~ubuntu-branches/ubuntu/raring/scilab/raring-proposed

<literal>P</literal>. A sum of each the rows in <literal>P</literal> is 1. Initial state is given by <literal>x0</literal>. If <literal>x0</literal> is a

364

matrix of size <literal>m=size(x0,"*")</literal> then

365

<literal>Y</literal> is a matrix of size <literal>m</literal>-by-<literal>n</literal>. <literal>Y(i,:)</literal> is the sample path

366

obtained from initial state <literal>x0(i)</literal>.

367

</para>

368

</listitem>

369

</varlistentry>

370

371

<term>multinomial</term>

372

373

<para>

374

<literal>Y=grand(n,"mul",nb,P)</literal> generates

375

<literal>n</literal> observations from the Multinomial

376

distribution : class <literal>nb</literal> events in

377

<literal>m</literal> categories (put <literal>nb</literal>

378

"balls" in <literal>m</literal>

379

"boxes"). <literal>P(i)</literal> is the probability that

380

an event will be classified into category

381

<literal>i</literal>. <literal>P</literal> the vector of probabilities is of

382

size <literal>m-1</literal> (the probability of category

383

<literal>m</literal> is <literal>1-sum(P)</literal>).

384

<literal>Y</literal> is of size <literal>m</literal>-by-<literal>n</literal>.

385

Each column <literal>Y(:,j)</literal> is an observation

386

from multinomial distribution and

387

<literal>Y(i,j)</literal> is the number of events falling in

388

category <literal>i</literal> (for the

389

<literal>j</literal>th observation) (<literal>sum(Y(:,j)) = nb</literal>).

390

</para>

391

</listitem>

392

</varlistentry>

393

394

<term>Poisson</term>

395

396

<para>

397

<literal>Y=grand(m,n,"poi",mu)</literal> generates

398

random variates from the Poisson distribution with mean

399

<literal>mu</literal> (real >= 0.0). Related function

400

: <link linkend="cdfpoi">cdfpoi</link>.

401

</para>

402

</listitem>

403

</varlistentry>

404

405

<term>random permutations</term>

406

407

<para>

408

<literal>Y=grand(n,"prm",vect)</literal> generate

409

<literal>n</literal> random permutations of the column

410

vector (<literal>m x 1</literal>)

411

<literal>vect</literal>.

412

</para>

413

</listitem>

414

</varlistentry>

415

416

<term>uniform (def)</term>

417

418

<para>

419

<literal>Y=grand(m,n,"def")</literal> generates

420

random variates from the uniform distribution over

421

<literal>[0,1)</literal> (1 is never return).

422

</para>

423

</listitem>

424

</varlistentry>

425

426

<term>uniform (unf)</term>

427

428

<para>

429

<literal>Y=grand(m,n,"unf",Low,High)</literal>

430

generates random reals uniformly distributed in

431

<literal>[Low, High)</literal> (<literal>High</literal> is never return).

432

</para>

433

</listitem>

434

</varlistentry>

435

436

<term>uniform (uin)</term>

437

438

<para>

439

<literal>Y=grand(m,n,"uin",Low,High)</literal>

440

generates random integers uniformly distributed between

441

442

(included). <literal>High</literal> and

443

<literal>Low</literal> must be integers such that

444

<literal>(High-Low+1) < 2,147,483,561</literal>.

445

</para>

446

</listitem>

447

</varlistentry>

448

449

<term>uniform (lgi)</term>

450

451

<para>

452

<literal>Y=grand(m,n,"lgi")</literal> returns the

453

basic output of the current generator : random integers

454

following a uniform distribution over :

455

</para>

456

457

458

<para>

459

<literal>[0, 2^32 - 1]</literal> for mt, kiss and fsultra;

460

</para>

461

</listitem>

462

463

<para>

464

<literal>[0, 2147483561]</literal> for clcg2;

465

</para>

466

</listitem>

467

468

<para>

469

<literal>[0, 2^31 - 2]</literal> for clcg4;

470

</para>

471

</listitem>

472

473

<para>

474

<literal>[0, 2^31 - 1]</literal> for urand.

475

</para>

476

</listitem>

477

</itemizedlist>

478

</listitem>

479

</varlistentry>

480

</variablelist>

481

</refsection>

482

483

<title>Set/get the current generator and its state</title>

484

<para>

485

The user has the possibility to choose between different base

486

generators (which give random integers following the "lgi" distribution, the others

487

being gotten from it).

488

</para>

489

490

491

492

493

<para>

494

The Mersenne-Twister of M. Matsumoto and T. Nishimura, period about <literal>2^19937</literal>,

495

state given by an array of <literal>624</literal> integers (plus an index onto this array); this

496

is the default generator.

497

</para>

498

</listitem>

499

</varlistentry>

500

501

502

503

<para>

504

The "Keep It Simple Stupid" of G. Marsaglia, period about <literal>2^123</literal>,

505

state given by <literal>4</literal> integers.

506

</para>

507

</listitem>

508

</varlistentry>

509

510

511

512

<para>

513

A Combined 2 Linear Congruential Generator of P. L'Ecuyer,

514

period about <literal>2^61</literal>, state given by <literal>2</literal> integers.

515

</para>

516

</listitem>

517

</varlistentry>

518

519

520

521

<para>

522

A Combined 4 Linear Congruential Generator of P. L'Ecuyer,

523

period about <literal>2^121</literal>, state given by 4 integers ; this one is

524

split in <literal>101</literal> different virtual (non-overlapping) generators

525

which may be useful for different tasks (see "Actions specific to clcg4" and

526

"Test example for clcg4").

527

</para>

528

</listitem>

529

</varlistentry>

530

531

<term>fsultra</term>

532

533

<para>

534

A Subtract-with-Borrow generator mixing with a congruential

535

generator of Arif Zaman and George Marsaglia, period more than <literal>10^356</literal>,

536

state given by an array of 37 integers (plus an index onto this array, a flag (0 or 1)

537

and another integer).

538

</para>

539

</listitem>

540

</varlistentry>

541

542

<term>urand</term>

543

544

<para>

545

The generator used by the scilab function <link linkend="rand">rand</link>, state

546

given by <literal>1</literal> integer, period of <literal>2^31</literal>.

547

This generator is based on "Urand, A Universal Random Number Generator" By,

548

Michael A. Malcolm, Cleve B. Moler, Stan-Cs-73-334, January 1973, Computer

549

Science Department, School Of Humanities And Sciences, Stanford University.

550

This is the faster of this list but its statistical qualities are less

551

satisfactory than the other generators.

552

</para>

553

</listitem>

554

</varlistentry>

555

</variablelist>

556

<para>

557

The differents actions common to all the generators, are:

558

</para>

559

560

561

<term>action= "getgen"</term>

562

563

<para>

564

<literal>S=grand("getgen")</literal> returns the current base generator.

565

In this case <literal>S</literal> is

566

a string among "mt", "kiss", "clcg2", "clcg4", "urand", "fsultra".

567

</para>

568

</listitem>

569

</varlistentry>

570

571

<term>action= "setgen"</term>

572

573

<para>

574

<literal>grand("setgen",gen)</literal> sets the current base generator to be <literal>gen</literal>

575

a string among "mt", "kiss", "clcg2", "clcg4", "urand", "fsultra".

576

Notice that this call returns the new current generator, i.e. <literal>gen</literal>.

577

</para>

578

</listitem>

579

</varlistentry>

580

581

<term>action= "getsd"</term>

582

583

<para>

584

<literal>S=grand("getsd")</literal> gets the current state (the current seeds) of the current base

585

generator ; <literal>S</literal> is given as a column vector (of integers) of dimension <literal>625</literal>

586

for mt (the first being an index in <literal>[1,624]</literal>), <literal>4</literal> for kiss, <literal>2</literal>

587

for clcg2, <literal>40</literal> for fsultra, <literal>4</literal> for clcg4

588

(for this last one you get the current state of the current virtual generator) and <literal>1</literal>

589

for urand.

590

</para>

591

</listitem>

592

</varlistentry>

593

594

<term>action= "setsd"</term>

595

596

<para>

597

<literal>grand("setsd",S), grand("setsd",s1[,s2,s3,s4])</literal> sets the state of the current

598

base generator (the new seeds) :

599

</para>

600

601

602

603

604

<para>

605

<literal>S</literal> is a vector of integers of dim <literal>625</literal> (the first component is an index

606

and must be in <literal>[1,624]</literal>, the <literal>624</literal> last ones must be in

607

<literal>[0,2^32[</literal>) (but must not be all zeros) ; a simpler initialisation may be done

608

with only one integer <literal>s1</literal> (<literal>s1</literal> must be in <literal>[0,2^32[</literal>) ;

609

</para>

610

</listitem>

611

</varlistentry>

612

613

614

615

<para>

616

<literal>4</literal> integers <literal>s1,s2, s3,s4</literal> in <literal>[0,2^32[</literal> must be provided ;

617

</para>

618

</listitem>

619

</varlistentry>

620

621

622

623

<para>

624

<literal>2</literal> integers <literal>s1</literal> in <literal>[1,2147483562]</literal> and <literal>s2</literal>

625

in <literal>[1,2147483398]</literal> must be given ;

626

</para>

627

</listitem>

628

</varlistentry>

629

630

631

632

<para>

633

<literal>4</literal> integers <literal>s1</literal> in <literal>[1,2147483646]</literal>, <literal>s2</literal>

634

in <literal>[1,2147483542]</literal>, <literal>s3</literal> in <literal>[1,2147483422]</literal>,

635

<literal>s4</literal> in <literal>[1,2147483322]</literal> are required ;

636

</para>

637

<para>

638

<literal>CAUTION</literal> : with clcg4 you set the seeds of the current virtual

639

generator but you may lost the synchronisation between this one

640

and the others virtuals generators (i.e. the sequence generated

641

is not warranty to be non-overlapping with a sequence generated

642

by another virtual generator)=> use instead the "setall" option.

643

</para>

644

</listitem>

645

</varlistentry>

646

647

<term>for urand</term>

648

649

<para>

650

<literal>1</literal> integer <literal>s1</literal> in <literal>[0,2^31[</literal> must be given.

651

</para>

652

</listitem>

653

</varlistentry>

654

655

<term>for fsultra</term>

656

657

<para>

658

<literal>S</literal> is a vector of integers of dim <literal>40</literal> (the first component

659

is an index and must be in <literal>[0,37]</literal>, the 2nd component is a flag (0 or 1), the 3rd component is

660

an integer in <literal>[1,2^32[</literal> and the 37 others integers are in <literal>[0,2^32[</literal>) ; a simpler (and recommanded)

661

initialisation may be done with two integers <literal>s1</literal> and <literal>s2</literal> in

662

<literal>[0,2^32[</literal>.

663

</para>

664

</listitem>

665

</varlistentry>

666

</variablelist>

667

</listitem>

668

</varlistentry>

669

670

<term>action= "phr2sd"</term>

671

672

<para>

673

<literal>Sd=grand("phr2sd", phrase)</literal> given a <literal>phrase</literal> (character string) generates

674

a <literal>1 x 2</literal> vector <literal>Sd</literal> which may be used as seeds to change the state of a

675

base generator (initialy suited for clcg2).

676

</para>

677

</listitem>

678

</varlistentry>

679

</variablelist>

680

</refsection>

681

682

<title>Options specific to clcg4</title>

683

<para>

684

The clcg4 generator may be used as the others generators but it offers the advantage

685

to be split in several (<literal>101</literal>) virtual generators with non-overlapping

686

sequences (when you use a classic generator you may change the initial state (seeds)

687

in order to get another sequence but you are not warranty to get a complete different one).

688

Each virtual generator corresponds to a sequence of <literal>2^72</literal> values which is

689

further split into <literal>V=2^31</literal> segments (or blocks) of length <literal>W=2^41</literal>.

690

For a given virtual generator you have the possibility to return at the beginning of the

691

sequence or at the beginning of the current segment or to go directly at the next segment.

692

You may also change the initial state (seed) of the generator <literal>0</literal> with the

693

"setall" option which then change also the initial state of the other virtual generators

694

so as to get synchronisation, i.e. in function of the new initial state of gen <literal>0</literal>

695

the initial state of gen <literal>1..100</literal> are recomputed so as to get <literal>101</literal>

696

non-overlapping sequences.

697

</para>

698

699

700

<term>action= "setcgn"</term>

701

702

<para>

703

<literal>grand("setcgn",G)</literal> sets the current virtual generator for clcg4 (when clcg4

704

is set, this is the virtual (clcg4) generator number <literal>G</literal> which is used); the virtual clcg4

705

generators are numbered from <literal>0,1,...,100</literal> (and so <literal>G</literal> must be an integer

706

in <literal>[0,100]</literal>) ; by default the current virtual generator is <literal>0</literal>.

707

</para>

708

</listitem>

709

</varlistentry>

710

711

<term>action= "getcgn"</term>

712

713

<para>

714

<literal>S=grand("getcgn")</literal> returns the number of the current virtual clcg4 generator.

715

</para>

716

</listitem>

717

</varlistentry>

718

719

<term>action= "initgn"</term>

720

721

<para>

722

<literal>grand("initgn",I)</literal> reinitializes the state of the current virtual generator

723

</para>

724

725

726

727

728

<para>sets the state to its initial seed</para>

729

</listitem>

730

</varlistentry>

731

732

733

734

<para>sets the state to its last (previous) seed (i.e. to the beginning of the current segment)</para>

735

</listitem>

736

</varlistentry>

737

738

739

740

<para>

741

sets the state to a new seed <literal>W</literal> values from its last seed (i.e. to the beginning

742

of the next segment) and resets the current segment parameters.

743

</para>

744

</listitem>

745

</varlistentry>

746

</variablelist>

747

</listitem>

748

</varlistentry>

749

750

<term>action= "setall"</term>

751

752

<para>

753

<literal>grand("setall",s1,s2,s3,s4)</literal> sets the initial state of generator <literal>0</literal>

754

to <literal>s1,s2,s3,s4</literal>. The initial seeds of the other generators are set accordingly

755

to have synchronisation. For constraints on <literal>s1, s2, s3, s4</literal> see the "setsd" action.

756

</para>

757

</listitem>

758

</varlistentry>

759

760

<term>action= "advnst"</term>

761

762

<para>

763

<literal>grand("advnst",K)</literal> advances the state of the current generator by <literal>2^K</literal> values

764

and resets the initial seed to that value.

765

</para>

766

</listitem>

767

</varlistentry>

768

</variablelist>

769

</refsection>

770

771

<title>Examples</title>

772

<para>

773

In the following example, we generate random numbers from various distributions and

774

plot the associated histograms.

775

</para>

776

<programlisting role="example"><![CDATA[

<refname>grand</refname>

<refpurpose>Random numbers</refpurpose>

</refnamediv>

<title>Calling Sequence</title>

Y=grand(m,n,"bet",A,B)

Y=grand(m,n,"bin",N,p)

Y=grand(m,n,"nbn",N,p)

Y=grand(m,n,"chi", Df)

Y=grand(m,n,"nch",Df,Xnon)

Y=grand(m,n,"exp",Av)

Y=grand(m,n,"f",Dfn,Dfd)

Y=grand(m,n,"nf",Dfn,Dfd,Xnon)

Y=grand(m,n,"gam",shape,scale)

Y=grand(m,n,"nor",Av,Sd)

Y=grand(m,n,"geom", p)

Y=grand(m,n,"poi",mu)

Y=grand(m,n,"def")

Y=grand(m,n,"unf",Low,High)

Y=grand(m,n,"uin",Low,High)

Y=grand(m,n,"lgi")

Y=grand(X,...)

Y=grand(n,"mn",Mean,Cov)

Y=grand(n,"markov",P,x0)

Y=grand(n,"mul",nb,P)

Y=grand(n,"prm",vect)

S=grand("getgen")

grand("setgen",gen)

S=grand("getsd")

grand("setsd",S)

S=grand("phr2sd",phrase)

grand("setcgn",G)

S=grand("getcgn")

grand("initgn",I)

grand("setall",s1,s2,s3,s4)

grand("advnst",K)

</synopsis>

</refsynopsisdiv>

<title>Arguments</title>

<para>

integers, size of the wanted matrix <literal>Y</literal>

</para>

</listitem>

</varlistentry>

<para>

a matrix whom only the dimensions (say <literal>m</literal>-by-<literal>n</literal>)

are used

</para>

</listitem>

</varlistentry>

<para>

a <literal>m</literal>-by-<literal>n</literal> matrix of doubles, with random entries

</para>

</listitem>

</varlistentry>

<para>output of the action (a string or a real column vector)</para>

</listitem>

</varlistentry>

</variablelist>

</refsection>

100

101

<title>Description</title>

102

<para>

103

This function generates random numbers from various

104

distributions.

105

</para>

106

<para>

107

The calling sequences:

108

</para>

109

110

Y=grand(m,n,"bet",A,B)

111

Y=grand(m,n,"bin",N,p)

112

Y=grand(m,n,"nbn",N,p)

113

Y=grand(m,n,"chi", Df)

114

Y=grand(m,n,"nch",Df,Xnon)

115

Y=grand(m,n,"exp",Av)

116

Y=grand(m,n,"f",Dfn,Dfd)

117

Y=grand(m,n,"nf",Dfn,Dfd,Xnon)

118

Y=grand(m,n,"gam",shape,scale)

119

Y=grand(m,n,"nor",Av,Sd)

120

Y=grand(m,n,"geom", p)

121

Y=grand(m,n,"poi",mu)

122

Y=grand(m,n,"def")

123

Y=grand(m,n,"unf",Low,High)

124

Y=grand(m,n,"uin",Low,High)

125

Y=grand(m,n,"lgi")

126

</screen>

127

<para>

128

produce a <literal>m</literal>-by-<literal>n</literal> matrix

129

with random entries.

130

</para>

131

<para>

132

The calling sequence:

133

</para>

134

135

Y=grand(X,...)

136

</screen>

137

<para>

138

where <literal>X</literal> is a <literal>m</literal>-by-<literal>n</literal> matrix,

139

produce the same effect.

140

In this case, only the size of <literal>X</literal> is used.

141

</para>

142

<para>

143

The calling sequences:

144

</para>

145

146

Y=grand(n,"mn",Mean,Cov)

147

Y=grand(n,"markov",P,x0)

148

Y=grand(n,"mul",nb,P)

149

Y=grand(n,"prm",vect)

150

</screen>

151

<para>

152

produce a <literal>m</literal>-by-<literal>n</literal> matrix

153

with random entries, where <literal>m</literal> is the size of the argument <literal>Mean</literal>,

154

<literal>Cov</literal>, <literal>P</literal> or <literal>vect</literal> depending

155

on the case (see below for details).

156

</para>

157

<para>

158

The calling sequences:

159

</para>

160

161

S=grand("getgen")

162

grand("setgen",gen)

163

164

S=grand("getsd")

165

grand("setsd",S)

166

167

grand("setcgn",G)

168

S=grand("getcgn")

169

170

grand("initgn",I)

171

172

grand("setall",s1,s2,s3,s4)

173

174

grand("advnst",K)

175

</screen>

176

<para>

177

configure or quiery the state of the underlying random number

178

generators.

179

</para>

180

</refsection>

181

182

<title>Getting random numbers from a given distribution</title>

183

184

185

186

187

<para>

188

<literal>Y=grand(m,n,"bet",A,B)</literal> generates

189

random variates from the beta distribution with parameters

190

<literal>A</literal> and <literal>B</literal>. The density

191

of the beta distribution is <latex><![CDATA[(0 < x < 1)]]></latex>:

192

</para>

193

<para>

194

<latex>

195

\dfrac{x^{A-1}(1-x)^{B-1}}{\beta(A,B)}

196

</latex>

197

</para>

198

<para>

199

<literal>A</literal> and <literal>B</literal> must be

200

reals <latex><![CDATA[>10^{-37}]]></latex>. Related function: <link linkend="cdfbet">cdfbet</link>.

201

</para>

202

</listitem>

203

</varlistentry>

204

205

<term>binomial</term>

206

207

<para>

208

<literal>Y=grand(m,n,"bin",N,p)</literal> generates random

209

variates from the binomial distribution with parameters

210

<literal>N</literal> (positive integer) and <literal>p</literal>

211

(real in [0,1]) : number of successes in <literal>N</literal>

212

independent Bernouilli trials with probability <literal>p</literal>

213

of success. Related functions: <link linkend="binomial">binomial</link>,

214

215

</para>

216

</listitem>

217

</varlistentry>

218

219

<term>negative binomial</term>

220

221

<para>

222

<literal>Y=grand(m,n,"nbn",N,p)</literal> generates random variates from the

223

negative binomial distribution with parameters <literal>N</literal> (positive integer) and <literal>p</literal> (real

224

in (0,1)) : number of failures occurring before <literal>N</literal> successes

225

in independent Bernouilli trials with probability <literal>p</literal> of success.

226

Related function: <link linkend="cdfnbn">cdfnbn</link>.

227

</para>

228

</listitem>

229

</varlistentry>

230

231

<term>chisquare</term>

232

233

<para>

234

<literal>Y=grand(m,n,"chi", Df)</literal> generates random

235

variates from the chi-square distribution with <literal>Df</literal>

236

(real > 0.0) degrees of freedom. Related function:

237

238

</para>

239

</listitem>

240

</varlistentry>

241

242

<term>non-central chi-square</term>

243

244

<para>

245

<literal>Y=grand(m,n,"nch",Df,Xnonc)</literal> generates

246

random variates from the non-central chisquare

247

distribution with <literal>Df</literal> degrees of freedom

248

(real >= 1.0) and noncentrality parameter

249

<literal>Xnonc</literal> (real >= 0.0). Related

250

function: <link linkend="cdfchn">cdfchn</link>.

251

</para>

252

</listitem>

253

</varlistentry>

254

255

<term>exponential</term>

256

257

<para>

258

<literal>Y=grand(m,n,"exp",Av)</literal> generates

259

random variates from the exponential distribution with mean

260

<literal>Av</literal> (real > 0.0).

261

</para>

262

</listitem>

263

</varlistentry>

264

265

<term>F variance ratio</term>

266

267

<para>

268

<literal>Y=grand(m,n,"f",Dfn,Dfd)</literal> generates

269

random variates from the F (variance ratio) distribution

270

with <literal>Dfn</literal> (real > 0.0) degrees of

271

freedom in the numerator and <literal>Dfd</literal> (real

272

> 0.0) degrees of freedom in the denominator. Related

273

function : <link linkend="cdff">cdff</link>.

274

</para>

275

</listitem>

276

</varlistentry>

277

278

<term>non-central F variance ratio</term>

279

280

<para>

281

<literal>Y=grand(m,n,"nf",Dfn,Dfd,Xnonc)</literal>

282

generates random variates from the noncentral F (variance

283

ratio) distribution with <literal>Dfn</literal> (real

284

>= 1) degrees of freedom in the numerator, and

285

<literal>Dfd</literal> (real > 0) degrees of freedom in

286

the denominator, and noncentrality parameter

287

<literal>Xnonc</literal> (real >= 0). Related

288

function : <link linkend="cdffnc">cdffnc</link>.

289

</para>

290

</listitem>

291

</varlistentry>

292

293

<term>gamma</term>

294

295

<para>

296

<literal>Y=grand(m,n,"gam",shape,scale)</literal>

297

generates random variates from the gamma distribution with

298

parameters <literal>shape</literal> (real > 0) and

299

<literal>scale</literal> (real > 0). The density of the

300

gamma distribution is :

301

</para>

302

<para>

303

<latex>

304

\dfrac{ \textrm{scale}^{\textrm{shape}} x^{\textrm{shape}-1} e^{-\textrm{scale} x}}{\gamma(\textrm{shape}) }

305

</latex>

306

</para>

307

<para>

308

Related functions : <link linkend="gamma">gamma</link>,

309

310

</para>

311

</listitem>

312

</varlistentry>

313

314

<term>Gauss Laplace (normal)</term>

315

316

<para>

317

<literal>Y=grand(m,n,"nor",Av,Sd)</literal> generates

318

random variates from the normal distribution with mean

319

<literal>Av</literal> (real) and standard deviation

320

<literal>Sd</literal> (real >= 0). Related function

321

: <link linkend="cdfnor">cdfnor</link>.

322

</para>

323

</listitem>

324

</varlistentry>

325

326

<term>multivariate gaussian (multivariate normal)</term>

327

328

<para>

329

<literal>Y=grand(n,"mn",Mean,Cov)</literal>

330

generates multivariate normal random

331

variates; <literal>Mean</literal> must be a <literal>m x 1</literal>

332

column vector and <literal>Cov</literal> a <literal>m</literal>-by-<literal>m</literal>

333

symmetric positive definite matrix (<literal>Y</literal> is then

334

a <literal>m</literal>-by-<literal>n</literal> matrix).

335

</para>

336

</listitem>

337

</varlistentry>

338

339

<term>geometric</term>

340

341

<para>

342

<literal>Y=grand(m,n,"geom", p)</literal> generates

343

random variates from the geometric distribution with

344

parameter <literal>p</literal> : number of Bernouilli trials

345

(with probability succes of <literal>p</literal>) until a

346

succes is met. <literal>p</literal> must be in

347

<latex>[p_{min},1]</latex> (with <latex> p_{min} = 1{,}3\times 10^{-307} </latex>).

348

</para>

349

<para>

350

<literal>Y</literal> contains positive real numbers

351

with integer values, whiсh are the "number of trials to get

352

a success".

353

</para>

354

</listitem>

355

</varlistentry>

356

357

<term>markov</term>

358

359

<para>

360

<literal>Y=grand(n,"markov",P,x0)</literal> generate

361

<literal>n</literal> successive states of a Markov chain

362

described by the transition matrix

363

<literal>P</literal>. A sum of each the rows in <literal>P</literal> is 1. Initial state is given by <literal>x0</literal>. If <literal>x0</literal> is a

364

matrix of size <literal>m=size(x0,"*")</literal> then

365

<literal>Y</literal> is a matrix of size <literal>m</literal>-by-<literal>n</literal>. <literal>Y(i,:)</literal> is the sample path

366

obtained from initial state <literal>x0(i)</literal>.

367

</para>

368

</listitem>

369

</varlistentry>

370

371

<term>multinomial</term>

372

373

<para>

374

<literal>Y=grand(n,"mul",nb,P)</literal> generates

375

<literal>n</literal> observations from the Multinomial

376

distribution : class <literal>nb</literal> events in

377

<literal>m</literal> categories (put <literal>nb</literal>

378

"balls" in <literal>m</literal>

379

"boxes"). <literal>P(i)</literal> is the probability that

380

an event will be classified into category

381

<literal>i</literal>. <literal>P</literal> the vector of probabilities is of

382

size <literal>m-1</literal> (the probability of category

383

<literal>m</literal> is <literal>1-sum(P)</literal>).

384

<literal>Y</literal> is of size <literal>m</literal>-by-<literal>n</literal>.

385

Each column <literal>Y(:,j)</literal> is an observation

386

from multinomial distribution and

387

<literal>Y(i,j)</literal> is the number of events falling in

388

category <literal>i</literal> (for the

389

<literal>j</literal>th observation) (<literal>sum(Y(:,j)) = nb</literal>).

390

</para>

391

</listitem>

392

</varlistentry>

393

394

<term>Poisson</term>

395

396

<para>

397

<literal>Y=grand(m,n,"poi",mu)</literal> generates

398

random variates from the Poisson distribution with mean

399

<literal>mu</literal> (real >= 0.0). Related function

400

: <link linkend="cdfpoi">cdfpoi</link>.

401

</para>

402

</listitem>

403

</varlistentry>

404

405

<term>random permutations</term>

406

407

<para>

408

<literal>Y=grand(n,"prm",vect)</literal> generate

409

<literal>n</literal> random permutations of the column

410

vector (<literal>m x 1</literal>)

411

<literal>vect</literal>.

412

</para>

413

</listitem>

414

</varlistentry>

415

416

<term>uniform (def)</term>

417

418

<para>

419

<literal>Y=grand(m,n,"def")</literal> generates

420

random variates from the uniform distribution over

421

<literal>[0,1)</literal> (1 is never return).

422

</para>

423

</listitem>

424

</varlistentry>

425

426

<term>uniform (unf)</term>

427

428

<para>

429

<literal>Y=grand(m,n,"unf",Low,High)</literal>

430

generates random reals uniformly distributed in

431

<literal>[Low, High)</literal> (<literal>High</literal> is never return).

432

</para>

433

</listitem>

434

</varlistentry>

435

436

<term>uniform (uin)</term>

437

438

<para>

439

<literal>Y=grand(m,n,"uin",Low,High)</literal>

440

generates random integers uniformly distributed between

441

442

(included). <literal>High</literal> and

443

<literal>Low</literal> must be integers such that

444

<literal>(High-Low+1) < 2,147,483,561</literal>.

445

</para>

446

</listitem>

447

</varlistentry>

448

449

<term>uniform (lgi)</term>

450

451

<para>

452

<literal>Y=grand(m,n,"lgi")</literal> returns the

453

basic output of the current generator : random integers

454

following a uniform distribution over :

455

</para>

456

457

458

<para>

459

<literal>[0, 2^32 - 1]</literal> for mt, kiss and fsultra;

460

</para>

461

</listitem>

462

463

<para>

464

<literal>[0, 2147483561]</literal> for clcg2;

465

</para>

466

</listitem>

467

468

<para>

469

<literal>[0, 2^31 - 2]</literal> for clcg4;

470

</para>

471

</listitem>

472

473

<para>

474

<literal>[0, 2^31 - 1]</literal> for urand.

475

</para>

476

</listitem>

477

</itemizedlist>

478

</listitem>

479

</varlistentry>

480

</variablelist>

481

</refsection>

482

483

<title>Set/get the current generator and its state</title>

484

<para>

485

The user has the possibility to choose between different base

486

generators (which give random integers following the "lgi" distribution, the others

487

being gotten from it).

488

</para>

489

490

491

492

493

<para>

494

The Mersenne-Twister of M. Matsumoto and T. Nishimura, period about <literal>2^19937</literal>,

495

state given by an array of <literal>624</literal> integers (plus an index onto this array); this

496

is the default generator.

497

</para>

498

</listitem>

499

</varlistentry>

500

501

502

503

<para>

504

The "Keep It Simple Stupid" of G. Marsaglia, period about <literal>2^123</literal>,

505

state given by <literal>4</literal> integers.

506

</para>

507

</listitem>

508

</varlistentry>

509

510

511

512

<para>

513

A Combined 2 Linear Congruential Generator of P. L'Ecuyer,

514

period about <literal>2^61</literal>, state given by <literal>2</literal> integers.

515

</para>

516

</listitem>

517

</varlistentry>

518

519

520

521

<para>

522

A Combined 4 Linear Congruential Generator of P. L'Ecuyer,

523

period about <literal>2^121</literal>, state given by 4 integers ; this one is

524

split in <literal>101</literal> different virtual (non-overlapping) generators

525

which may be useful for different tasks (see "Actions specific to clcg4" and

526

"Test example for clcg4").

527

</para>

528

</listitem>

529

</varlistentry>

530

531

<term>fsultra</term>

532

533

<para>

534

A Subtract-with-Borrow generator mixing with a congruential

535

generator of Arif Zaman and George Marsaglia, period more than <literal>10^356</literal>,

536

state given by an array of 37 integers (plus an index onto this array, a flag (0 or 1)

537

and another integer).

538

</para>

539

</listitem>

540

</varlistentry>

541

542

<term>urand</term>

543

544

<para>

545

The generator used by the scilab function <link linkend="rand">rand</link>, state

546

given by <literal>1</literal> integer, period of <literal>2^31</literal>.

547

This generator is based on "Urand, A Universal Random Number Generator" By,

548

Michael A. Malcolm, Cleve B. Moler, Stan-Cs-73-334, January 1973, Computer

549

Science Department, School Of Humanities And Sciences, Stanford University.

550

This is the faster of this list but its statistical qualities are less

551

satisfactory than the other generators.

552

</para>

553

</listitem>

554

</varlistentry>

555

</variablelist>

556

<para>

557

The differents actions common to all the generators, are:

558

</para>

559

560

561

<term>action= "getgen"</term>

562

563

<para>

564

<literal>S=grand("getgen")</literal> returns the current base generator.

565

In this case <literal>S</literal> is

566

a string among "mt", "kiss", "clcg2", "clcg4", "urand", "fsultra".

567

</para>

568

</listitem>

569

</varlistentry>

570

571

<term>action= "setgen"</term>

572

573

<para>

574

<literal>grand("setgen",gen)</literal> sets the current base generator to be <literal>gen</literal>

575

a string among "mt", "kiss", "clcg2", "clcg4", "urand", "fsultra".

576

Notice that this call returns the new current generator, i.e. <literal>gen</literal>.

577

</para>

578

</listitem>

579

</varlistentry>

580

581

<term>action= "getsd"</term>

582

583

<para>

584

<literal>S=grand("getsd")</literal> gets the current state (the current seeds) of the current base

585

generator ; <literal>S</literal> is given as a column vector (of integers) of dimension <literal>625</literal>

586

for mt (the first being an index in <literal>[1,624]</literal>), <literal>4</literal> for kiss, <literal>2</literal>

587

for clcg2, <literal>40</literal> for fsultra, <literal>4</literal> for clcg4

588

(for this last one you get the current state of the current virtual generator) and <literal>1</literal>

589

for urand.

590

</para>

591

</listitem>

592

</varlistentry>

593

594

<term>action= "setsd"</term>

595

596

<para>

597

<literal>grand("setsd",S), grand("setsd",s1[,s2,s3,s4])</literal> sets the state of the current

598

base generator (the new seeds) :

599

</para>

600

601

602

603

604

<para>

605

<literal>S</literal> is a vector of integers of dim <literal>625</literal> (the first component is an index

606

and must be in <literal>[1,624]</literal>, the <literal>624</literal> last ones must be in

607

<literal>[0,2^32[</literal>) (but must not be all zeros) ; a simpler initialisation may be done

608

with only one integer <literal>s1</literal> (<literal>s1</literal> must be in <literal>[0,2^32[</literal>) ;

609

</para>

610

</listitem>

611

</varlistentry>

612

613

614

615

<para>

616

<literal>4</literal> integers <literal>s1,s2, s3,s4</literal> in <literal>[0,2^32[</literal> must be provided ;

617

</para>

618

</listitem>

619

</varlistentry>

620

621

622

623

<para>

624

<literal>2</literal> integers <literal>s1</literal> in <literal>[1,2147483562]</literal> and <literal>s2</literal>

625

in <literal>[1,2147483398]</literal> must be given ;

626

</para>

627

</listitem>

628

</varlistentry>

629

630

631

632

<para>

633

<literal>4</literal> integers <literal>s1</literal> in <literal>[1,2147483646]</literal>, <literal>s2</literal>

634

in <literal>[1,2147483542]</literal>, <literal>s3</literal> in <literal>[1,2147483422]</literal>,

635

<literal>s4</literal> in <literal>[1,2147483322]</literal> are required ;

636

</para>

637

<para>

638

<literal>CAUTION</literal> : with clcg4 you set the seeds of the current virtual

639

generator but you may lost the synchronisation between this one

640

and the others virtuals generators (i.e. the sequence generated

641

is not warranty to be non-overlapping with a sequence generated

642

by another virtual generator)=> use instead the "setall" option.

643

</para>

644

</listitem>

645

</varlistentry>

646

647

<term>for urand</term>

648

649

<para>

650

<literal>1</literal> integer <literal>s1</literal> in <literal>[0,2^31[</literal> must be given.

651

</para>

652

</listitem>

653

</varlistentry>

654

655

<term>for fsultra</term>

656

657

<para>

658

<literal>S</literal> is a vector of integers of dim <literal>40</literal> (the first component

659

is an index and must be in <literal>[0,37]</literal>, the 2nd component is a flag (0 or 1), the 3rd component is

660

an integer in <literal>[1,2^32[</literal> and the 37 others integers are in <literal>[0,2^32[</literal>) ; a simpler (and recommanded)

661

initialisation may be done with two integers <literal>s1</literal> and <literal>s2</literal> in

662

<literal>[0,2^32[</literal>.

663

</para>

664

</listitem>

665

</varlistentry>

666

</variablelist>

667

</listitem>

668

</varlistentry>

669

670

<term>action= "phr2sd"</term>

671

672

<para>

673

<literal>Sd=grand("phr2sd", phrase)</literal> given a <literal>phrase</literal> (character string) generates

674

a <literal>1 x 2</literal> vector <literal>Sd</literal> which may be used as seeds to change the state of a

675

base generator (initialy suited for clcg2).

676

</para>

677

</listitem>

678

</varlistentry>

679

</variablelist>

680

</refsection>

681

682

<title>Options specific to clcg4</title>

683

<para>

684

The clcg4 generator may be used as the others generators but it offers the advantage

685

to be split in several (<literal>101</literal>) virtual generators with non-overlapping

686

sequences (when you use a classic generator you may change the initial state (seeds)

687

in order to get another sequence but you are not warranty to get a complete different one).

688

Each virtual generator corresponds to a sequence of <literal>2^72</literal> values which is

689

further split into <literal>V=2^31</literal> segments (or blocks) of length <literal>W=2^41</literal>.

690

For a given virtual generator you have the possibility to return at the beginning of the

691

sequence or at the beginning of the current segment or to go directly at the next segment.

692

You may also change the initial state (seed) of the generator <literal>0</literal> with the

693

"setall" option which then change also the initial state of the other virtual generators

694

so as to get synchronisation, i.e. in function of the new initial state of gen <literal>0</literal>

695

the initial state of gen <literal>1..100</literal> are recomputed so as to get <literal>101</literal>

696

non-overlapping sequences.

697

</para>

698

699

700

<term>action= "setcgn"</term>

701

702

<para>

703

<literal>grand("setcgn",G)</literal> sets the current virtual generator for clcg4 (when clcg4

704

is set, this is the virtual (clcg4) generator number <literal>G</literal> which is used); the virtual clcg4

705

generators are numbered from <literal>0,1,...,100</literal> (and so <literal>G</literal> must be an integer

706

in <literal>[0,100]</literal>) ; by default the current virtual generator is <literal>0</literal>.

707

</para>

708

</listitem>

709

</varlistentry>

710

711

<term>action= "getcgn"</term>

712

713

<para>

714

<literal>S=grand("getcgn")</literal> returns the number of the current virtual clcg4 generator.

715

</para>

716

</listitem>

717

</varlistentry>

718

719

<term>action= "initgn"</term>

720

721

<para>

722

<literal>grand("initgn",I)</literal> reinitializes the state of the current virtual generator

723

</para>

724

725

726

727

728

<para>sets the state to its initial seed</para>

729

</listitem>

730

</varlistentry>

731

732

733

734

<para>sets the state to its last (previous) seed (i.e. to the beginning of the current segment)</para>

735

</listitem>

736

</varlistentry>

737

738

739

740

<para>

741

sets the state to a new seed <literal>W</literal> values from its last seed (i.e. to the beginning

742

of the next segment) and resets the current segment parameters.

743

</para>

744

</listitem>

745

</varlistentry>

746

</variablelist>

747

</listitem>

748

</varlistentry>

749

750

<term>action= "setall"</term>

751

752

<para>

753

<literal>grand("setall",s1,s2,s3,s4)</literal> sets the initial state of generator <literal>0</literal>

754

to <literal>s1,s2,s3,s4</literal>. The initial seeds of the other generators are set accordingly

755

to have synchronisation. For constraints on <literal>s1, s2, s3, s4</literal> see the "setsd" action.

756

</para>

757

</listitem>

758

</varlistentry>

759

760

<term>action= "advnst"</term>

761

762

<para>

763

<literal>grand("advnst",K)</literal> advances the state of the current generator by <literal>2^K</literal> values

764

and resets the initial seed to that value.

765

</para>

766

</listitem>

767

</varlistentry>

768

</variablelist>

769

</refsection>

770

771

<title>Examples</title>

772

<para>

773

In the following example, we generate random numbers from various distributions and

774

plot the associated histograms.

775

</para>

776

<programlisting role="example"><![CDATA[

777

// Returns a 400-by-800 matrix of random doubles,

778

// with normal distribution and average 0 and standard deviation 1.

779

R = grand(400,800,"nor",0,1);

780

scf();

781

histplot(10,R);

782

xtitle("Normal random numbers from grand","X","Frequency");

783

]]></programlisting>

784

<scilab:image localized="true">

785

R = grand(400,800,"nor",0,1);

786

histplot(10,R);

787

xtitle("Normal random numbers from grand","X","Frequency");

788

</scilab:image>

789

790

<programlisting role="example"><![CDATA[

783

791

// Returns a 400-by-800 matrix of random doubles,

784

792

// uniform in [0,1).

785

793

R = grand(400,800,"def");

786

794

scf();

787

795

histplot(10,R);

788

796

xtitle("Uniform random numbers from grand","X","Frequency");

797

]]></programlisting>

798

<scilab:image localized="true">

799

R = grand(400,800,"def");

800

histplot(10,R);

801

xtitle("Uniform random numbers from grand","X","Frequency");

802

</scilab:image>

803

804

<programlisting role="example"><![CDATA[

789

805

// Returns a 400-by-800 matrix of random doubles,

790

806

// with Poisson distribution and average equal to 5.

791

807

R = grand(400,800,"poi",5);

793

809

histplot(10,R);

794

810

xtitle("Poisson random numbers from grand","X","Frequency");

795

811

]]></programlisting>

796

<para>

797

In the following example, we generate random numbers from the exponential distribution and

798

then compare the empirical with the theoretical distribution.

799

</para>

800

<programlisting role="example"><![CDATA[

801

lambda=1.6;

802

N=100000;

803

X = grand(1,N,"exp",lambda);

804

scf();

805

classes = linspace(0,12,25);

806

histplot(classes,X)

807

x=linspace(0,12,25);

808

y = (1/lambda)*exp(-(1/lambda)*x);

809

plot(x,y,"ro-");

810

legend(["Empirical" "Theory"]);

811

xtitle("Exponential random numbers from grand","X","Frequency");

812

]]></programlisting>

813

<para>

814

In the following example, we generate random numbers from the gamma distribution and

815

then compare the empirical with the theoretical distribution.

816

</para>

817

<programlisting role="example"><![CDATA[

818

N=10000;

819

A=10;

820

B=4;

821

R=grand(1,N,"gam",A,B);

822

XS=gsort(R,"g","i")';

823

PS=(1:N)'/N;

824

P=cdfgam("PQ",XS,A*ones(XS),B*ones(XS));

825

scf();

826

plot(XS,PS,"b-"); // Empirical distribution

827

plot(XS,P,"r-"); // Theoretical distribution

828

legend(["Empirical" "Theory"]);

829

xtitle("Cumulative distribution function of Gamma random numbers","X","F");

830

]]></programlisting>

831

<para>

832

In the following example, we generate 10 random integers in the [1,365] interval.

833

</para>

834

<programlisting role="example"><![CDATA[

812

<scilab:image localized="true">

813

R = grand(400,800,"poi",5);

814

histplot(10,R);

815

xtitle("Poisson random numbers from grand","X","Frequency");

816

</scilab:image>

817

818

<para>

819

In the following example, we generate random numbers from the exponential distribution and

820

then compare the empirical with the theoretical distribution.

821

</para>

822

<programlisting role="example"><![CDATA[

823

lambda=1.6;

824

N=100000;

825

X = grand(1,N,"exp",lambda);

826

scf();

827

classes = linspace(0,12,25);

828

histplot(classes,X)

829

x=linspace(0,12,25);

830

y = (1/lambda)*exp(-(1/lambda)*x);

831

plot(x,y,"ro-");

832

legend(["Empirical" "Theory"]);

833

xtitle("Exponential random numbers from grand","X","Frequency");

834

]]></programlisting>

835

<scilab:image localized="true">

836

lambda=1.6;

837

N=100000;

838

X = grand(1,N,"exp",lambda);

839

classes = linspace(0,12,25);

840

histplot(classes,X)

841

x=linspace(0,12,25);

842

y = (1/lambda)*exp(-(1/lambda)*x);

843

plot(x,y,"ro-");

844

legend(["Empirical" "Theory"]);

845

xtitle("Exponential random numbers from grand","X","Frequency");

846

</scilab:image>

847

<para>

848

In the following example, we generate random numbers from the gamma distribution and

849

then compare the empirical with the theoretical distribution.

850

</para>

851

<programlisting role="example"><![CDATA[

852

N=10000;

853

A=10;

854

B=4;

855

R=grand(1,N,"gam",A,B);

856

XS=gsort(R,"g","i")';

857

PS=(1:N)'/N;

858

P=cdfgam("PQ",XS,A*ones(XS),B*ones(XS));

859

scf();

860

plot(XS,PS,"b-"); // Empirical distribution

861

plot(XS,P,"r-"); // Theoretical distribution

862

legend(["Empirical" "Theory"]);

863

xtitle("Cumulative distribution function of Gamma random numbers","X","F");

864

]]></programlisting>

865

<scilab:image localized="true">

866

N=10000;

867

A=10;

868

B=4;

869

R=grand(1,N,"gam",A,B);

870

XS=gsort(R,"g","i")';

871

PS=(1:N)'/N;

872

P=cdfgam("PQ",XS,A*ones(XS),B*ones(XS));

873

scf();

874

plot(XS,PS,"b-"); // Empirical distribution

875

plot(XS,P,"r-"); // Theoretical distribution

876

legend(["Empirical" "Theory"]);

877

xtitle("Cumulative distribution function of Gamma random numbers","X","F");

878

</scilab:image>

879

<para>

880

In the following example, we generate 10 random integers in the [1,365] interval.

881

</para>

882

<programlisting role="example"><![CDATA[

835

883

grand(10,1,"uin",1,365)

836

884

]]></programlisting>

837

<para>

838

In the following example, we generate 12 permutations of the [1,2,...,7] set.

839

The 12 permutations are stored column-by-column.

840

</para>

841

<programlisting role="example"><![CDATA[

885

<para>

886

In the following example, we generate 12 permutations of the [1,2,...,7] set.

887

The 12 permutations are stored column-by-column.

888

</para>

889

<programlisting role="example"><![CDATA[

842

890

grand(12,"prm",(1:7)')

843

891

]]></programlisting>

844

</refsection>

845

846

<title>Get predictible or less predictible numbers</title>

847

<para>

848

The pseudo random number generators are based on deterministic sequences.

849

In order to get reproducible simulations, the initial seed of the generator is constant,

850

such that the sequence will remain the same from a session to the other.

851

Hence, by default, the first numbers produced by <literal>grand</literal> are always the same.

852

</para>

853

<para>

854

In some situations, we may want to initialize the seed of the generator in

855

order to produce less predictible numbers.

856

In this case, we may initialize the seed with the output of the <literal>getdate</literal> function:

857

</para>

858

<programlisting role="example"><![CDATA[

892

</refsection>

893

894

<title>Get predictible or less predictible numbers</title>

895

<para>

896

The pseudo random number generators are based on deterministic sequences.

897

In order to get reproducible simulations, the initial seed of the generator is constant,

898

such that the sequence will remain the same from a session to the other.

899

Hence, by default, the first numbers produced by <literal>grand</literal> are always the same.

900

</para>

901

<para>

902

In some situations, we may want to initialize the seed of the generator in

903

order to produce less predictible numbers.

904

In this case, we may initialize the seed with the output of the <literal>getdate</literal> function:

905

</para>

906

<programlisting role="example"><![CDATA[

859

907

n=getdate("s");

860

908

grand("setsd",n)

861

909

]]></programlisting>

862

</refsection>

863

864

<title>Test example for clcg4</title>

865

<para>

866

An example of the need of the splitting capabilities of clcg4 is as follows.

867

Two statistical techniques are being compared on data of different sizes. The first

868

technique uses bootstrapping and is thought to be as accurate using less data

869

than the second method which employs only brute force. For the first method, a data

870

set of size uniformly distributed between 25 and 50 will be generated. Then the data set

871

of the specified size will be generated and analyzed. The second method will choose a

872

data set size between 100 and 200, generate the data and analyze it. This process will

873

be repeated 1000 times. For variance reduction, we want the random numbers used in the

874

two methods to be the same for each of the 1000 comparisons. But method two will use more

875

random numbers than method one and without this package, synchronization might be difficult.

876

With clcg4, it is a snap. Use generator 0 to obtain the sample size for method one and

877

generator 1 to obtain the data. Then reset the state to the beginning of the current block

878

and do the same for the second method. This assures that the initial data for method two is

879

that used by method one. When both have concluded, advance the block for both generators.

880

</para>

881

</refsection>

882

883

884

885

886

887

</member>

888

889

890

</member>

891

892

893

</member>

894

</simplelist>

895

</refsection>

910

</refsection>

911

912

<title>Test example for clcg4</title>

913

<para>

914

An example of the need of the splitting capabilities of clcg4 is as follows.

915

Two statistical techniques are being compared on data of different sizes. The first

916

technique uses bootstrapping and is thought to be as accurate using less data

917

than the second method which employs only brute force. For the first method, a data

918

set of size uniformly distributed between 25 and 50 will be generated. Then the data set

919

of the specified size will be generated and analyzed. The second method will choose a

920

data set size between 100 and 200, generate the data and analyze it. This process will

921

be repeated 1000 times. For variance reduction, we want the random numbers used in the

922

two methods to be the same for each of the 1000 comparisons. But method two will use more

923

random numbers than method one and without this package, synchronization might be difficult.

924

With clcg4, it is a snap. Use generator 0 to obtain the sample size for method one and

925

generator 1 to obtain the data. Then reset the state to the beginning of the current block

926

and do the same for the second method. This assures that the initial data for method two is

927

that used by method one. When both have concluded, advance the block for both generators.

928

</para>

929

</refsection>

930

931

932

933

934

935

</member>

936

937

938

</member>

939

940

941

</member>

942

</simplelist>

943

</refsection>

896

944

</refentry>

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