
2053
                                </mml:semantics>
 
2054
                            </mml:math>
 
2055
                        </imagedata>
 
2056
                    </imageobject>
 
2057
                </mediaobject>
 
2058
                <para>境界条件のもとで</para>
 
2059
                <mediaobject>
 
2060
                    <imageobject>
 
2061
                        <imagedata align="left">
 
2062
                            <mml:math>
 
2063
                                <mml:semantics>
 
2064
                                    <mml:mrow>
 
2065
                                        <mml:mo mml:stretchy="true">{</mml:mo>
 
2066
                                        <mml:mtable>
 
2067
                                            <mml:mtr>
 
2068
                                                <mml:mrow>
 
2069
                                                    <mml:mo mml:stretchy="false">φ</mml:mo>
 
2070
                                                    <mml:mo mml:stretchy="false">=</mml:mo>
 
2071
                                                    <mml:mn>0</mml:mn>
 
2072
                                                </mml:mrow>
 
2073
                                            </mml:mtr>
 
2074
                                            <mml:mtr>
 
2075
                                                <mml:mrow>
 
2076
                                                    <mml:mi>x</mml:mi>
 
2077
                                                    <mml:mi/>
 
2078
                                                    <mml:mo mml:stretchy="false">ψ</mml:mo>
 
2079
                                                    <mml:mrow>
 
2080
                                                        <mml:mi>'</mml:mi>
 
2081
                                                        <mml:mo mml:stretchy="false">−</mml:mo>
 
2082
                                                        <mml:mn>0.3</mml:mn>
 
2083
                                                    </mml:mrow>
 
2084
                                                    <mml:mrow>
 
2085
                                                        <mml:mrow>
 
2086
                                                            <mml:mo mml:stretchy="false">ψ</mml:mo>
 
2087
                                                            <mml:mo mml:stretchy="false">+</mml:mo>
 
2088
                                                            <mml:mn>0.7x</mml:mn>
 
2089
                                                        </mml:mrow>
 
2090
                                                        <mml:mo mml:stretchy="false">=</mml:mo>
 
2091
                                                        <mml:mn>0</mml:mn>
 
2092
                                                    </mml:mrow>
 
2093
                                                </mml:mrow>
 
2094
                                            </mml:mtr>
 
2095
                                        </mml:mtable>
 
2096
                                    </mml:mrow>
 
2097
                                    <mml:annotation mml:encoding="StarMath 5.0">left lbrace
 
2098
                                        stack{ %phi=0# x`%psi'-0.3%psi+0.7x=0} right
 
2099
                                        none
 
2100
                                    </mml:annotation>
 
2101
                                </mml:semantics>
 
2102
                            </mml:math>
 
2103
                        </imagedata>
 
2104
                    </imageobject>
 
2105
                </mediaobject>
 
2106
                <para>
 
2107
                    <literal>x=0</literal> かつ <literal>x=1</literal>の場合
 
2108
                </para>
 
2109
                <programlisting role="example"><![CDATA[ 
2110
2110
N=2;// two differential equations
2111
2111
m=[2 2];//each differential equation is of second  order
2112
2112
M=sum(m);
2200
2200
        fsub2,dfsub2,gsub2,dgsub2,guess2);
2201
2201
scf(2);clf();plot(xpoints,zu2([1 3],:)) // the evolution of the solution phi and psi
2202
2202
 ]]></programlisting>
2203
 
      </listitem>
2204
 
      <listitem>
2205
 
        <para>
2206
 
          <emphasis role="bold">固有値問題:</emphasis>
2207
 
        </para>
2208
 
        <programlisting role="example"><![CDATA[ 
 
2203
            </listitem>
 
2204
            <listitem>
 
2205
                <para>
 
2206
                    <emphasis role="bold">固有値問題:</emphasis>
 
2207
                </para>
 
2208
                <programlisting role="example"><![CDATA[ 
2209
2209
// y''(x)=-la*y(x)
2210
2210
// BV: y(0)=y'(0); y(1)=0
2211
2211
// Eigenfunctions and eigenvalues are y(x,n)=sin(s(n)*(1-x)), la(n)=s(n)^2,
2250
2250
xtitle(['Startvalue =  '+string(la0);'Eigenvalue = '+string(z(3,1))],'x',' ')
2251
2251
legend(['y(x)';'y''(x)'])
2252
2252
 ]]></programlisting>
2253
 
      </listitem>
2254
 
      <listitem>
2255
 
        <para>
2256
 
          <emphasis role="bold">一つ以上の解がある多点境界値問題.</emphasis>
2257
 
        </para>
2258
 
        <programlisting role="example"><![CDATA[ 
 
2253
            </listitem>
 
2254
            <listitem>
 
2255
                <para>
 
2256
                    <emphasis role="bold">一つ以上の解がある多点境界値問題.</emphasis>
 
2257
                </para>
 
2258
                <programlisting role="example"><![CDATA[ 
2259
2259
// DE: y''(x)=-exp(y(x))
2260
2260
// BV: y(0)=0; y(1)=0
2261
2261
// This boundary value problem has more than one solution.
2317
2317
plot2d(x,z1(1,:),style=[5])
2318
2318
xtitle(' ','x',' ')
2319
2319
 ]]></programlisting>
2320
 
      </listitem>
2321
 
      <listitem>
2322
 
        <para>
2323
 
          <emphasis role="bold">多点境界値問題.</emphasis>
2324
 
        </para>
2325
 
        <programlisting role="example"><![CDATA[ 
 
2320
            </listitem>
 
2321
            <listitem>
 
2322
                <para>
 
2323
                    <emphasis role="bold">多点境界値問題.</emphasis>
 
2324
                </para>
 
2325
                <programlisting role="example"><![CDATA[ 
2326
2326
// DE y'''(x)=1
2327
2327
// z=[y(x);y'(x);y''(x)]
2328
2328
// BV: y(-1)=2 y(1)=2
2354
2354
 
2355
2355
disp(norm(yex(x)-z(1,:)),'norm(yex(x)-z(1,:))= ')
2356
2356
 ]]></programlisting>
2357
 
      </listitem>
2358
 
    </itemizedlist>
2359
 
  </refsection>
2360
 
  <refsection role="see also">
2361
 
    <title>参照</title>
2362
 
    <simplelist type="inline">
2363
 
      <member>
2364
 
        <link linkend="link">link</link>
2365
 
      </member>
2366
 
      <member>
2367
 
        <link linkend="external">external</link>
2368
 
      </member>
2369
 
      <member>
2370
 
        <link linkend="ode">ode</link>
2371
 
      </member>
2372
 
      <member>
2373
 
        <link linkend="dassl">dassl</link>
2374
 
      </member>
2375
 
    </simplelist>
2376
 
  </refsection>
2377
 
  <refsection>
2378
 
    <title>使用される関数</title>
2379
 
    <para>この関数は, 以下の作者により開発された
2380
 
      Fortran ルーチン<literal>colnew</literal>
2381
 
      に基づいています.
2382
 
    </para>
2383
 
    <para>U. Ascher, Department of Computer Science, University of British
2384
 
      Columbia, Vancouver, B.C. V6T 1W5, Canada
2385
 
    </para>
2386
 
    <para>G. Bader, institut f. Angewandte mathematik university of
2387
 
      Heidelberg; im Neuenheimer feld 294d-6900 Heidelberg 1
2388
 
    </para>
2389
 
  </refsection>
2390
 
  <refsection>
2391
 
    <title>参考文献</title>
2392
 
    <orderedlist>
2393
 
      <listitem>
2394
 
        <para>U. Ascher, J. Christiansen and R.D. Russell, collocation
2395
 
          software for boundary-value ODEs, acm trans. math software 7 (1981),
2396
 
          209-222. this paper contains EXAMPLES where use of the code is
2397
 
          demonstrated.
2398
 
        </para>
2399
 
      </listitem>
2400
 
      <listitem>
2401
 
        <para>G. Bader and U. Ascher, a new basis implementation for a mixed
2402
 
          order boundary value ode solver, siam j. scient. stat. comput.
2403
 
          (1987).
2404
 
        </para>
2405
 
      </listitem>
2406
 
      <listitem>
2407
 
        <para>U. Ascher, J. Christiansen and R.D. Russell, a collocation
2408
 
          solver for mixed order systems of boundary value problems, math. comp.
2409
 
          33 (1979), 659-679.
2410
 
        </para>
2411
 
      </listitem>
2412
 
      <listitem>
2413
 
        <para>U. Ascher, J. Christiansen and R.D. russell, colsys - a
2414
 
          collocation code for boundary value problems, lecture notes comp.sc.
2415
 
          76, springer verlag, b. childs et. al. (eds.) (1979), 164-185.
2416
 
        </para>
2417
 
      </listitem>
2418
 
      <listitem>
2419
 
        <para>C. Deboor and R. Weiss, solveblok: a package for solving almost
2420
 
          block diagonal linear systems, acm trans. math. software 6 (1980),
2421
 
          80-87.
2422
 
        </para>
2423
 
      </listitem>
2424
 
    </orderedlist>
2425
 
  </refsection>
 
2357
            </listitem>
 
2358
        </itemizedlist>
 
2359
    </refsection>
 
2360
    <refsection role="see also">
 
2361
        <title>参照</title>
 
2362
        <simplelist type="inline">
 
2363
            <member>
 
2364
                <link linkend="link">link</link>
 
2365
            </member>
 
2366
            <member>
 
2367
                <link linkend="external">external</link>
 
2368
            </member>
 
2369
            <member>
 
2370
                <link linkend="ode">ode</link>
 
2371
            </member>
 
2372
            <member>
 
2373
                <link linkend="dassl">dassl</link>
 
2374
            </member>
 
2375
        </simplelist>
 
2376
    </refsection>
 
2377
    <refsection>
 
2378
        <title>使用される関数</title>
 
2379
        <para>この関数は, 以下の作者により開発された
 
2380
            Fortran ルーチン<literal>colnew</literal>
 
2381
            に基づいています.
 
2382
        </para>
 
2383
        <para>U. Ascher, Department of Computer Science, University of British
 
2384
            Columbia, Vancouver, B.C. V6T 1W5, Canada
 
2385
        </para>
 
2386
        <para>G. Bader, institut f. Angewandte mathematik university of
 
2387
            Heidelberg; im Neuenheimer feld 294d-6900 Heidelberg 1
 
2388
        </para>
 
2389
    </refsection>
 
2390
    <refsection>
 
2391
        <title>参考文献</title>
 
2392
        <orderedlist>
 
2393
            <listitem>
 
2394
                <para>U. Ascher, J. Christiansen and R.D. Russell, collocation
 
2395
                    software for boundary-value ODEs, acm trans. math software 7 (1981),
 
2396
                    209-222. this paper contains EXAMPLES where use of the code is
 
2397
                    demonstrated.
 
2398
                </para>
 
2399
            </listitem>
 
2400
            <listitem>
 
2401
                <para>G. Bader and U. Ascher, a new basis implementation for a mixed
 
2402
                    order boundary value ode solver, siam j. scient. stat. comput.
 
2403
                    (1987).
 
2404
                </para>
 
2405
            </listitem>
 
2406
            <listitem>
 
2407
                <para>U. Ascher, J. Christiansen and R.D. Russell, a collocation
 
2408
                    solver for mixed order systems of boundary value problems, math. comp.
 
2409
                    33 (1979), 659-679.
 
2410
                </para>
 
2411
            </listitem>
 
2412
            <listitem>
 
2413
                <para>U. Ascher, J. Christiansen and R.D. russell, colsys - a
 
2414
                    collocation code for boundary value problems, lecture notes comp.sc.
 
2415
                    76, springer verlag, b. childs et. al. (eds.) (1979), 164-185.
 
2416
                </para>
 
2417
            </listitem>
 
2418
            <listitem>
 
2419
                <para>C. Deboor and R. Weiss, solveblok: a package for solving almost
 
2420
                    block diagonal linear systems, acm trans. math. software 6 (1980),
 
2421
                    80-87.
 
2422
                </para>
 
2423
            </listitem>
 
2424
        </orderedlist>
 
2425
    </refsection>
2426
2426
</refentry>