1
subroutine lsodes (f, neq, y, t, tout, itol, rtol, atol, itask,
2
1 istate, iopt, rwork, lrw, iwork, liw, jac, mf)
4
integer neq, itol, itask, istate, iopt, lrw, iwork, liw, mf
5
double precision y, t, tout, rtol, atol, rwork
6
dimension neq(1), y(1), rtol(1), atol(1), rwork(lrw), iwork(liw)
7
c-----------------------------------------------------------------------
8
c this is the march 30, 1987 version of
9
c lsodes.. livermore solver for ordinary differential equations
10
c with general sparse jacobian matrices.
11
c this version is in double precision.
13
c lsodes solves the initial value problem for stiff or nonstiff
14
c systems of first order ode-s,
15
c dy/dt = f(t,y) , or, in component form,
16
c dy(i)/dt = f(i) = f(i,t,y(1),y(2),...,y(neq)) (i = 1,...,neq).
17
c lsodes is a variant of the lsode package, and is intended for
18
c problems in which the jacobian matrix df/dy has an arbitrary
19
c sparse structure (when the problem is stiff).
21
c authors.. alan c. hindmarsh,
22
c computing and mathematics research division, l-316
23
c lawrence livermore national laboratory
24
c livermore, ca 94550.
26
c and andrew h. sherman
27
c j. s. nolen and associates
29
c-----------------------------------------------------------------------
31
c 1. alan c. hindmarsh, odepack, a systematized collection of ode
32
c solvers, in scientific computing, r. s. stepleman et al. (eds.),
33
c north-holland, amsterdam, 1983, pp. 55-64.
35
c 2. s. c. eisenstat, m. c. gursky, m. h. schultz, and a. h. sherman,
36
c yale sparse matrix package.. i. the symmetric codes,
37
c int. j. num. meth. eng., 18 (1982), pp. 1145-1151.
39
c 3. s. c. eisenstat, m. c. gursky, m. h. schultz, and a. h. sherman,
40
c yale sparse matrix package.. ii. the nonsymmetric codes,
41
c research report no. 114, dept. of computer sciences, yale
43
c-----------------------------------------------------------------------
46
c communication between the user and the lsodes package, for normal
47
c situations, is summarized here. this summary describes only a subset
48
c of the full set of options available. see the full description for
49
c details, including optional communication, nonstandard options,
50
c and instructions for special situations. see also the example
51
c problem (with program and output) following this summary.
53
c a. first provide a subroutine of the form..
54
c subroutine f (neq, t, y, ydot)
55
c dimension y(neq), ydot(neq)
56
c which supplies the vector function f by loading ydot(i) with f(i).
58
c b. next determine (or guess) whether or not the problem is stiff.
59
c stiffness occurs when the jacobian matrix df/dy has an eigenvalue
60
c whose real part is negative and large in magnitude, compared to the
61
c reciprocal of the t span of interest. if the problem is nonstiff,
62
c use a method flag mf = 10. if it is stiff, there are two standard
63
c for the method flag, mf = 121 and mf = 222. in both cases, lsodes
64
c requires the jacobian matrix in some form, and it treats this matrix
65
c in general sparse form, with sparsity structure determined internally.
66
c (for options where the user supplies the sparsity structure, see
67
c the full description of mf below.)
69
c c. if the problem is stiff, you are encouraged to supply the jacobian
70
c directly (mf = 121), but if this is not feasible, lsodes will
71
c compute it internally by difference quotients (mf = 222).
72
c if you are supplying the jacobian, provide a subroutine of the form..
73
c subroutine jac (neq, t, y, j, ian, jan, pdj)
74
c dimension y(1), ian(1), jan(1), pdj(1)
75
c here neq, t, y, and j are input arguments, and the jac routine is to
76
c load the array pdj (of length neq) with the j-th column of df/dy.
77
c i.e., load pdj(i) with df(i)/dy(j) for all relevant values of i.
78
c the arguments ian and jan should be ignored for normal situations.
79
c lsodes will call the jac routine with j = 1,2,...,neq.
80
c only nonzero elements need be loaded. usually, a crude approximation
81
c to df/dy, possibly with fewer nonzero elements, will suffice.
83
c d. write a main program which calls subroutine lsodes once for
84
c each point at which answers are desired. this should also provide
85
c for possible use of logical unit 6 for output of error messages
86
c by lsodes. on the first call to lsodes, supply arguments as follows..
87
c f = name of subroutine for right-hand side vector f.
88
c this name must be declared external in calling program.
89
c neq = number of first order ode-s.
90
c y = array of initial values, of length neq.
91
c t = the initial value of the independent variable.
92
c tout = first point where output is desired (.ne. t).
93
c itol = 1 or 2 according as atol (below) is a scalar or array.
94
c rtol = relative tolerance parameter (scalar).
95
c atol = absolute tolerance parameter (scalar or array).
96
c the estimated local error in y(i) will be controlled so as
97
c to be roughly less (in magnitude) than
98
c ewt(i) = rtol*abs(y(i)) + atol if itol = 1, or
99
c ewt(i) = rtol*abs(y(i)) + atol(i) if itol = 2.
100
c thus the local error test passes if, in each component,
101
c either the absolute error is less than atol (or atol(i)),
102
c or the relative error is less than rtol.
103
c use rtol = 0.0 for pure absolute error control, and
104
c use atol = 0.0 (or atol(i) = 0.0) for pure relative error
105
c control. caution.. actual (global) errors may exceed these
106
c local tolerances, so choose them conservatively.
107
c itask = 1 for normal computation of output values of y at t = tout.
108
c istate = integer flag (input and output). set istate = 1.
109
c iopt = 0 to indicate no optional inputs used.
110
c rwork = real work array of length at least..
111
c 20 + 16*neq for mf = 10,
112
c 20 + (2 + 1./lenrat)*nnz + (11 + 9./lenrat)*neq
113
c for mf = 121 or 222,
115
c nnz = the number of nonzero elements in the sparse
116
c jacobian (if this is unknown, use an estimate), and
117
c lenrat = the real to integer wordlength ratio (usually 1 in
118
c single precision and 2 in double precision).
119
c in any case, the required size of rwork cannot generally
120
c be predicted in advance if mf = 121 or 222, and the value
121
c above is a rough estimate of a crude lower bound. some
122
c experimentation with this size may be necessary.
123
c (when known, the correct required length is an optional
124
c output, available in iwork(17).)
125
c lrw = declared length of rwork (in user-s dimension).
126
c iwork = integer work array of length at least 30.
127
c liw = declared length of iwork (in user-s dimension).
128
c jac = name of subroutine for jacobian matrix (mf = 121).
129
c if used, this name must be declared external in calling
130
c program. if not used, pass a dummy name.
131
c mf = method flag. standard values are..
132
c 10 for nonstiff (adams) method, no jacobian used.
133
c 121 for stiff (bdf) method, user-supplied sparse jacobian.
134
c 222 for stiff method, internally generated sparse jacobian.
135
c note that the main program must declare arrays y, rwork, iwork,
138
c e. the output from the first call (or any call) is..
139
c y = array of computed values of y(t) vector.
140
c t = corresponding value of independent variable (normally tout).
141
c istate = 2 if lsodes was successful, negative otherwise.
142
c -1 means excess work done on this call (perhaps wrong mf).
143
c -2 means excess accuracy requested (tolerances too small).
144
c -3 means illegal input detected (see printed message).
145
c -4 means repeated error test failures (check all inputs).
146
c -5 means repeated convergence failures (perhaps bad jacobian
147
c supplied or wrong choice of mf or tolerances).
148
c -6 means error weight became zero during problem. (solution
149
c component i vanished, and atol or atol(i) = 0.)
150
c -7 means a fatal error return flag came from the sparse
151
c solver cdrv by way of prjs or slss. should never happen.
152
c a return with istate = -1, -4, or -5 may result from using
153
c an inappropriate sparsity structure, one that is quite
154
c different from the initial structure. consider calling
155
c lsodes again with istate = 3 to force the structure to be
156
c reevaluated. see the full description of istate below.
158
c f. to continue the integration after a successful return, simply
159
c reset tout and call lsodes again. no other parameters need be reset.
161
c-----------------------------------------------------------------------
164
c the following is a simple example problem, with the coding
165
c needed for its solution by lsodes. the problem is from chemical
166
c kinetics, and consists of the following 12 rate equations..
168
c dy2/dt = rk1*y1 + rk11*rk14*y4 + rk19*rk14*y5
169
c - rk3*y2*y3 - rk15*y2*y12 - rk2*y2
170
c dy3/dt = rk2*y2 - rk5*y3 - rk3*y2*y3 - rk7*y10*y3
171
c + rk11*rk14*y4 + rk12*rk14*y6
172
c dy4/dt = rk3*y2*y3 - rk11*rk14*y4 - rk4*y4
173
c dy5/dt = rk15*y2*y12 - rk19*rk14*y5 - rk16*y5
174
c dy6/dt = rk7*y10*y3 - rk12*rk14*y6 - rk8*y6
175
c dy7/dt = rk17*y10*y12 - rk20*rk14*y7 - rk18*y7
176
c dy8/dt = rk9*y10 - rk13*rk14*y8 - rk10*y8
177
c dy9/dt = rk4*y4 + rk16*y5 + rk8*y6 + rk18*y7
178
c dy10/dt = rk5*y3 + rk12*rk14*y6 + rk20*rk14*y7
179
c + rk13*rk14*y8 - rk7*y10*y3 - rk17*y10*y12
180
c - rk6*y10 - rk9*y10
182
c dy12/dt = rk6*y10 + rk19*rk14*y5 + rk20*rk14*y7
183
c - rk15*y2*y12 - rk17*y10*y12
185
c with rk1 = rk5 = 0.1, rk4 = rk8 = rk16 = rk18 = 2.5,
186
c rk10 = 5.0, rk2 = rk6 = 10.0, rk14 = 30.0,
187
c rk3 = rk7 = rk9 = rk11 = rk12 = rk13 = rk19 = rk20 = 50.0,
188
c rk15 = rk17 = 100.0.
190
c the t interval is from 0 to 1000, and the initial conditions
191
c are y1 = 1, y2 = y3 = ... = y12 = 0. the problem is stiff.
193
c the following coding solves this problem with lsodes, using mf = 121
194
c and printing results at t = .1, 1., 10., 100., 1000. it uses
195
c itol = 1 and mixed relative/absolute tolerance controls.
196
c during the run and at the end, statistical quantities of interest
197
c are printed (see optional outputs in the full description below).
200
c double precision atol, rtol, rwork, t, tout, y
201
c dimension y(12), rwork(500), iwork(30)
202
c data lrw/500/, liw/30/
217
c call lsodes (fex, neq, y, t, tout, itol, rtol, atol,
218
c 1 itask, istate, iopt, rwork, lrw, iwork, liw, jex, mf)
219
c write(6,30)t,iwork(11),rwork(11),(y(i),i=1,neq)
220
c 30 format(//' at t =',e11.3,4x,
221
c 1 ' no. steps =',i5,4x,' last step =',e11.3/
222
c 2 ' y array = ',4e14.5/13x,4e14.5/13x,4e14.5)
223
c if (istate .lt. 0) go to 80
233
c nnzlu = iwork(25) + iwork(26) + neq
234
c write (6,70) lenrw,leniw,nst,nfe,nje,nlu,nnz,nnzlu
235
c 70 format(//' required rwork size =',i4,' iwork size =',i4/
236
c 1 ' no. steps =',i4,' no. f-s =',i4,' no. j-s =',i4,
237
c 2 ' no. lu-s =',i4/' no. of nonzeros in j =',i5,
238
c 3 ' no. of nonzeros in lu =',i5)
240
c 80 write(6,90)istate
241
c 90 format(///' error halt.. istate =',i3)
245
c subroutine fex (neq, t, y, ydot)
246
c double precision t, y, ydot
247
c double precision rk1, rk2, rk3, rk4, rk5, rk6, rk7, rk8, rk9,
248
c 1 rk10, rk11, rk12, rk13, rk14, rk15, rk16, rk17
249
c dimension y(12), ydot(12)
250
c data rk1/0.1d0/, rk2/10.0d0/, rk3/50.0d0/, rk4/2.5d0/, rk5/0.1d0/,
251
c 1 rk6/10.0d0/, rk7/50.0d0/, rk8/2.5d0/, rk9/50.0d0/, rk10/5.0d0/,
252
c 2 rk11/50.0d0/, rk12/50.0d0/, rk13/50.0d0/, rk14/30.0d0/,
253
c 3 rk15/100.0d0/, rk16/2.5d0/, rk17/100.0d0/, rk18/2.5d0/,
254
c 4 rk19/50.0d0/, rk20/50.0d0/
255
c ydot(1) = -rk1*y(1)
256
c ydot(2) = rk1*y(1) + rk11*rk14*y(4) + rk19*rk14*y(5)
257
c 1 - rk3*y(2)*y(3) - rk15*y(2)*y(12) - rk2*y(2)
258
c ydot(3) = rk2*y(2) - rk5*y(3) - rk3*y(2)*y(3) - rk7*y(10)*y(3)
259
c 1 + rk11*rk14*y(4) + rk12*rk14*y(6)
260
c ydot(4) = rk3*y(2)*y(3) - rk11*rk14*y(4) - rk4*y(4)
261
c ydot(5) = rk15*y(2)*y(12) - rk19*rk14*y(5) - rk16*y(5)
262
c ydot(6) = rk7*y(10)*y(3) - rk12*rk14*y(6) - rk8*y(6)
263
c ydot(7) = rk17*y(10)*y(12) - rk20*rk14*y(7) - rk18*y(7)
264
c ydot(8) = rk9*y(10) - rk13*rk14*y(8) - rk10*y(8)
265
c ydot(9) = rk4*y(4) + rk16*y(5) + rk8*y(6) + rk18*y(7)
266
c ydot(10) = rk5*y(3) + rk12*rk14*y(6) + rk20*rk14*y(7)
267
c 1 + rk13*rk14*y(8) - rk7*y(10)*y(3) - rk17*y(10)*y(12)
268
c 2 - rk6*y(10) - rk9*y(10)
269
c ydot(11) = rk10*y(8)
270
c ydot(12) = rk6*y(10) + rk19*rk14*y(5) + rk20*rk14*y(7)
271
c 1 - rk15*y(2)*y(12) - rk17*y(10)*y(12)
275
c subroutine jex (neq, t, y, j, ia, ja, pdj)
276
c double precision t, y, pdj
277
c double precision rk1, rk2, rk3, rk4, rk5, rk6, rk7, rk8, rk9,
278
c 1 rk10, rk11, rk12, rk13, rk14, rk15, rk16, rk17
279
c dimension y(1), ia(1), ja(1), pdj(1)
280
c data rk1/0.1d0/, rk2/10.0d0/, rk3/50.0d0/, rk4/2.5d0/, rk5/0.1d0/,
281
c 1 rk6/10.0d0/, rk7/50.0d0/, rk8/2.5d0/, rk9/50.0d0/, rk10/5.0d0/,
282
c 2 rk11/50.0d0/, rk12/50.0d0/, rk13/50.0d0/, rk14/30.0d0/,
283
c 3 rk15/100.0d0/, rk16/2.5d0/, rk17/100.0d0/, rk18/2.5d0/,
284
c 4 rk19/50.0d0/, rk20/50.0d0/
285
c go to (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), j
289
c 2 pdj(2) = -rk3*y(3) - rk15*y(12) - rk2
290
c pdj(3) = rk2 - rk3*y(3)
292
c pdj(5) = rk15*y(12)
293
c pdj(12) = -rk15*y(12)
295
c 3 pdj(2) = -rk3*y(2)
296
c pdj(3) = -rk5 - rk3*y(2) - rk7*y(10)
299
c pdj(10) = rk5 - rk7*y(10)
301
c 4 pdj(2) = rk11*rk14
303
c pdj(4) = -rk11*rk14 - rk4
306
c 5 pdj(2) = rk19*rk14
307
c pdj(5) = -rk19*rk14 - rk16
309
c pdj(12) = rk19*rk14
311
c 6 pdj(3) = rk12*rk14
312
c pdj(6) = -rk12*rk14 - rk8
314
c pdj(10) = rk12*rk14
316
c 7 pdj(7) = -rk20*rk14 - rk18
318
c pdj(10) = rk20*rk14
319
c pdj(12) = rk20*rk14
321
c 8 pdj(8) = -rk13*rk14 - rk10
322
c pdj(10) = rk13*rk14
325
c 10 pdj(3) = -rk7*y(3)
327
c pdj(7) = rk17*y(12)
329
c pdj(10) = -rk7*y(3) - rk17*y(12) - rk6 - rk9
330
c pdj(12) = rk6 - rk17*y(12)
332
c 12 pdj(2) = -rk15*y(2)
334
c pdj(7) = rk17*y(10)
335
c pdj(10) = -rk17*y(10)
336
c pdj(12) = -rk15*y(2) - rk17*y(10)
340
c the output of this program (on a cray-1 in single precision)
344
c at t = 1.000e-01 no. steps = 12 last step = 1.515e-02
345
c y array = 9.90050e-01 6.28228e-03 3.65313e-03 7.51934e-07
346
c 1.12167e-09 1.18458e-09 1.77291e-12 3.26476e-07
347
c 5.46720e-08 9.99500e-06 4.48483e-08 2.76398e-06
350
c at t = 1.000e+00 no. steps = 33 last step = 7.880e-02
351
c y array = 9.04837e-01 9.13105e-03 8.20622e-02 2.49177e-05
352
c 1.85055e-06 1.96797e-06 1.46157e-07 2.39557e-05
353
c 3.26306e-05 7.21621e-04 5.06433e-05 3.05010e-03
356
c at t = 1.000e+01 no. steps = 48 last step = 1.239e+00
357
c y array = 3.67876e-01 3.68958e-03 3.65133e-01 4.48325e-05
358
c 6.10798e-05 4.33148e-05 5.90211e-05 1.18449e-04
359
c 3.15235e-03 3.56531e-03 4.15520e-03 2.48741e-01
362
c at t = 1.000e+02 no. steps = 91 last step = 3.764e+00
363
c y array = 4.44981e-05 4.42666e-07 4.47273e-04 -3.53257e-11
364
c 2.81577e-08 -9.67741e-11 2.77615e-07 1.45322e-07
365
c 1.56230e-02 4.37394e-06 1.60104e-02 9.52246e-01
368
c at t = 1.000e+03 no. steps = 111 last step = 4.156e+02
369
c y array = -2.65492e-13 2.60539e-14 -8.59563e-12 6.29355e-14
370
c -1.78066e-13 5.71471e-13 -1.47561e-12 4.58078e-15
371
c 1.56314e-02 1.37878e-13 1.60184e-02 9.52719e-01
374
c required rwork size = 442 iwork size = 30
375
c no. steps = 111 no. f-s = 142 no. j-s = 2 no. lu-s = 20
376
c no. of nonzeros in j = 44 no. of nonzeros in lu = 50
377
c-----------------------------------------------------------------------
378
c full description of user interface to lsodes.
380
c the user interface to lsodes consists of the following parts.
382
c i. the call sequence to subroutine lsodes, which is a driver
383
c routine for the solver. this includes descriptions of both
384
c the call sequence arguments and of user-supplied routines.
385
c following these descriptions is a description of
386
c optional inputs available through the call sequence, and then
387
c a description of optional outputs (in the work arrays).
389
c ii. descriptions of other routines in the lsodes package that may be
390
c (optionally) called by the user. these provide the ability to
391
c alter error message handling, save and restore the internal
392
c common, and obtain specified derivatives of the solution y(t).
394
c iii. descriptions of common blocks to be declared in overlay
395
c or similar environments, or to be saved when doing an interrupt
396
c of the problem and continued solution later.
398
c iv. description of two routines in the lsodes package, either of
399
c which the user may replace with his own version, if desired.
400
c these relate to the measurement of errors.
402
c-----------------------------------------------------------------------
403
c part i. call sequence.
405
c the call sequence parameters used for input only are
406
c f, neq, tout, itol, rtol, atol, itask, iopt, lrw, liw, jac, mf,
407
c and those used for both input and output are
409
c the work arrays rwork and iwork are also used for conditional and
410
c optional inputs and optional outputs. (the term output here refers
411
c to the return from subroutine lsodes to the user-s calling program.)
413
c the legality of input parameters will be thoroughly checked on the
414
c initial call for the problem, but not checked thereafter unless a
415
c change in input parameters is flagged by istate = 3 on input.
417
c the descriptions of the call arguments are as follows.
419
c f = the name of the user-supplied subroutine defining the
420
c ode system. the system must be put in the first-order
421
c form dy/dt = f(t,y), where f is a vector-valued function
422
c of the scalar t and the vector y. subroutine f is to
423
c compute the function f. it is to have the form
424
c subroutine f (neq, t, y, ydot)
425
c dimension y(1), ydot(1)
426
c where neq, t, and y are input, and the array ydot = f(t,y)
427
c is output. y and ydot are arrays of length neq.
428
c (in the dimension statement above, 1 is a dummy
429
c dimension.. it can be replaced by any value.)
430
c subroutine f should not alter y(1),...,y(neq).
431
c f must be declared external in the calling program.
433
c subroutine f may access user-defined quantities in
434
c neq(2),... and/or in y(neq(1)+1),... if neq is an array
435
c (dimensioned in f) and/or y has length exceeding neq(1).
436
c see the descriptions of neq and y below.
438
c if quantities computed in the f routine are needed
439
c externally to lsodes, an extra call to f should be made
440
c for this purpose, for consistent and accurate results.
441
c if only the derivative dy/dt is needed, use intdy instead.
443
c neq = the size of the ode system (number of first order
444
c ordinary differential equations). used only for input.
445
c neq may be decreased, but not increased, during the problem.
446
c if neq is decreased (with istate = 3 on input), the
447
c remaining components of y should be left undisturbed, if
448
c these are to be accessed in f and/or jac.
450
c normally, neq is a scalar, and it is generally referred to
451
c as a scalar in this user interface description. however,
452
c neq may be an array, with neq(1) set to the system size.
453
c (the lsodes package accesses only neq(1).) in either case,
454
c this parameter is passed as the neq argument in all calls
455
c to f and jac. hence, if it is an array, locations
456
c neq(2),... may be used to store other integer data and pass
457
c it to f and/or jac. subroutines f and/or jac must include
458
c neq in a dimension statement in that case.
460
c y = a real array for the vector of dependent variables, of
461
c length neq or more. used for both input and output on the
462
c first call (istate = 1), and only for output on other calls.
463
c on the first call, y must contain the vector of initial
464
c values. on output, y contains the computed solution vector,
465
c evaluated at t. if desired, the y array may be used
466
c for other purposes between calls to the solver.
468
c this array is passed as the y argument in all calls to
469
c f and jac. hence its length may exceed neq, and locations
470
c y(neq+1),... may be used to store other real data and
471
c pass it to f and/or jac. (the lsodes package accesses only
474
c t = the independent variable. on input, t is used only on the
475
c first call, as the initial point of the integration.
476
c on output, after each call, t is the value at which a
477
c computed solution y is evaluated (usually the same as tout).
478
c on an error return, t is the farthest point reached.
480
c tout = the next value of t at which a computed solution is desired.
481
c used only for input.
483
c when starting the problem (istate = 1), tout may be equal
484
c to t for one call, then should .ne. t for the next call.
485
c for the initial t, an input value of tout .ne. t is used
486
c in order to determine the direction of the integration
487
c (i.e. the algebraic sign of the step sizes) and the rough
488
c scale of the problem. integration in either direction
489
c (forward or backward in t) is permitted.
491
c if itask = 2 or 5 (one-step modes), tout is ignored after
492
c the first call (i.e. the first call with tout .ne. t).
493
c otherwise, tout is required on every call.
495
c if itask = 1, 3, or 4, the values of tout need not be
496
c monotone, but a value of tout which backs up is limited
497
c to the current internal t interval, whose endpoints are
498
c tcur - hu and tcur (see optional outputs, below, for
501
c itol = an indicator for the type of error control. see
502
c description below under atol. used only for input.
504
c rtol = a relative error tolerance parameter, either a scalar or
505
c an array of length neq. see description below under atol.
508
c atol = an absolute error tolerance parameter, either a scalar or
509
c an array of length neq. input only.
511
c the input parameters itol, rtol, and atol determine
512
c the error control performed by the solver. the solver will
513
c control the vector e = (e(i)) of estimated local errors
514
c in y, according to an inequality of the form
515
c rms-norm of ( e(i)/ewt(i) ) .le. 1,
516
c where ewt(i) = rtol(i)*abs(y(i)) + atol(i),
517
c and the rms-norm (root-mean-square norm) here is
518
c rms-norm(v) = sqrt(sum v(i)**2 / neq). here ewt = (ewt(i))
519
c is a vector of weights which must always be positive, and
520
c the values of rtol and atol should all be non-negative.
521
c the following table gives the types (scalar/array) of
522
c rtol and atol, and the corresponding form of ewt(i).
524
c itol rtol atol ewt(i)
525
c 1 scalar scalar rtol*abs(y(i)) + atol
526
c 2 scalar array rtol*abs(y(i)) + atol(i)
527
c 3 array scalar rtol(i)*abs(y(i)) + atol
528
c 4 array array rtol(i)*abs(y(i)) + atol(i)
530
c when either of these parameters is a scalar, it need not
531
c be dimensioned in the user-s calling program.
533
c if none of the above choices (with itol, rtol, and atol
534
c fixed throughout the problem) is suitable, more general
535
c error controls can be obtained by substituting
536
c user-supplied routines for the setting of ewt and/or for
537
c the norm calculation. see part iv below.
539
c if global errors are to be estimated by making a repeated
540
c run on the same problem with smaller tolerances, then all
541
c components of rtol and atol (i.e. of ewt) should be scaled
544
c itask = an index specifying the task to be performed.
545
c input only. itask has the following values and meanings.
546
c 1 means normal computation of output values of y(t) at
547
c t = tout (by overshooting and interpolating).
548
c 2 means take one step only and return.
549
c 3 means stop at the first internal mesh point at or
550
c beyond t = tout and return.
551
c 4 means normal computation of output values of y(t) at
552
c t = tout but without overshooting t = tcrit.
553
c tcrit must be input as rwork(1). tcrit may be equal to
554
c or beyond tout, but not behind it in the direction of
555
c integration. this option is useful if the problem
556
c has a singularity at or beyond t = tcrit.
557
c 5 means take one step, without passing tcrit, and return.
558
c tcrit must be input as rwork(1).
560
c note.. if itask = 4 or 5 and the solver reaches tcrit
561
c (within roundoff), it will return t = tcrit (exactly) to
562
c indicate this (unless itask = 4 and tout comes before tcrit,
563
c in which case answers at t = tout are returned first).
565
c istate = an index used for input and output to specify the
566
c the state of the calculation.
568
c on input, the values of istate are as follows.
569
c 1 means this is the first call for the problem
570
c (initializations will be done). see note below.
571
c 2 means this is not the first call, and the calculation
572
c is to continue normally, with no change in any input
573
c parameters except possibly tout and itask.
574
c (if itol, rtol, and/or atol are changed between calls
575
c with istate = 2, the new values will be used but not
576
c tested for legality.)
577
c 3 means this is not the first call, and the
578
c calculation is to continue normally, but with
579
c a change in input parameters other than
580
c tout and itask. changes are allowed in
581
c neq, itol, rtol, atol, iopt, lrw, liw, mf,
582
c the conditional inputs ia and ja,
583
c and any of the optional inputs except h0.
584
c in particular, if miter = 1 or 2, a call with istate = 3
585
c will cause the sparsity structure of the problem to be
586
c recomputed (or reread from ia and ja if moss = 0).
587
c note.. a preliminary call with tout = t is not counted
588
c as a first call here, as no initialization or checking of
589
c input is done. (such a call is sometimes useful for the
590
c purpose of outputting the initial conditions.)
591
c thus the first call for which tout .ne. t requires
592
c istate = 1 on input.
594
c on output, istate has the following values and meanings.
595
c 1 means nothing was done, as tout was equal to t with
596
c istate = 1 on input. (however, an internal counter was
597
c set to detect and prevent repeated calls of this type.)
598
c 2 means the integration was performed successfully.
599
c -1 means an excessive amount of work (more than mxstep
600
c steps) was done on this call, before completing the
601
c requested task, but the integration was otherwise
602
c successful as far as t. (mxstep is an optional input
603
c and is normally 500.) to continue, the user may
604
c simply reset istate to a value .gt. 1 and call again
605
c (the excess work step counter will be reset to 0).
606
c in addition, the user may increase mxstep to avoid
607
c this error return (see below on optional inputs).
608
c -2 means too much accuracy was requested for the precision
609
c of the machine being used. this was detected before
610
c completing the requested task, but the integration
611
c was successful as far as t. to continue, the tolerance
612
c parameters must be reset, and istate must be set
613
c to 3. the optional output tolsf may be used for this
614
c purpose. (note.. if this condition is detected before
615
c taking any steps, then an illegal input return
616
c (istate = -3) occurs instead.)
617
c -3 means illegal input was detected, before taking any
618
c integration steps. see written message for details.
619
c note.. if the solver detects an infinite loop of calls
620
c to the solver with illegal input, it will cause
622
c -4 means there were repeated error test failures on
623
c one attempted step, before completing the requested
624
c task, but the integration was successful as far as t.
625
c the problem may have a singularity, or the input
626
c may be inappropriate.
627
c -5 means there were repeated convergence test failures on
628
c one attempted step, before completing the requested
629
c task, but the integration was successful as far as t.
630
c this may be caused by an inaccurate jacobian matrix,
631
c if one is being used.
632
c -6 means ewt(i) became zero for some i during the
633
c integration. pure relative error control (atol(i)=0.0)
634
c was requested on a variable which has now vanished.
635
c the integration was successful as far as t.
636
c -7 means a fatal error return flag came from the sparse
637
c solver cdrv by way of prjs or slss (numerical
638
c factorization or backsolve). this should never happen.
639
c the integration was successful as far as t.
641
c note.. an error return with istate = -1, -4, or -5 and with
642
c miter = 1 or 2 may mean that the sparsity structure of the
643
c problem has changed significantly since it was last
644
c determined (or input). in that case, one can attempt to
645
c complete the integration by setting istate = 3 on the next
646
c call, so that a new structure determination is done.
648
c note.. since the normal output value of istate is 2,
649
c it does not need to be reset for normal continuation.
650
c also, since a negative input value of istate will be
651
c regarded as illegal, a negative output value requires the
652
c user to change it, and possibly other inputs, before
653
c calling the solver again.
655
c iopt = an integer flag to specify whether or not any optional
656
c inputs are being used on this call. input only.
657
c the optional inputs are listed separately below.
658
c iopt = 0 means no optional inputs are being used.
659
c default values will be used in all cases.
660
c iopt = 1 means one or more optional inputs are being used.
662
c rwork = a work array used for a mixture of real (double precision)
663
c and integer work space.
664
c the length of rwork (in real words) must be at least
665
c 20 + nyh*(maxord + 1) + 3*neq + lwm where
666
c nyh = the initial value of neq,
667
c maxord = 12 (if meth = 1) or 5 (if meth = 2) (unless a
668
c smaller value is given as an optional input),
669
c lwm = 0 if miter = 0,
670
c lwm = 2*nnz + 2*neq + (nnz+9*neq)/lenrat if miter = 1,
671
c lwm = 2*nnz + 2*neq + (nnz+10*neq)/lenrat if miter = 2,
672
c lwm = neq + 2 if miter = 3.
673
c in the above formulas,
674
c nnz = number of nonzero elements in the jacobian matrix.
675
c lenrat = the real to integer wordlength ratio (usually 1 in
676
c single precision and 2 in double precision).
677
c (see the mf description for meth and miter.)
678
c thus if maxord has its default value and neq is constant,
679
c the minimum length of rwork is..
680
c 20 + 16*neq for mf = 10,
681
c 20 + 16*neq + lwm for mf = 11, 111, 211, 12, 112, 212,
682
c 22 + 17*neq for mf = 13,
683
c 20 + 9*neq for mf = 20,
684
c 20 + 9*neq + lwm for mf = 21, 121, 221, 22, 122, 222,
685
c 22 + 10*neq for mf = 23.
686
c if miter = 1 or 2, the above formula for lwm is only a
687
c crude lower bound. the required length of rwork cannot
688
c be readily predicted in general, as it depends on the
689
c sparsity structure of the problem. some experimentation
692
c the first 20 words of rwork are reserved for conditional
693
c and optional inputs and optional outputs.
695
c the following word in rwork is a conditional input..
696
c rwork(1) = tcrit = critical value of t which the solver
697
c is not to overshoot. required if itask is
698
c 4 or 5, and ignored otherwise. (see itask.)
700
c lrw = the length of the array rwork, as declared by the user.
701
c (this will be checked by the solver.)
703
c iwork = an integer work array. the length of iwork must be at least
704
c 31 + neq + nnz if moss = 0 and miter = 1 or 2, or
706
c (nnz is the number of nonzero elements in df/dy.)
708
c in lsodes, iwork is used only for conditional and
709
c optional inputs and optional outputs.
711
c the following two blocks of words in iwork are conditional
712
c inputs, required if moss = 0 and miter = 1 or 2, but not
713
c otherwise (see the description of mf for moss).
714
c iwork(30+j) = ia(j) (j=1,...,neq+1)
715
c iwork(31+neq+k) = ja(k) (k=1,...,nnz)
716
c the two arrays ia and ja describe the sparsity structure
717
c to be assumed for the jacobian matrix. ja contains the row
718
c indices where nonzero elements occur, reading in columnwise
719
c order, and ia contains the starting locations in ja of the
720
c descriptions of columns 1,...,neq, in that order, with
721
c ia(1) = 1. thus, for each column index j = 1,...,neq, the
722
c values of the row index i in column j where a nonzero
723
c element may occur are given by
724
c i = ja(k), where ia(j) .le. k .lt. ia(j+1).
725
c if nnz is the total number of nonzero locations assumed,
726
c then the length of the ja array is nnz, and ia(neq+1) must
727
c be nnz + 1. duplicate entries are not allowed.
729
c liw = the length of the array iwork, as declared by the user.
730
c (this will be checked by the solver.)
732
c note.. the work arrays must not be altered between calls to lsodes
733
c for the same problem, except possibly for the conditional and
734
c optional inputs, and except for the last 3*neq words of rwork.
735
c the latter space is used for internal scratch space, and so is
736
c available for use by the user outside lsodes between calls, if
737
c desired (but not for use by f or jac).
739
c jac = name of user-supplied routine (miter = 1 or moss = 1) to
740
c compute the jacobian matrix, df/dy, as a function of
741
c the scalar t and the vector y. it is to have the form
742
c subroutine jac (neq, t, y, j, ian, jan, pdj)
743
c dimension y(1), ian(1), jan(1), pdj(1)
744
c where neq, t, y, j, ian, and jan are input, and the array
745
c pdj, of length neq, is to be loaded with column j
746
c of the jacobian on output. thus df(i)/dy(j) is to be
747
c loaded into pdj(i) for all relevant values of i.
748
c here t and y have the same meaning as in subroutine f,
749
c and j is a column index (1 to neq). ian and jan are
750
c undefined in calls to jac for structure determination
751
c (moss = 1). otherwise, ian and jan are structure
752
c descriptors, as defined under optional outputs below, and
753
c so can be used to determine the relevant row indices i, if
754
c desired. (in the dimension statement above, 1 is a
755
c dummy dimension.. it can be replaced by any value.)
756
c jac need not provide df/dy exactly. a crude
757
c approximation (possibly with greater sparsity) will do.
758
c in any case, pdj is preset to zero by the solver,
759
c so that only the nonzero elements need be loaded by jac.
760
c calls to jac are made with j = 1,...,neq, in that order, and
761
c each such set of calls is preceded by a call to f with the
762
c same arguments neq, t, and y. thus to gain some efficiency,
763
c intermediate quantities shared by both calculations may be
764
c saved in a user common block by f and not recomputed by jac,
765
c if desired. jac must not alter its input arguments.
766
c jac must be declared external in the calling program.
767
c subroutine jac may access user-defined quantities in
768
c neq(2),... and y(neq(1)+1),... if neq is an array
769
c (dimensioned in jac) and y has length exceeding neq(1).
770
c see the descriptions of neq and y above.
772
c mf = the method flag. used only for input.
773
c mf has three decimal digits-- moss, meth, miter--
774
c mf = 100*moss + 10*meth + miter.
775
c moss indicates the method to be used to obtain the sparsity
776
c structure of the jacobian matrix if miter = 1 or 2..
777
c moss = 0 means the user has supplied ia and ja
778
c (see descriptions under iwork above).
779
c moss = 1 means the user has supplied jac (see below)
780
c and the structure will be obtained from neq
781
c initial calls to jac.
782
c moss = 2 means the structure will be obtained from neq+1
783
c initial calls to f.
784
c meth indicates the basic linear multistep method..
785
c meth = 1 means the implicit adams method.
786
c meth = 2 means the method based on backward
787
c differentiation formulas (bdf-s).
788
c miter indicates the corrector iteration method..
789
c miter = 0 means functional iteration (no jacobian matrix
791
c miter = 1 means chord iteration with a user-supplied
792
c sparse jacobian, given by subroutine jac.
793
c miter = 2 means chord iteration with an internally
794
c generated (difference quotient) sparse jacobian
795
c (using ngp extra calls to f per df/dy value,
796
c where ngp is an optional output described below.)
797
c miter = 3 means chord iteration with an internally
798
c generated diagonal jacobian approximation.
799
c (using 1 extra call to f per df/dy evaluation).
800
c if miter = 1 or moss = 1, the user must supply a subroutine
801
c jac (the name is arbitrary) as described above under jac.
802
c otherwise, a dummy argument can be used.
804
c the standard choices for mf are..
805
c mf = 10 for a nonstiff problem,
806
c mf = 21 or 22 for a stiff problem with ia/ja supplied
807
c (21 if jac is supplied, 22 if not),
808
c mf = 121 for a stiff problem with jac supplied,
810
c mf = 222 for a stiff problem with neither ia/ja nor
812
c the sparseness structure can be changed during the
813
c problem by making a call to lsodes with istate = 3.
814
c-----------------------------------------------------------------------
817
c the following is a list of the optional inputs provided for in the
818
c call sequence. (see also part ii.) for each such input variable,
819
c this table lists its name as used in this documentation, its
820
c location in the call sequence, its meaning, and the default value.
821
c the use of any of these inputs requires iopt = 1, and in that
822
c case all of these inputs are examined. a value of zero for any
823
c of these optional inputs will cause the default value to be used.
824
c thus to use a subset of the optional inputs, simply preload
825
c locations 5 to 10 in rwork and iwork to 0.0 and 0 respectively, and
826
c then set those of interest to nonzero values.
828
c name location meaning and default value
830
c h0 rwork(5) the step size to be attempted on the first step.
831
c the default value is determined by the solver.
833
c hmax rwork(6) the maximum absolute step size allowed.
834
c the default value is infinite.
836
c hmin rwork(7) the minimum absolute step size allowed.
837
c the default value is 0. (this lower bound is not
838
c enforced on the final step before reaching tcrit
839
c when itask = 4 or 5.)
841
c seth rwork(8) the element threshhold for sparsity determination
842
c when moss = 1 or 2. if the absolute value of
843
c an estimated jacobian element is .le. seth, it
844
c will be assumed to be absent in the structure.
845
c the default value of seth is 0.
847
c maxord iwork(5) the maximum order to be allowed. the default
848
c value is 12 if meth = 1, and 5 if meth = 2.
849
c if maxord exceeds the default value, it will
850
c be reduced to the default value.
851
c if maxord is changed during the problem, it may
852
c cause the current order to be reduced.
854
c mxstep iwork(6) maximum number of (internally defined) steps
855
c allowed during one call to the solver.
856
c the default value is 500.
858
c mxhnil iwork(7) maximum number of messages printed (per problem)
859
c warning that t + h = t on a step (h = step size).
860
c this must be positive to result in a non-default
861
c value. the default value is 10.
862
c-----------------------------------------------------------------------
865
c as optional additional output from lsodes, the variables listed
866
c below are quantities related to the performance of lsodes
867
c which are available to the user. these are communicated by way of
868
c the work arrays, but also have internal mnemonic names as shown.
869
c except where stated otherwise, all of these outputs are defined
870
c on any successful return from lsodes, and on any return with
871
c istate = -1, -2, -4, -5, or -6. on an illegal input return
872
c (istate = -3), they will be unchanged from their existing values
873
c (if any), except possibly for tolsf, lenrw, and leniw.
874
c on any error return, outputs relevant to the error will be defined,
877
c name location meaning
879
c hu rwork(11) the step size in t last used (successfully).
881
c hcur rwork(12) the step size to be attempted on the next step.
883
c tcur rwork(13) the current value of the independent variable
884
c which the solver has actually reached, i.e. the
885
c current internal mesh point in t. on output, tcur
886
c will always be at least as far as the argument
887
c t, but may be farther (if interpolation was done).
889
c tolsf rwork(14) a tolerance scale factor, greater than 1.0,
890
c computed when a request for too much accuracy was
891
c detected (istate = -3 if detected at the start of
892
c the problem, istate = -2 otherwise). if itol is
893
c left unaltered but rtol and atol are uniformly
894
c scaled up by a factor of tolsf for the next call,
895
c then the solver is deemed likely to succeed.
896
c (the user may also ignore tolsf and alter the
897
c tolerance parameters in any other way appropriate.)
899
c nst iwork(11) the number of steps taken for the problem so far.
901
c nfe iwork(12) the number of f evaluations for the problem so far,
902
c excluding those for structure determination
905
c nje iwork(13) the number of jacobian evaluations for the problem
906
c so far, excluding those for structure determination
909
c nqu iwork(14) the method order last used (successfully).
911
c nqcur iwork(15) the order to be attempted on the next step.
913
c imxer iwork(16) the index of the component of largest magnitude in
914
c the weighted local error vector ( e(i)/ewt(i) ),
915
c on an error return with istate = -4 or -5.
917
c lenrw iwork(17) the length of rwork actually required.
918
c this is defined on normal returns and on an illegal
919
c input return for insufficient storage.
921
c leniw iwork(18) the length of iwork actually required.
922
c this is defined on normal returns and on an illegal
923
c input return for insufficient storage.
925
c nnz iwork(19) the number of nonzero elements in the jacobian
926
c matrix, including the diagonal (miter = 1 or 2).
927
c (this may differ from that given by ia(neq+1)-1
928
c if moss = 0, because of added diagonal entries.)
930
c ngp iwork(20) the number of groups of column indices, used in
931
c difference quotient jacobian aproximations if
932
c miter = 2. this is also the number of extra f
933
c evaluations needed for each jacobian evaluation.
935
c nlu iwork(21) the number of sparse lu decompositions for the
938
c lyh iwork(22) the base address in rwork of the history array yh,
939
c described below in this list.
941
c ipian iwork(23) the base address of the structure descriptor array
942
c ian, described below in this list.
944
c ipjan iwork(24) the base address of the structure descriptor array
945
c jan, described below in this list.
947
c nzl iwork(25) the number of nonzero elements in the strict lower
948
c triangle of the lu factorization used in the chord
949
c iteration (miter = 1 or 2).
951
c nzu iwork(26) the number of nonzero elements in the strict upper
952
c triangle of the lu factorization used in the chord
953
c iteration (miter = 1 or 2).
954
c the total number of nonzeros in the factorization
955
c is therefore nzl + nzu + neq.
957
c the following four arrays are segments of the rwork array which
958
c may also be of interest to the user as optional outputs.
959
c for each array, the table below gives its internal name,
960
c its base address, and its description.
961
c for yh and acor, the base addresses are in rwork (a real array).
962
c the integer arrays ian and jan are to be obtained by declaring an
963
c integer array iwk and identifying iwk(1) with rwork(21), using either
964
c an equivalence statement or a subroutine call. then the base
965
c addresses ipian (of ian) and ipjan (of jan) in iwk are to be obtained
966
c as optional outputs iwork(23) and iwork(24), respectively.
967
c thus ian(1) is iwk(ipian), etc.
969
c name base address description
971
c ian ipian (in iwk) structure descriptor array of size neq + 1.
972
c jan ipjan (in iwk) structure descriptor array of size nnz.
973
c (see above) ian and jan together describe the sparsity
974
c structure of the jacobian matrix, as used by
975
c lsodes when miter = 1 or 2.
976
c jan contains the row indices of the nonzero
977
c locations, reading in columnwise order, and
978
c ian contains the starting locations in jan of
979
c the descriptions of columns 1,...,neq, in
980
c that order, with ian(1) = 1. thus for each
981
c j = 1,...,neq, the row indices i of the
982
c nonzero locations in column j are
983
c i = jan(k), ian(j) .le. k .lt. ian(j+1).
984
c note that ian(neq+1) = nnz + 1.
985
c (if moss = 0, ian/jan may differ from the
986
c input ia/ja because of a different ordering
987
c in each column, and added diagonal entries.)
989
c yh lyh the nordsieck history array, of size nyh by
990
c (optional (nqcur + 1), where nyh is the initial value
991
c output) of neq. for j = 0,1,...,nqcur, column j+1
992
c of yh contains hcur**j/factorial(j) times
993
c the j-th derivative of the interpolating
994
c polynomial currently representing the solution,
995
c evaluated at t = tcur. the base address lyh
996
c is another optional output, listed above.
998
c acor lenrw-neq+1 array of size neq used for the accumulated
999
c corrections on each step, scaled on output
1000
c to represent the estimated local error in y
1001
c on the last step. this is the vector e in
1002
c the description of the error control. it is
1003
c defined only on a successful return from
1006
c-----------------------------------------------------------------------
1007
c part ii. other routines callable.
1009
c the following are optional calls which the user may make to
1010
c gain additional capabilities in conjunction with lsodes.
1011
c (the routines xsetun and xsetf are designed to conform to the
1012
c slatec error handling package.)
1014
c form of call function
1015
c call xsetun(lun) set the logical unit number, lun, for
1016
c output of messages from lsodes, if
1017
c the default is not desired.
1018
c the default value of lun is 6.
1020
c call xsetf(mflag) set a flag to control the printing of
1021
c messages by lsodes.
1022
c mflag = 0 means do not print. (danger..
1023
c this risks losing valuable information.)
1024
c mflag = 1 means print (the default).
1026
c either of the above calls may be made at
1027
c any time and will take effect immediately.
1029
c call srcms(rsav,isav,job) saves and restores the contents of
1030
c the internal common blocks used by
1031
c lsodes (see part iii below).
1032
c rsav must be a real array of length 224
1033
c or more, and isav must be an integer
1034
c array of length 75 or more.
1035
c job=1 means save common into rsav/isav.
1036
c job=2 means restore common from rsav/isav.
1037
c srcms is useful if one is
1038
c interrupting a run and restarting
1039
c later, or alternating between two or
1040
c more problems solved with lsodes.
1042
c call intdy(,,,,,) provide derivatives of y, of various
1043
c (see below) orders, at a specified point t, if
1044
c desired. it may be called only after
1045
c a successful return from lsodes.
1047
c the detailed instructions for using intdy are as follows.
1048
c the form of the call is..
1051
c call intdy (t, k, rwork(lyh), nyh, dky, iflag)
1053
c the input parameters are..
1055
c t = value of independent variable where answers are desired
1056
c (normally the same as the t last returned by lsodes).
1057
c for valid results, t must lie between tcur - hu and tcur.
1058
c (see optional outputs for tcur and hu.)
1059
c k = integer order of the derivative desired. k must satisfy
1060
c 0 .le. k .le. nqcur, where nqcur is the current order
1061
c (see optional outputs). the capability corresponding
1062
c to k = 0, i.e. computing y(t), is already provided
1063
c by lsodes directly. since nqcur .ge. 1, the first
1064
c derivative dy/dt is always available with intdy.
1065
c lyh = the base address of the history array yh, obtained
1066
c as an optional output as shown above.
1067
c nyh = column length of yh, equal to the initial value of neq.
1069
c the output parameters are..
1071
c dky = a real array of length neq containing the computed value
1072
c of the k-th derivative of y(t).
1073
c iflag = integer flag, returned as 0 if k and t were legal,
1074
c -1 if k was illegal, and -2 if t was illegal.
1075
c on an error return, a message is also written.
1076
c-----------------------------------------------------------------------
1077
c part iii. common blocks.
1079
c if lsodes is to be used in an overlay situation, the user
1080
c must declare, in the primary overlay, the variables in..
1081
c (1) the call sequence to lsodes,
1082
c (2) the three internal common blocks
1083
c /ls0001/ of length 257 (218 double precision words
1084
c followed by 39 integer words),
1085
c /lss001/ of length 40 ( 6 double precision words
1086
c followed by 34 integer words),
1087
c /eh0001/ of length 2 (integer words).
1089
c if lsodes is used on a system in which the contents of internal
1090
c common blocks are not preserved between calls, the user should
1091
c declare the above three common blocks in his main program to insure
1092
c that their contents are preserved.
1094
c if the solution of a given problem by lsodes is to be interrupted
1095
c and then later continued, such as when restarting an interrupted run
1096
c or alternating between two or more problems, the user should save,
1097
c following the return from the last lsodes call prior to the
1098
c interruption, the contents of the call sequence variables and the
1099
c internal common blocks, and later restore these values before the
1100
c next lsodes call for that problem. to save and restore the common
1101
c blocks, use subroutine srcms (see part ii above).
1103
c-----------------------------------------------------------------------
1104
c part iv. optionally replaceable solver routines.
1106
c below are descriptions of two routines in the lsodes package which
1107
c relate to the measurement of errors. either routine can be
1108
c replaced by a user-supplied version, if desired. however, since such
1109
c a replacement may have a major impact on performance, it should be
1110
c done only when absolutely necessary, and only with great caution.
1111
c (note.. the means by which the package version of a routine is
1112
c superseded by the user-s version may be system-dependent.)
1115
c the following subroutine is called just before each internal
1116
c integration step, and sets the array of error weights, ewt, as
1117
c described under itol/rtol/atol above..
1118
c subroutine ewset (neq, itol, rtol, atol, ycur, ewt)
1119
c where neq, itol, rtol, and atol are as in the lsodes call sequence,
1120
c ycur contains the current dependent variable vector, and
1121
c ewt is the array of weights set by ewset.
1123
c if the user supplies this subroutine, it must return in ewt(i)
1124
c (i = 1,...,neq) a positive quantity suitable for comparing errors
1125
c in y(i) to. the ewt array returned by ewset is passed to the
1126
c vnorm routine (see below), and also used by lsodes in the computation
1127
c of the optional output imxer, the diagonal jacobian approximation,
1128
c and the increments for difference quotient jacobians.
1130
c in the user-supplied version of ewset, it may be desirable to use
1131
c the current values of derivatives of y. derivatives up to order nq
1132
c are available from the history array yh, described above under
1133
c optional outputs. in ewset, yh is identical to the ycur array,
1134
c extended to nq + 1 columns with a column length of nyh and scale
1135
c factors of h**j/factorial(j). on the first call for the problem,
1136
c given by nst = 0, nq is 1 and h is temporarily set to 1.0.
1137
c the quantities nq, nyh, h, and nst can be obtained by including
1138
c in ewset the statements..
1139
c double precision h, rls
1140
c common /ls0001/ rls(218),ils(39)
1145
c thus, for example, the current value of dy/dt can be obtained as
1146
c ycur(nyh+i)/h (i=1,...,neq) (and the division by h is
1147
c unnecessary when nst = 0).
1150
c the following is a real function routine which computes the weighted
1151
c root-mean-square norm of a vector v..
1152
c d = vnorm (n, v, w)
1154
c n = the length of the vector,
1155
c v = real array of length n containing the vector,
1156
c w = real array of length n containing weights,
1157
c d = sqrt( (1/n) * sum(v(i)*w(i))**2 ).
1158
c vnorm is called with n = neq and with w(i) = 1.0/ewt(i), where
1159
c ewt is as set by subroutine ewset.
1161
c if the user supplies this function, it should return a non-negative
1162
c value of vnorm suitable for use in the error control in lsodes.
1163
c none of the arguments should be altered by vnorm.
1164
c for example, a user-supplied vnorm routine might..
1165
c -substitute a max-norm of (v(i)*w(i)) for the rms-norm, or
1166
c -ignore some components of v in the norm, with the effect of
1167
c suppressing the error control on those components of y.
1168
c-----------------------------------------------------------------------
1169
c-----------------------------------------------------------------------
1170
c other routines in the lsodes package.
1172
c in addition to subroutine lsodes, the lsodes package includes the
1173
c following subroutines and function routines..
1174
c iprep acts as an iterface between lsodes and prep, and also does
1175
c adjusting of work space pointers and work arrays.
1176
c prep is called by iprep to compute sparsity and do sparse matrix
1177
c preprocessing if miter = 1 or 2.
1178
c jgroup is called by prep to compute groups of jacobian column
1179
c indices for use when miter = 2.
1180
c adjlr adjusts the length of required sparse matrix work space.
1181
c it is called by prep.
1182
c cntnzu is called by prep and counts the nonzero elements in the
1183
c strict upper triangle of j + j-transpose, where j = df/dy.
1184
c intdy computes an interpolated value of the y vector at t = tout.
1185
c stode is the core integrator, which does one step of the
1186
c integration and the associated error control.
1187
c cfode sets all method coefficients and test constants.
1188
c prjs computes and preprocesses the jacobian matrix j = df/dy
1189
c and the newton iteration matrix p = i - h*l0*j.
1190
c slss manages solution of linear system in chord iteration.
1191
c ewset sets the error weight vector ewt before each step.
1192
c vnorm computes the weighted r.m.s. norm of a vector.
1193
c srcms is a user-callable routine to save and restore
1194
c the contents of the internal common blocks.
1195
c odrv constructs a reordering of the rows and columns of
1196
c a matrix by the minimum degree algorithm. odrv is a
1197
c driver routine which calls subroutines md, mdi, mdm,
1198
c mdp, mdu, and sro. see ref. 2 for details. (the odrv
1199
c module has been modified since ref. 2, however.)
1200
c cdrv performs reordering, symbolic factorization, numerical
1201
c factorization, or linear system solution operations,
1202
c depending on a path argument ipath. cdrv is a
1203
c driver routine which calls subroutines nroc, nsfc,
1204
c nnfc, nnsc, and nntc. see ref. 3 for details.
1205
c lsodes uses cdrv to solve linear systems in which the
1206
c coefficient matrix is p = i - con*j, where i is the
1207
c identity, con is a scalar, and j is an approximation to
1208
c the jacobian df/dy. because cdrv deals with rowwise
1209
c sparsity descriptions, cdrv works with p-transpose, not p.
1210
c d1mach computes the unit roundoff in a machine-independent manner.
1211
c xerrwv, xsetun, and xsetf handle the printing of all error
1212
c messages and warnings. xerrwv is machine-dependent.
1213
c note.. vnorm and d1mach are function routines.
1214
c all the others are subroutines.
1216
c the intrinsic and external routines used by lsodes are..
1217
c dabs, dmax1, dmin1, dfloat, max0, min0, mod, dsign, dsqrt, and write.
1219
c a block data subprogram is also included with the package,
1220
c for loading some of the variables in internal common.
1222
c-----------------------------------------------------------------------
1223
c the following card is for optimized compilation on lll compilers.
1225
c-----------------------------------------------------------------------
1227
integer illin, init, lyh, lewt, lacor, lsavf, lwm, liwm,
1228
1 mxstep, mxhnil, nhnil, ntrep, nslast, nyh, iowns
1229
integer icf, ierpj, iersl, jcur, jstart, kflag, l, meth, miter,
1230
1 maxord, maxcor, msbp, mxncf, n, nq, nst, nfe, nje, nqu
1231
integer iplost, iesp, istatc, iys, iba, ibian, ibjan, ibjgp,
1232
1 ipian, ipjan, ipjgp, ipigp, ipr, ipc, ipic, ipisp, iprsp, ipa,
1233
2 lenyh, lenyhm, lenwk, lreq, lrat, lrest, lwmin, moss, msbj,
1234
3 nslj, ngp, nlu, nnz, nsp, nzl, nzu
1235
integer i, i1, i2, iflag, imax, imul, imxer, ipflag, ipgo, irem,
1236
1 j, kgo, lenrat, lenyht, leniw, lenrw, lf0, lia, lja,
1237
2 lrtem, lwtem, lyhd, lyhn, mf1, mord, mxhnl0, mxstp0, ncolm
1238
double precision rowns,
1239
1 ccmax, el0, h, hmin, hmxi, hu, rc, tn, uround
1240
double precision con0, conmin, ccmxj, psmall, rbig, seth
1241
double precision atoli, ayi, big, ewti, h0, hmax, hmx, rh, rtoli,
1242
1 tcrit, tdist, tnext, tol, tolsf, tp, size, sum, w0,
1246
c-----------------------------------------------------------------------
1247
c the following two internal common blocks contain
1248
c (a) variables which are local to any subroutine but whose values must
1249
c be preserved between calls to the routine (own variables), and
1250
c (b) variables which are communicated between subroutines.
1251
c the structure of each block is as follows.. all real variables are
1252
c listed first, followed by all integers. within each type, the
1253
c variables are grouped with those local to subroutine lsodes first,
1254
c then those local to subroutine stode or subroutine prjs
1255
c (no other routines have own variables), and finally those used
1256
c for communication. the block ls0001 is declared in subroutines
1257
c lsodes, iprep, prep, intdy, stode, prjs, and slss. the block lss001
1258
c is declared in subroutines lsodes, iprep, prep, prjs, and slss.
1259
c groups of variables are replaced by dummy arrays in the common
1260
c declarations in routines where those variables are not used.
1261
c-----------------------------------------------------------------------
1262
common /ls0001/ rowns(209),
1263
1 ccmax, el0, h, hmin, hmxi, hu, rc, tn, uround,
1264
2 illin, init, lyh, lewt, lacor, lsavf, lwm, liwm,
1265
3 mxstep, mxhnil, nhnil, ntrep, nslast, nyh, iowns(6),
1266
4 icf, ierpj, iersl, jcur, jstart, kflag, l, meth, miter,
1267
5 maxord, maxcor, msbp, mxncf, n, nq, nst, nfe, nje, nqu
1269
common /lss001/ con0, conmin, ccmxj, psmall, rbig, seth,
1270
1 iplost, iesp, istatc, iys, iba, ibian, ibjan, ibjgp,
1271
2 ipian, ipjan, ipjgp, ipigp, ipr, ipc, ipic, ipisp, iprsp, ipa,
1272
3 lenyh, lenyhm, lenwk, lreq, lrat, lrest, lwmin, moss, msbj,
1273
4 nslj, ngp, nlu, nnz, nsp, nzl, nzu
1275
data mord(1),mord(2)/12,5/, mxstp0/500/, mxhnl0/10/
1276
c-----------------------------------------------------------------------
1277
c in the data statement below, set lenrat equal to the ratio of
1278
c the wordlength for a real number to that for an integer. usually,
1279
c lenrat = 1 for single precision and 2 for double precision. if the
1280
c true ratio is not an integer, use the next smaller integer (.ge. 1).
1281
c-----------------------------------------------------------------------
1283
c-----------------------------------------------------------------------
1285
c this code block is executed on every call.
1286
c it tests istate and itask for legality and branches appropriately.
1287
c if istate .gt. 1 but the flag init shows that initialization has
1288
c not yet been done, an error return occurs.
1289
c if istate = 1 and tout = t, jump to block g and return immediately.
1290
c-----------------------------------------------------------------------
1291
if (istate .lt. 1 .or. istate .gt. 3) go to 601
1292
if (itask .lt. 1 .or. itask .gt. 5) go to 602
1293
if (istate .eq. 1) go to 10
1294
if (init .eq. 0) go to 603
1295
if (istate .eq. 2) go to 200
1298
if (tout .eq. t) go to 430
1300
c-----------------------------------------------------------------------
1302
c the next code block is executed for the initial call (istate = 1),
1303
c or for a continuation call with parameter changes (istate = 3).
1304
c it contains checking of all inputs and various initializations.
1305
c if istate = 1, the final setting of work space pointers, the matrix
1306
c preprocessing, and other initializations are done in block c.
1308
c first check legality of the non-optional inputs neq, itol, iopt,
1310
c-----------------------------------------------------------------------
1311
if (neq(1) .le. 0) go to 604
1312
if (istate .eq. 1) go to 25
1313
if (neq(1) .gt. n) go to 605
1315
if (itol .lt. 1 .or. itol .gt. 4) go to 606
1316
if (iopt .lt. 0 .or. iopt .gt. 1) go to 607
1320
miter = mf1 - 10*meth
1321
if (moss .lt. 0 .or. moss .gt. 2) go to 608
1322
if (meth .lt. 1 .or. meth .gt. 2) go to 608
1323
if (miter .lt. 0 .or. miter .gt. 3) go to 608
1324
if (miter .eq. 0 .or. miter .eq. 3) moss = 0
1325
c next process and check the optional inputs. --------------------------
1326
if (iopt .eq. 1) go to 40
1330
if (istate .eq. 1) h0 = 0.0d0
1335
40 maxord = iwork(5)
1336
if (maxord .lt. 0) go to 611
1337
if (maxord .eq. 0) maxord = 100
1338
maxord = min0(maxord,mord(meth))
1340
if (mxstep .lt. 0) go to 612
1341
if (mxstep .eq. 0) mxstep = mxstp0
1343
if (mxhnil .lt. 0) go to 613
1344
if (mxhnil .eq. 0) mxhnil = mxhnl0
1345
if (istate .ne. 1) go to 50
1347
if ((tout - t)*h0 .lt. 0.0d0) go to 614
1349
if (hmax .lt. 0.0d0) go to 615
1351
if (hmax .gt. 0.0d0) hmxi = 1.0d0/hmax
1353
if (hmin .lt. 0.0d0) go to 616
1355
if (seth .lt. 0.0d0) go to 609
1356
c check rtol and atol for legality. ------------------------------------
1360
if (itol .ge. 3) rtoli = rtol(i)
1361
if (itol .eq. 2 .or. itol .eq. 4) atoli = atol(i)
1362
if (rtoli .lt. 0.0d0) go to 619
1363
if (atoli .lt. 0.0d0) go to 620
1365
c-----------------------------------------------------------------------
1366
c compute required work array lengths, as far as possible, and test
1367
c these against lrw and liw. then set tentative pointers for work
1368
c arrays. pointers to rwork/iwork segments are named by prefixing l to
1369
c the name of the segment. e.g., the segment yh starts at rwork(lyh).
1370
c segments of rwork (in order) are denoted wm, yh, savf, ewt, acor.
1371
c if miter = 1 or 2, the required length of the matrix work space wm
1372
c is not yet known, and so a crude minimum value is used for the
1373
c initial tests of lrw and liw, and yh is temporarily stored as far
1374
c to the right in rwork as possible, to leave the maximum amount
1375
c of space for wm for matrix preprocessing. thus if miter = 1 or 2
1376
c and moss .ne. 2, some of the segments of rwork are temporarily
1377
c omitted, as they are not needed in the preprocessing. these
1378
c omitted segments are.. acor if istate = 1, ewt and acor if istate = 3
1379
c and moss = 1, and savf, ewt, and acor if istate = 3 and moss = 0.
1380
c-----------------------------------------------------------------------
1382
if (istate .eq. 1) nyh = n
1384
if (miter .eq. 1) lwmin = 4*n + 10*n/lrat
1385
if (miter .eq. 2) lwmin = 4*n + 11*n/lrat
1386
if (miter .eq. 3) lwmin = n + 2
1387
lenyh = (maxord+1)*nyh
1389
lenrw = 20 + lwmin + lrest
1392
if (moss .eq. 0 .and. miter .ne. 0 .and. miter .ne. 3)
1393
1 leniw = leniw + n + 1
1395
if (lenrw .gt. lrw) go to 617
1396
if (leniw .gt. liw) go to 618
1398
if (moss .eq. 0 .and. miter .ne. 0 .and. miter .ne. 3)
1399
1 leniw = leniw + iwork(lia+n) - 1
1401
if (leniw .gt. liw) go to 618
1406
if (istate .eq. 1) nq = 1
1407
ncolm = min0(nq+1,maxord+2)
1410
if (miter .eq. 1 .or. miter .eq. 2) lenyht = lenyhm
1412
if (istate .eq. 3) imul = moss
1413
if (moss .eq. 2) imul = 3
1414
lrtem = lenyht + imul*n
1416
if (miter .eq. 1 .or. miter .eq. 2) lwtem = lrw - 20 - lrtem
1419
lsavf = lyhn + lenyht
1423
if (istate .eq. 1) go to 100
1424
c-----------------------------------------------------------------------
1425
c istate = 3. move yh to its new location.
1426
c note that only the part of yh needed for the next step, namely
1427
c min(nq+1,maxord+2) columns, is actually moved.
1428
c a temporary error weight array ewt is loaded if moss = 2.
1429
c sparse matrix processing is done in iprep/prep if miter = 1 or 2.
1430
c if maxord was reduced below nq, then the pointers are finally set
1431
c so that savf is identical to yh(*,maxord+2).
1432
c-----------------------------------------------------------------------
1434
imax = lyhn - 1 + lenyhm
1435
c move yh. branch for move right, no move, or move left. --------------
1436
if (lyhd.lt.0) go to 70
1437
if (lyhd.eq.0) go to 80
1439
70 do 72 i = lyhn,imax
1441
72 rwork(j) = rwork(j+lyhd)
1443
74 do 76 i = lyhn,imax
1444
76 rwork(i) = rwork(i+lyhd)
1447
if (miter .eq. 0 .or. miter .eq. 3) go to 92
1448
if (moss .ne. 2) go to 85
1449
c temporarily load ewt if miter = 1 or 2 and moss = 2. -----------------
1450
call ewset (n, itol, rtol, atol, rwork(lyh), rwork(lewt))
1452
if (rwork(i+lewt-1) .le. 0.0d0) go to 621
1453
82 rwork(i+lewt-1) = 1.0d0/rwork(i+lewt-1)
1455
c iprep and prep do sparse matrix preprocessing if miter = 1 or 2. -----
1456
lsavf = min0(lsavf,lrw)
1457
lewt = min0(lewt,lrw)
1458
lacor = min0(lacor,lrw)
1459
call iprep (neq, y, rwork, iwork(lia), iwork(lja), ipflag, f, jac)
1460
lenrw = lwm - 1 + lenwk + lrest
1462
if (ipflag .ne. -1) iwork(23) = ipian
1463
if (ipflag .ne. -1) iwork(24) = ipjan
1465
go to (90, 628, 629, 630, 631, 632, 633), ipgo
1467
if (lenrw .gt. lrw) go to 617
1468
c set flag to signal parameter changes to stode. -----------------------
1470
if (n .eq. nyh) go to 200
1471
c neq was reduced. zero part of yh to avoid undefined references. -----
1473
i2 = lyh + (maxord + 1)*nyh - 1
1474
if (i1 .gt. i2) go to 200
1478
c-----------------------------------------------------------------------
1480
c the next block is for the initial call only (istate = 1).
1481
c it contains all remaining initializations, the initial call to f,
1482
c the sparse matrix preprocessing (miter = 1 or 2), and the
1483
c calculation of the initial step size.
1484
c the error weights in ewt are inverted after being loaded.
1485
c-----------------------------------------------------------------------
1496
c load the initial value vector in yh. ---------------------------------
1498
105 rwork(i+lyh-1) = y(i)
1499
c initial call to f. (lf0 points to yh(*,2).) -------------------------
1501
call f (neq, t, y, rwork(lf0))
1503
c load and invert the ewt array. (h is temporarily set to 1.0.) -------
1504
call ewset (n, itol, rtol, atol, rwork(lyh), rwork(lewt))
1506
if (rwork(i+lewt-1) .le. 0.0d0) go to 621
1507
110 rwork(i+lewt-1) = 1.0d0/rwork(i+lewt-1)
1508
if (miter .eq. 0 .or. miter .eq. 3) go to 120
1509
c iprep and prep do sparse matrix preprocessing if miter = 1 or 2. -----
1510
lacor = min0(lacor,lrw)
1511
call iprep (neq, y, rwork, iwork(lia), iwork(lja), ipflag, f, jac)
1512
lenrw = lwm - 1 + lenwk + lrest
1514
if (ipflag .ne. -1) iwork(23) = ipian
1515
if (ipflag .ne. -1) iwork(24) = ipjan
1517
go to (115, 628, 629, 630, 631, 632, 633), ipgo
1519
if (lenrw .gt. lrw) go to 617
1520
c check tcrit for legality (itask = 4 or 5). ---------------------------
1522
if (itask .ne. 4 .and. itask .ne. 5) go to 125
1524
if ((tcrit - tout)*(tout - t) .lt. 0.0d0) go to 625
1525
if (h0 .ne. 0.0d0 .and. (t + h0 - tcrit)*h0 .gt. 0.0d0)
1527
c initialize all remaining parameters. ---------------------------------
1528
125 uround = d1mach(4)
1530
if (miter .ne. 0) rwork(lwm) = dsqrt(uround)
1534
psmall = 1000.0d0*uround
1535
rbig = 0.01d0/psmall
1546
c-----------------------------------------------------------------------
1547
c the coding below computes the step size, h0, to be attempted on the
1548
c first step, unless the user has supplied a value for this.
1549
c first check that tout - t differs significantly from zero.
1550
c a scalar tolerance quantity tol is computed, as max(rtol(i))
1551
c if this is positive, or max(atol(i)/abs(y(i))) otherwise, adjusted
1552
c so as to be between 100*uround and 1.0e-3.
1553
c then the computed value h0 is given by..
1555
c h0**2 = tol / ( w0**-2 + (1/neq) * sum ( f(i)/ywt(i) )**2 )
1557
c where w0 = max ( abs(t), abs(tout) ),
1558
c f(i) = i-th component of initial value of f,
1559
c ywt(i) = ewt(i)/tol (a weight for y(i)).
1560
c the sign of h0 is inferred from the initial values of tout and t.
1561
c-----------------------------------------------------------------------
1563
if (h0 .ne. 0.0d0) go to 180
1564
tdist = dabs(tout - t)
1565
w0 = dmax1(dabs(t),dabs(tout))
1566
if (tdist .lt. 2.0d0*uround*w0) go to 622
1568
if (itol .le. 2) go to 140
1570
130 tol = dmax1(tol,rtol(i))
1571
140 if (tol .gt. 0.0d0) go to 160
1574
if (itol .eq. 2 .or. itol .eq. 4) atoli = atol(i)
1576
if (ayi .ne. 0.0d0) tol = dmax1(tol,atoli/ayi)
1578
160 tol = dmax1(tol,100.0d0*uround)
1579
tol = dmin1(tol,0.001d0)
1580
sum = vnorm (n, rwork(lf0), rwork(lewt))
1581
sum = 1.0d0/(tol*w0*w0) + tol*sum**2
1582
h0 = 1.0d0/dsqrt(sum)
1583
h0 = dmin1(h0,tdist)
1584
h0 = dsign(h0,tout-t)
1585
c adjust h0 if necessary to meet hmax bound. ---------------------------
1586
180 rh = dabs(h0)*hmxi
1587
if (rh .gt. 1.0d0) h0 = h0/rh
1588
c load h with h0 and scale yh(*,2) by h0. ------------------------------
1591
190 rwork(i+lf0-1) = h0*rwork(i+lf0-1)
1593
c-----------------------------------------------------------------------
1595
c the next code block is for continuation calls only (istate = 2 or 3)
1596
c and is to check stop conditions before taking a step.
1597
c-----------------------------------------------------------------------
1599
go to (210, 250, 220, 230, 240), itask
1600
210 if ((tn - tout)*h .lt. 0.0d0) go to 250
1601
call intdy (tout, 0, rwork(lyh), nyh, y, iflag)
1602
if (iflag .ne. 0) go to 627
1605
220 tp = tn - hu*(1.0d0 + 100.0d0*uround)
1606
if ((tp - tout)*h .gt. 0.0d0) go to 623
1607
if ((tn - tout)*h .lt. 0.0d0) go to 250
1609
230 tcrit = rwork(1)
1610
if ((tn - tcrit)*h .gt. 0.0d0) go to 624
1611
if ((tcrit - tout)*h .lt. 0.0d0) go to 625
1612
if ((tn - tout)*h .lt. 0.0d0) go to 245
1613
call intdy (tout, 0, rwork(lyh), nyh, y, iflag)
1614
if (iflag .ne. 0) go to 627
1617
240 tcrit = rwork(1)
1618
if ((tn - tcrit)*h .gt. 0.0d0) go to 624
1619
245 hmx = dabs(tn) + dabs(h)
1620
ihit = dabs(tn - tcrit) .le. 100.0d0*uround*hmx
1622
tnext = tn + h*(1.0d0 + 4.0d0*uround)
1623
if ((tnext - tcrit)*h .le. 0.0d0) go to 250
1624
h = (tcrit - tn)*(1.0d0 - 4.0d0*uround)
1625
if (istate .eq. 2) jstart = -2
1626
c-----------------------------------------------------------------------
1628
c the next block is normally executed for all calls and contains
1629
c the call to the one-step core integrator stode.
1631
c this is a looping point for the integration steps.
1633
c first check for too many steps being taken, update ewt (if not at
1634
c start of problem), check for too much accuracy being requested, and
1635
c check for h below the roundoff level in t.
1636
c-----------------------------------------------------------------------
1638
if ((nst-nslast) .ge. mxstep) go to 500
1639
call ewset (n, itol, rtol, atol, rwork(lyh), rwork(lewt))
1641
if (rwork(i+lewt-1) .le. 0.0d0) go to 510
1642
260 rwork(i+lewt-1) = 1.0d0/rwork(i+lewt-1)
1643
270 tolsf = uround*vnorm (n, rwork(lyh), rwork(lewt))
1644
if (tolsf .le. 1.0d0) go to 280
1646
if (nst .eq. 0) go to 626
1648
280 if ((tn + h) .ne. tn) go to 290
1650
if (nhnil .gt. mxhnil) go to 290
1651
call xerrwv('lsodes-- warning..internal t (=r1) and h (=r2) are',
1652
1 50, 101, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1654
1 ' such that in the machine, t + h = t on the next step ',
1655
1 60, 101, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1656
call xerrwv(' (h = step size). solver will continue anyway',
1657
1 50, 101, 0, 0, 0, 0, 2, tn, h)
1658
if (nhnil .lt. mxhnil) go to 290
1659
call xerrwv('lsodes-- above warning has been issued i1 times. ',
1660
1 50, 102, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1661
call xerrwv(' it will not be issued again for this problem',
1662
1 50, 102, 0, 1, mxhnil, 0, 0, 0.0d0, 0.0d0)
1664
c-----------------------------------------------------------------------
1665
c call stode(neq,y,yh,nyh,yh,ewt,savf,acor,wm,wm,f,jac,prjs,slss)
1666
c-----------------------------------------------------------------------
1667
call stode (neq, y, rwork(lyh), nyh, rwork(lyh), rwork(lewt),
1668
1 rwork(lsavf), rwork(lacor), rwork(lwm), rwork(lwm),
1669
2 f, jac, prjs, slss)
1671
go to (300, 530, 540, 550), kgo
1672
c-----------------------------------------------------------------------
1674
c the following block handles the case of a successful return from the
1675
c core integrator (kflag = 0). test for stop conditions.
1676
c-----------------------------------------------------------------------
1678
go to (310, 400, 330, 340, 350), itask
1679
c itask = 1. if tout has been reached, interpolate. -------------------
1680
310 if ((tn - tout)*h .lt. 0.0d0) go to 250
1681
call intdy (tout, 0, rwork(lyh), nyh, y, iflag)
1684
c itask = 3. jump to exit if tout was reached. ------------------------
1685
330 if ((tn - tout)*h .ge. 0.0d0) go to 400
1687
c itask = 4. see if tout or tcrit was reached. adjust h if necessary.
1688
340 if ((tn - tout)*h .lt. 0.0d0) go to 345
1689
call intdy (tout, 0, rwork(lyh), nyh, y, iflag)
1692
345 hmx = dabs(tn) + dabs(h)
1693
ihit = dabs(tn - tcrit) .le. 100.0d0*uround*hmx
1695
tnext = tn + h*(1.0d0 + 4.0d0*uround)
1696
if ((tnext - tcrit)*h .le. 0.0d0) go to 250
1697
h = (tcrit - tn)*(1.0d0 - 4.0d0*uround)
1700
c itask = 5. see if tcrit was reached and jump to exit. ---------------
1701
350 hmx = dabs(tn) + dabs(h)
1702
ihit = dabs(tn - tcrit) .le. 100.0d0*uround*hmx
1703
c-----------------------------------------------------------------------
1705
c the following block handles all successful returns from lsodes.
1706
c if itask .ne. 1, y is loaded from yh and t is set accordingly.
1707
c istate is set to 2, the illegal input counter is zeroed, and the
1708
c optional outputs are loaded into the work arrays before returning.
1709
c if istate = 1 and tout = t, there is a return with no action taken,
1710
c except that if this has happened repeatedly, the run is terminated.
1711
c-----------------------------------------------------------------------
1713
410 y(i) = rwork(i+lyh-1)
1715
if (itask .ne. 4 .and. itask .ne. 5) go to 420
1734
430 ntrep = ntrep + 1
1735
if (ntrep .lt. 5) return
1737
1 'lsodes-- repeated calls with istate = 1 and tout = t (=r1) ',
1738
1 60, 301, 0, 0, 0, 0, 1, t, 0.0d0)
1740
c-----------------------------------------------------------------------
1742
c the following block handles all unsuccessful returns other than
1743
c those for illegal input. first the error message routine is called.
1744
c if there was an error test or convergence test failure, imxer is set.
1745
c then y is loaded from yh, t is set to tn, and the illegal input
1746
c counter illin is set to 0. the optional outputs are loaded into
1747
c the work arrays before returning.
1748
c-----------------------------------------------------------------------
1749
c the maximum number of steps was taken before reaching tout. ----------
1750
500 call xerrwv('lsodes-- at current t (=r1), mxstep (=i1) steps ',
1751
1 50, 201, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1752
call xerrwv(' taken on this call before reaching tout ',
1753
1 50, 201, 0, 1, mxstep, 0, 1, tn, 0.0d0)
1756
c ewt(i) .le. 0.0 for some i (not at start of problem). ----------------
1757
510 ewti = rwork(lewt+i-1)
1758
call xerrwv('lsodes-- at t (=r1), ewt(i1) has become r2 .le. 0.',
1759
1 50, 202, 0, 1, i, 0, 2, tn, ewti)
1762
c too much accuracy requested for machine precision. -------------------
1763
520 call xerrwv('lsodes-- at t (=r1), too much accuracy requested ',
1764
1 50, 203, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1765
call xerrwv(' for precision of machine.. see tolsf (=r2) ',
1766
1 50, 203, 0, 0, 0, 0, 2, tn, tolsf)
1770
c kflag = -1. error test failed repeatedly or with abs(h) = hmin. -----
1771
530 call xerrwv('lsodes-- at t(=r1) and step size h(=r2), the error',
1772
1 50, 204, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1773
call xerrwv(' test failed repeatedly or with abs(h) = hmin',
1774
1 50, 204, 0, 0, 0, 0, 2, tn, h)
1777
c kflag = -2. convergence failed repeatedly or with abs(h) = hmin. ----
1778
540 call xerrwv('lsodes-- at t (=r1) and step size h (=r2), the ',
1779
1 50, 205, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1780
call xerrwv(' corrector convergence failed repeatedly ',
1781
1 50, 205, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1782
call xerrwv(' or with abs(h) = hmin ',
1783
1 30, 205, 0, 0, 0, 0, 2, tn, h)
1786
c kflag = -3. fatal error flag returned by prjs or slss (cdrv). -------
1787
550 call xerrwv('lsodes-- at t (=r1) and step size h (=r2), a fatal',
1788
1 50, 207, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1789
call xerrwv(' error flag was returned by cdrv (by way of ',
1790
1 50, 207, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1791
call xerrwv(' subroutine prjs or slss)',
1792
1 30, 207, 0, 0, 0, 0, 2, tn, h)
1795
c compute imxer if relevant. -------------------------------------------
1799
size = dabs(rwork(i+lacor-1)*rwork(i+lewt-1))
1800
if (big .ge. size) go to 570
1805
c set y vector, t, illin, and optional outputs. ------------------------
1807
590 y(i) = rwork(i+lyh-1)
1824
c-----------------------------------------------------------------------
1826
c the following block handles all error returns due to illegal input
1827
c (istate = -3), as detected before calling the core integrator.
1828
c first the error message routine is called. then if there have been
1829
c 5 consecutive such returns just before this call to the solver,
1830
c the run is halted.
1831
c-----------------------------------------------------------------------
1832
601 call xerrwv('lsodes-- istate (=i1) illegal ',
1833
1 30, 1, 0, 1, istate, 0, 0, 0.0d0, 0.0d0)
1835
602 call xerrwv('lsodes-- itask (=i1) illegal ',
1836
1 30, 2, 0, 1, itask, 0, 0, 0.0d0, 0.0d0)
1838
603 call xerrwv('lsodes-- istate .gt. 1 but lsodes not initialized ',
1839
1 50, 3, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1841
604 call xerrwv('lsodes-- neq (=i1) .lt. 1 ',
1842
1 30, 4, 0, 1, neq(1), 0, 0, 0.0d0, 0.0d0)
1844
605 call xerrwv('lsodes-- istate = 3 and neq increased (i1 to i2) ',
1845
1 50, 5, 0, 2, n, neq(1), 0, 0.0d0, 0.0d0)
1847
606 call xerrwv('lsodes-- itol (=i1) illegal ',
1848
1 30, 6, 0, 1, itol, 0, 0, 0.0d0, 0.0d0)
1850
607 call xerrwv('lsodes-- iopt (=i1) illegal ',
1851
1 30, 7, 0, 1, iopt, 0, 0, 0.0d0, 0.0d0)
1853
608 call xerrwv('lsodes-- mf (=i1) illegal ',
1854
1 30, 8, 0, 1, mf, 0, 0, 0.0d0, 0.0d0)
1856
609 call xerrwv('lsodes-- seth (=r1) .lt. 0.0 ',
1857
1 30, 9, 0, 0, 0, 0, 1, seth, 0.0d0)
1859
611 call xerrwv('lsodes-- maxord (=i1) .lt. 0 ',
1860
1 30, 11, 0, 1, maxord, 0, 0, 0.0d0, 0.0d0)
1862
612 call xerrwv('lsodes-- mxstep (=i1) .lt. 0 ',
1863
1 30, 12, 0, 1, mxstep, 0, 0, 0.0d0, 0.0d0)
1865
613 call xerrwv('lsodes-- mxhnil (=i1) .lt. 0 ',
1866
1 30, 13, 0, 1, mxhnil, 0, 0, 0.0d0, 0.0d0)
1868
614 call xerrwv('lsodes-- tout (=r1) behind t (=r2) ',
1869
1 40, 14, 0, 0, 0, 0, 2, tout, t)
1870
call xerrwv(' integration direction is given by h0 (=r1) ',
1871
1 50, 14, 0, 0, 0, 0, 1, h0, 0.0d0)
1873
615 call xerrwv('lsodes-- hmax (=r1) .lt. 0.0 ',
1874
1 30, 15, 0, 0, 0, 0, 1, hmax, 0.0d0)
1876
616 call xerrwv('lsodes-- hmin (=r1) .lt. 0.0 ',
1877
1 30, 16, 0, 0, 0, 0, 1, hmin, 0.0d0)
1879
617 call xerrwv('lsodes-- rwork length is insufficient to proceed. ',
1880
1 50, 17, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1882
1 ' length needed is .ge. lenrw (=i1), exceeds lrw (=i2)',
1883
1 60, 17, 0, 2, lenrw, lrw, 0, 0.0d0, 0.0d0)
1885
618 call xerrwv('lsodes-- iwork length is insufficient to proceed. ',
1886
1 50, 18, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1888
1 ' length needed is .ge. leniw (=i1), exceeds liw (=i2)',
1889
1 60, 18, 0, 2, leniw, liw, 0, 0.0d0, 0.0d0)
1891
619 call xerrwv('lsodes-- rtol(i1) is r1 .lt. 0.0 ',
1892
1 40, 19, 0, 1, i, 0, 1, rtoli, 0.0d0)
1894
620 call xerrwv('lsodes-- atol(i1) is r1 .lt. 0.0 ',
1895
1 40, 20, 0, 1, i, 0, 1, atoli, 0.0d0)
1897
621 ewti = rwork(lewt+i-1)
1898
call xerrwv('lsodes-- ewt(i1) is r1 .le. 0.0 ',
1899
1 40, 21, 0, 1, i, 0, 1, ewti, 0.0d0)
1902
1 'lsodes-- tout (=r1) too close to t(=r2) to start integration',
1903
1 60, 22, 0, 0, 0, 0, 2, tout, t)
1906
1 'lsodes-- itask = i1 and tout (=r1) behind tcur - hu (= r2) ',
1907
1 60, 23, 0, 1, itask, 0, 2, tout, tp)
1910
1 'lsodes-- itask = 4 or 5 and tcrit (=r1) behind tcur (=r2) ',
1911
1 60, 24, 0, 0, 0, 0, 2, tcrit, tn)
1914
1 'lsodes-- itask = 4 or 5 and tcrit (=r1) behind tout (=r2) ',
1915
1 60, 25, 0, 0, 0, 0, 2, tcrit, tout)
1917
626 call xerrwv('lsodes-- at start of problem, too much accuracy ',
1918
1 50, 26, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1920
1 ' requested for precision of machine.. see tolsf (=r1) ',
1921
1 60, 26, 0, 0, 0, 0, 1, tolsf, 0.0d0)
1924
627 call xerrwv('lsodes-- trouble from intdy. itask = i1, tout = r1',
1925
1 50, 27, 0, 1, itask, 0, 1, tout, 0.0d0)
1928
1 'lsodes-- rwork length insufficient (for subroutine prep). ',
1929
1 60, 28, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1931
1 ' length needed is .ge. lenrw (=i1), exceeds lrw (=i2)',
1932
1 60, 28, 0, 2, lenrw, lrw, 0, 0.0d0, 0.0d0)
1935
1 'lsodes-- rwork length insufficient (for subroutine jgroup). ',
1936
1 60, 29, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1938
1 ' length needed is .ge. lenrw (=i1), exceeds lrw (=i2)',
1939
1 60, 29, 0, 2, lenrw, lrw, 0, 0.0d0, 0.0d0)
1942
1 'lsodes-- rwork length insufficient (for subroutine odrv). ',
1943
1 60, 30, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1945
1 ' length needed is .ge. lenrw (=i1), exceeds lrw (=i2)',
1946
1 60, 30, 0, 2, lenrw, lrw, 0, 0.0d0, 0.0d0)
1949
1 'lsodes-- error from odrv in yale sparse matrix package ',
1950
1 60, 31, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1954
1 ' at t (=r1), odrv returned error flag = i1*neq + i2. ',
1955
1 60, 31, 0, 2, imul, irem, 1, tn, 0.0d0)
1958
1 'lsodes-- rwork length insufficient (for subroutine cdrv). ',
1959
1 60, 32, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1961
1 ' length needed is .ge. lenrw (=i1), exceeds lrw (=i2)',
1962
1 60, 32, 0, 2, lenrw, lrw, 0, 0.0d0, 0.0d0)
1965
1 'lsodes-- error from cdrv in yale sparse matrix package ',
1966
1 60, 33, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1970
1 ' at t (=r1), cdrv returned error flag = i1*neq + i2. ',
1971
1 60, 33, 0, 2, imul, irem, 1, tn, 0.0d0)
1972
if (imul .eq. 2) call xerrwv(
1973
1 ' duplicate entry in sparsity structure descriptors ',
1974
1 60, 33, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1975
if (imul .eq. 3 .or. imul .eq. 6) call xerrwv(
1976
1 ' insufficient storage for nsfc (called by cdrv) ',
1977
1 60, 33, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1979
700 if (illin .eq. 5) go to 710
1983
710 call xerrwv('lsodes-- repeated occurrences of illegal input ',
1984
1 50, 302, 0, 0, 0, 0, 0, 0.0d0, 0.0d0)
1986
800 call xerrwv('lsodes-- run aborted.. apparent infinite loop ',
1987
1 50, 303, 2, 0, 0, 0, 0, 0.0d0, 0.0d0)
1989
c----------------------- end of subroutine lsodes ----------------------
added
removed
scipy/integrate/odepack/lsodes.f
