scilab/modules/differential_equations/help/en_US/daskr.xml

   1 <?xml version="1.0" encoding="UTF-8"?>
   2 <!--
   3  * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
   4  * Copyright (C) 2013 - Scilab Enterprises - Paul Bignier
   5  *
   6  * Copyright (C) 2012 - 2016 - Scilab Enterprises
   7  *
   8  * This file is hereby licensed under the terms of the GNU GPL v2.0,
   9  * pursuant to article 5.3.4 of the CeCILL v.2.1.
  10  * This file was originally licensed under the terms of the CeCILL v2.1,
  11  * and continues to be available under such terms.
  12  * For more information, see the COPYING file which you should have received
  13  * along with this program.
  14  *
  15  -->
  16 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns5="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="daskr" xml:lang="en">
  17     <refnamediv>
  18         <refname>daskr</refname>
  19         <refpurpose>DAE solver with zero crossing</refpurpose>
  20     </refnamediv>
  21     <refsynopsisdiv>
  22         <title>Syntax</title>
  23         <synopsis>[r, nn [, hd]] = daskr(x0, t0, t [, atol [, rtol]], res [, jac], ng, surf [, info [, psol] [, pjac]] [, hd])</synopsis>
  24     </refsynopsisdiv>
  25     <refsection>
  26         <title>Arguments</title>
  27         <variablelist>
  28             <varlistentry>
  29                 <term>x0</term>
  30                 <listitem>
  31                     <para>
  32                         is either <literal>y0</literal> (<literal>ydot0</literal> is
  33                         estimated by <literal>daskr</literal> with zero as first estimate)
  34                         or the matrix <literal>[y0 ydot0]</literal>.
  35                         <literal>g(t, y0, ydot0)</literal> must be equal to zero.
  36                         If you only know an estimate of <literal>ydot0</literal>, set
  37                         <literal>info(7)=1</literal>.
  38                     </para>
  39                     <variablelist>
  40                         <varlistentry>
  41                             <term>y0</term>
  42                             <listitem>
  43                                 <para>a real column vector of initial conditions.</para>
  44                             </listitem>
  45                         </varlistentry>
  46                         <varlistentry>
  47                             <term>ydot0</term>
  48                             <listitem>
  49                                 <para>a real column vector of the time derivative of
  50                                     <literal>y</literal> at <literal>t0</literal> (may be an estimate).
  51                                 </para>
  52                             </listitem>
  53                         </varlistentry>
  54                     </variablelist>
  55                 </listitem>
  56             </varlistentry>
  57             <varlistentry>
  58                 <term>t0</term>
  59                 <listitem>
  60                     <para>a real number, it is the initial instant.</para>
  61                 </listitem>
  62             </varlistentry>
  63             <varlistentry>
  64                 <term>t</term>
  65                 <listitem>
  66                     <para>a real scalar or vector. Gives instants for which you want the
  67                         solution. Note that you can get solution at each daskr's step point
  68                         by setting <literal>info(2)=1</literal>.
  69                     </para>
  70                 </listitem>
  71             </varlistentry>
  72             <varlistentry>
  73                 <term>atol, rtol</term>
  74                 <listitem>
  75                     <para>real scalars or column vectors of same size as
  76                         <literal>y</literal> or both of size 1. <literal>atol</literal> and <literal>rtol</literal> give respectively
  77                         absolute and relative error tolerances of solution. If vectors, the
  78                         tolerances are specified for each component of
  79                         <literal>y</literal>.
  80                     </para>
  81                 </listitem>
  82             </varlistentry>
  83             <varlistentry>
  84                 <term>res</term>
  85                 <listitem>
  86                     <para>
  87                         <link linkend="external">external</link> (function, list or string). Computes the value of
  88                         <literal>g(t, y, ydot)</literal>. It may be :
  89                     </para>
  90                     <itemizedlist>
  91                         <listitem>
  92                             <para>A Scilab function.</para>
  93                             <para>Its syntax must be
  94                                 <literal>[r, ires] = res(t, y, ydot)</literal> and must return the residual
  95                                 <literal>r = g(t, y, ydot)</literal> and error flag
  96                                 <literal>ires</literal>. <literal>ires = 0</literal> if
  97                                 <literal>res</literal> succeeds to compute <literal>r</literal>,
  98                                 <literal>ires = -1</literal> if residual is locally not defined for
  99                                 <literal>(t, y, ydot)</literal>, <literal>ires = -2</literal> if
 100                                 parameters are out of admissible range.
 101                             </para>
 102                         </listitem>
 103                         <listitem>
 104                             <para>A list.</para>
 105                             <para>This form allows to pass parameters other than t, y, ydot to the function.
 106                                 It must be as follows:
 107                             </para>
 108                             <programlisting role="no-scilab-exec"><![CDATA[
 109 list(res, x1, x2, ...)
 110  ]]></programlisting>
 111                             <para>where the syntax of the function
 112                                 <literal>res</literal> is now
 113                             </para>
 114                             <programlisting role="no-scilab-exec"><![CDATA[
 115 r = res(t, y, ydot, x1, x2, ...)
 116  ]]></programlisting>
 117                             <para>
 118                                 <literal>res</literal> still returns
 119                                 <literal>r = g(t, y, ydot)</literal> as a function of
 120                                 <literal>(t, y, ydot, x1, x2, ...)</literal>.
 121                             </para>
 122                             <para>Warning: this form must not be used if there is no extra
 123                                 argument to pass to the function.
 124                             </para>
 125                         </listitem>
 126                         <listitem>
 127                             <para>A string.</para>
 128                             <para>It must refer to the name of a C function or a Fortran subroutine
 129                                 linked with Scilab.
 130                             </para>
 131                             <para>In C, the syntax must be:</para>
 132                             <programlisting role="no-scilab-exec"><![CDATA[
 133 void res(double *t, double y[], double yd[], double r[],
 134          int *ires, double rpar[], int ipar[])
 135  ]]></programlisting>
 136                             <para>In Fortran, it must be:</para>
 137                             <programlisting role="no-scilab-exec"><![CDATA[
 138 subroutine res(t, y, yd, r, ires, rpar, ipar)
 139 double precision t, y(*), yd(*),r(*),rpar(*)
 140 integer ires, ipar(*)
 141  ]]></programlisting>
 142                             <para>
 143                                 The <literal>rpar</literal> and <literal>ipar</literal>
 144                                 arrays must be present but cannot be used.
 145                             </para>
 146                         </listitem>
 147                     </itemizedlist>
 148                 </listitem>
 149             </varlistentry>
 150             <varlistentry>
 151                 <term>jac</term>
 152                 <listitem>
 153                     <para>
 154                         <link linkend="external">external</link> (function, list or string).
 155                         Computes the value of <literal>dg/dy + cj*dg/dydot</literal> for a given value of parameter
 156                         <literal>cj</literal>.
 157                     </para>
 158                     <itemizedlist>
 159                         <listitem>
 160                             <para>A Scilab function.</para>
 161                             <para>Its syntax must be
 162                                 <literal>r = jac(t, y, ydot, cj)</literal> and must return
 163                                 <literal>r = dg(t, y, ydot)/dy + cj*dg(t, y, ydot)/dydot</literal> where
 164                                 <literal>cj</literal> is a real scalar.
 165                             </para>
 166                         </listitem>
 167                         <listitem>
 168                             <para>A list.</para>
 169                             <para>It must be as follows</para>
 170                             <programlisting role="no-scilab-exec"><![CDATA[
 171 list(jac, x1, x2, ...)
 172  ]]></programlisting>
 173                             <para>where the syntax of the function
 174                                 <literal>jac</literal> is now
 175                             </para>
 176                             <programlisting role="no-scilab-exec"><![CDATA[
 177 r=jac(t, y, ydot, cj, x1, x2,...)
 178  ]]></programlisting>
 179                             <para>
 180                                 <literal>jac</literal> still returns
 181                                 <literal>dg/dy + cj*dg/dydot</literal> as a function of
 182                                 <literal>(t, y, ydot, cj, x1, x2, ...)</literal>.
 183                             </para>
 184                         </listitem>
 185                         <listitem>
 186                             <para>A character string.</para>
 187                             <para>It must refer to the name of a C function or a Fortran subroutine linked with Scilab
 188                             </para>
 189                             <para>In C, the syntax must be:</para>
 190                             <programlisting role="no-scilab-exec"><![CDATA[
 191 void jac(double *t, double y[], double yd[], double pd[],
 192          double *cj, double rpar[], int ipar[])
 193  ]]></programlisting>
 194                             <para>In Fortran, it must be:</para>
 195                             <programlisting role="no-scilab-exec"><![CDATA[
 196 subroutine jac(t, y, yd, pd, cj, rpar, ipar)
 197 double precision t, y(*), yd(*), pd(*), cj, rpar(*)
 198 integer ipar(*)
 199  ]]></programlisting>
 200                         </listitem>
 201                     </itemizedlist>
 202                 </listitem>
 203             </varlistentry>
 204             <varlistentry>
 205                 <term>surf</term>
 206                 <listitem>
 207                     <para>
 208                         <link linkend="external">external</link> (function, list or string).
 209                         Computes the value of the column vector <literal>surf(t, y)</literal> with
 210                         <literal>ng</literal> components. Each component defines a surface.
 211                         It may be defined by:
 212                     </para>
 213                     <itemizedlist>
 214                         <listitem>
 215                             <para>A Scilab function.</para>
 216                             <para>Its syntax must be
 217                                 <literal>surf(t, y)</literal>
 218                             </para>
 219                         </listitem>
 220                         <listitem>
 221                             <para>A list.</para>
 222                             <para>It must be as follows</para>
 223                             <programlisting role="no-scilab-exec"><![CDATA[
 224 list(surf, x1, x2, ...)
 225  ]]></programlisting>
 226                             <para>where the syntax of the function
 227                                 <literal>surf</literal> is now
 228                             </para>
 229                             <programlisting role="no-scilab-exec"><![CDATA[
 230 r = surf(t, y, x1, x2, ...)
 231  ]]></programlisting>
 232                         </listitem>
 233                         <listitem>
 234                             <para>A character string.</para>
 235                             <para>It must refer to the name of a C function or a Fortran subroutine linked with Scilab.
 236                             </para>
 237                             <para>In C, the syntax must be:</para>
 238                             <programlisting role="no-scilab-exec"><![CDATA[
 239 void surf(int *ny, double *t, double y[], int *ng, double gout[])
 240  ]]></programlisting>
 241                             <para>In Fortran, it must be:</para>
 242                             <programlisting role="no-scilab-exec"><![CDATA[
 243 subroutine surf(ny, t, y, ng, gout)
 244 double precision t, y(*), gout(*)
 245 integer ny, ng
 246  ]]></programlisting>
 247                         </listitem>
 248                     </itemizedlist>
 249                 </listitem>
 250             </varlistentry>
 251             <varlistentry>
 252                 <term>info</term>
 253                 <listitem>
 254                     <para>
 255                         list which contains <literal>14</literal> elements. Default value is
 256                         <literal>list([], 0, [], [], [], 0, [], 0, [], 0, 0, [], [], 1)</literal>.
 257                     </para>
 258                     <variablelist>
 259                         <varlistentry>
 260                             <term>info(1)</term>
 261                             <listitem>
 262                                 <para>real scalar which gives the maximum time for which
 263                                     <literal>g</literal> is allowed to be evaluated or an empty matrix
 264                                     <literal>[]</literal> if no limits imposed for time.
 265                                 </para>
 266                             </listitem>
 267                         </varlistentry>
 268                         <varlistentry>
 269                             <term>info(2)</term>
 270                             <listitem>
 271                                 <para>
 272                                     flag which indicates if <literal>daskr</literal> returns
 273                                     its intermediate computed values (<literal>= 1</literal>)
 274                                     or only the user specified time point values
 275                                     (<literal>= 0</literal>).
 276                                 </para>
 277                             </listitem>
 278                         </varlistentry>
 279                         <varlistentry>
 280                             <term>info(3)</term>
 281                             <listitem>
 282                                 <para>
 283                                     <literal>2</literal> components vector which give the
 284                                     definition <literal>[ml,mu]</literal> of band matrix computed
 285                                     by <literal>jac</literal>; <literal>r(i - j + ml + mu + 1,j) = "dg(i)/dy(j)+cj*dg(i)/dydot(j)"</literal>.
 286                                     If <literal>jac</literal> returns a full matrix set
 287                                     <literal>info(3)=[]</literal>.
 288                                     Treat as dummy if
 289                                     <literal>info(8)=1</literal>.
 290                                 </para>
 291                             </listitem>
 292                         </varlistentry>
 293                         <varlistentry>
 294                             <term>info(4)</term>
 295                             <listitem>
 296                                 <para>real scalar which gives the maximum step size. Set
 297                                     <literal>info(4)=[]</literal> if no limitation.
 298                                 </para>
 299                             </listitem>
 300                         </varlistentry>
 301                         <varlistentry>
 302                             <term>info(5)</term>
 303                             <listitem>
 304                                 <para>real scalar which gives the initial step size. Set
 305                                     <literal>info(5)=[]</literal> if not specified.
 306                                 </para>
 307                             </listitem>
 308                         </varlistentry>
 309                         <varlistentry>
 310                             <term>info(6)</term>
 311                             <listitem>
 312                                 <para>
 313                                     set <literal>info(6)=1</literal> if the solution is
 314                                     known to be non negative, else set
 315                                     <literal>info(6)=0</literal>.
 316                                 </para>
 317                             </listitem>
 318                         </varlistentry>
 319                         <varlistentry>
 320                             <term>info(7)</term>
 321                             <listitem>
 322                                 <para>
 323                                     if ydot0 is set so that
 324                                     <literal>g(t0, y0, ydot0) = 0</literal> then set
 325                                     <literal>info(7)=[]</literal>. Otherwise, set
 326                                     <literal>info(7)=[+-1, ..., +-1]</literal>, with
 327                                     <literal>info(7)(i) = 1</literal> if y(i) is a differential variable and
 328                                     <literal>info(7)(i) = -1</literal> if y(i) is an algebraic variable
 329                                     (if its derivatives do not appear explicitly in the function g(t, y, ydot)).
 330                                 </para>
 331                             </listitem>
 332                         </varlistentry>
 333                         <varlistentry>
 334                             <term>info(8)</term>
 335                             <listitem>
 336                                 <para>
 337                                     direct / Krylov method. Set <literal>info(8)=1</literal> and provide a routine in <literal>psol</literal>
 338                                     if you want the solver to use Krylov iterations, else (daskr's direct method) set
 339                                     <literal>info(8)=0</literal>.
 340                                 </para>
 341                             </listitem>
 342                         </varlistentry>
 343                         <varlistentry>
 344                             <term>info(9)</term>
 345                             <listitem>
 346                                 <para>
 347                                     Krylov parameters. Treat as dummy argument if you have set
 348                                     <literal>info(8)=0</literal>. Otherwise, set
 349                                     <literal>info(9)=[]</literal> or
 350                                     <literal>info(9)=[maxl kmp nrmax epli]</literal>, where:
 351                                 </para>
 352                                 <para>
 353                                     - maxl = maximum number of iterations in the GMRes algorithm (default
 354                                     <literal>min(5, neq)</literal>),
 355                                 </para>
 356                                 <para>
 357                                     - kmp = number of vectors on which orthogonalization is done in the GMRes algorithm
 358                                     (default maxl),
 359                                 </para>
 360                                 <para>
 361                                     - nrmax = maximum number of restarts of the GMRes algorithm per nonlinear iteration
 362                                     (default <literal>5</literal>),
 363                                 </para>
 364                                 <para>
 365                                     - epli = convergence test constant in GMRes algorithm (default <literal>0.05</literal>).
 366                                 </para>
 367                             </listitem>
 368                         </varlistentry>
 369                         <varlistentry>
 370                             <term>info(10)</term>
 371                             <listitem>
 372                                 <para>
 373                                     initial conditions. Treat as dummy argument if
 374                                     <literal>info(7)=[]</literal>. Set
 375                                     <literal>info(10)=1</literal> if the solver should stop right after
 376                                     computation of the initial conditions, else set
 377                                     <literal>info(10)=0</literal>.
 378                                 </para>
 379                             </listitem>
 380                         </varlistentry>
 381                         <varlistentry>
 382                             <term>info(11)</term>
 383                             <listitem>
 384                                 <para>
 385                                     preconditioner computation and LU-factorization routine for <literal>psol</literal>.
 386                                     Treat as dummy argument if <literal>info(8)=0</literal>. Set
 387                                     <literal>info(11)=1</literal> and provide a <literal>pjac</literal> routine if the
 388                                     <link linkend="external">external</link> <literal>psol</literal> should use a specific routine, else set
 389                                     <literal>info(11)=0</literal>.
 390                                 </para>
 391                             </listitem>
 392                         </varlistentry>
 393                         <varlistentry>
 394                             <term>info(12)</term>
 395                             <listitem>
 396                                 <para>
 397                                     if you wish to control errors locally on all the variables then set
 398                                     <literal>info(12)=[]</literal>. Otherwise, set
 399                                     <literal>info(12)=[+-1, ..., +-1]</literal>, with
 400                                     <literal>info(12)(i) = 1</literal> if y(i) is a differential variable and
 401                                     <literal>info(12)(i) = -1</literal> if y(i) is an algebraic variable
 402                                     (if its derivatives do not appear explicitly in the function g(t, y, ydot)).
 403                                 </para>
 404                             </listitem>
 405                         </varlistentry>
 406                         <varlistentry>
 407                             <term>info(13)</term>
 408                             <listitem>
 409                                 <para>
 410                                     heuristic parameters. Treat as dummy argument if
 411                                     <literal>info(7)=[]</literal>. Otherwise, set
 412                                     <literal>info(13)=[]</literal> or
 413                                     <literal>info(13)=[mxnit mxnj mxnh lsoff stptol epinit]</literal>, where:
 414                                 </para>
 415                                 <para>
 416                                     - mxnit = maximum number of Newton iterations per Jacobian or preconditioner evaluation (default
 417                                     <literal>5</literal> if <literal>info(8)=0</literal>, <literal>15</literal> otherwise),
 418                                 </para>
 419                                 <para>
 420                                     - mxnj = maximum number of Jacobian or preconditioner evaluations (default
 421                                     <literal>6</literal> if <literal>info(8)=0</literal>, <literal>2</literal> otherwise),
 422                                 </para>
 423                                 <para>
 424                                     - mxnh = maximum number of values of the artificial stepsize
 425                                        parameter <literal>h</literal> to be tried if info(7) ≠ []
 426                                        (default <literal>5</literal>),
 427                                 </para>
 428                                 <para>
 429                                     - lsoff = flag to turn off the linesearch algorithm (lsoff = 0 means linesearch is on, lsoff = 1 means it is turned off)
 430                                     (default <literal>0</literal>),
 431                                 </para>
 432                                 <para>
 433                                     - stptol = minimum scaled step in linesearch algorithm (default <literal>(unit roundoff)^(2/3)</literal>),
 434                                 </para>
 435                                 <para>
 436                                     - epinit = swing factor in the Newton iteration convergence test (default <literal>0.01</literal>).
 437                                 </para>
 438                             </listitem>
 439                         </varlistentry>
 440                         <varlistentry>
 441                             <term>info(14)</term>
 442                             <listitem>
 443                                 <para>
 444                                     verbosity. Set <literal>info(14)=1</literal> for minimal extra printing,
 445                                     <literal>info(14)=2</literal> for full printing, else set
 446                                     <literal>info(14)=0</literal>.
 447                                 </para>
 448                             </listitem>
 449                         </varlistentry>
 450                     </variablelist>
 451                 </listitem>
 452             </varlistentry>
 453             <varlistentry>
 454                 <term>psol</term>
 455                 <listitem>
 456                     <para>
 457                         <link linkend="external">external</link> (function, list or string). Solves a linear system
 458                         <literal>P*x = b</literal>, with P being the factored preconditioner that routine <literal>pjac</literal>
 459                         computed beforehand and stored in <literal>wp</literal> and <literal>iwp</literal>.
 460                     </para>
 461                     <itemizedlist>
 462                         <listitem>
 463                             <para>A Scilab function.</para>
 464                             <para>Its syntax must be
 465                                 <literal>[r, ier] = psol(wp, iwp, b)</literal> and must return the solution of the system in
 466                                 <literal>r</literal> and an error flag <literal>ier</literal>.
 467                             </para>
 468                         </listitem>
 469                         <listitem>
 470                             <para>A list.</para>
 471                             <para>It must be as follows:</para>
 472                             <programlisting role="no-scilab-exec"><![CDATA[
 473 list(psol, x1, x2, ...)
 474  ]]></programlisting>
 475                             <para>
 476                                 where the syntax of <literal>psol</literal> is now
 477                             </para>
 478                             <programlisting role="no-scilab-exec"><![CDATA[
 479 psol(wp, iwp, b, x1, x2, ...)
 480  ]]></programlisting>
 481                             <para>
 482                                 <literal>psol</literal> still returns the solution in <literal>r</literal>.
 483                             </para>
 484                         </listitem>
 485                         <listitem>
 486                             <para>A character string.</para>
 487                             <para>It must refer to the name of a C function or a Fortran subroutine linked with Scilab
 488                             </para>
 489                             <para>In C, the syntax must be:</para>
 490                             <programlisting role="no-scilab-exec"><![CDATA[
 491 void psol (int*neq, double*t, double*y, double*ydot, double*savr,
 492   double*wk, double*cj, double*wght, double*wp, int*iwp, double*b, double*eplin, int*ier, double*rpar, int*ipar)
 493  ]]></programlisting>
 494                             where the arrays <literal>wp</literal> and <literal>iwp</literal> contain matrix elements of LU-factored preconditioner
 495                             <literal>P</literal>, <literal>wp</literal> being the values and
 496                             <literal>iwp</literal> the pivots used in the factorization.
 497                             <para>In Fortran, it must be:</para>
 498                             <programlisting role="no-scilab-exec"><![CDATA[
 499 subroutine psol (neq, t, y, ydot, savr, wk, cj, wght,
 500   wp, iwp, b, eplin, ier, rpar, ipar)
 501 double precision t,y(*), ydot(*), savr(*), wk(*), cj, wght(*), wp(*),
 502   b(*), eplin, rpar(*)
 503 integer neq, iwp(*), ier, ipar(*)
 504  ]]></programlisting>
 505                         </listitem>
 506                     </itemizedlist>
 507                 </listitem>
 508             </varlistentry>
 509             <varlistentry>
 510                 <term>pjac</term>
 511                 <listitem>
 512                     <para>
 513                         <link linkend="external">external</link> (function, list or string). Computes the value of
 514                         <literal>dg/dy + cj*dg/dydot</literal> for a given value of parameter
 515                         <literal>cj</literal> and LU-factorizes it in two arrays, real and integer.
 516                     </para>
 517                     <itemizedlist>
 518                         <listitem>
 519                             <para>A Scilab function.</para>
 520                             <para>Its syntax must be
 521                                 <literal>[wp, iwp, ires] = pjac(neq, t, y, ydot, h, cj, rewt, savr)</literal> and in return,
 522                                 the arrays <literal>wp</literal> and <literal>iwp</literal> must contain all factored preconditioner information.
 523                             </para>
 524                         </listitem>
 525                         <listitem>
 526                             <para>A list.</para>
 527                             <para>It must be as follows</para>
 528                             <programlisting role="no-scilab-exec"><![CDATA[
 529 list(pjac, x1, x2, ...)
 530  ]]></programlisting>
 531                             <para>
 532                                 where the syntax of <literal>pjac</literal> is
 533                             </para>
 534                             <programlisting role="no-scilab-exec"><![CDATA[
 535 pjac(neq, t, y, ydot, h, cj, rewt, savr, x1, x2,...)
 536  ]]></programlisting>
 537                             <para>
 538                                 <literal>pjac</literal> still returns factorized
 539                                 <literal>dg/dy + cj*dg/dydot</literal> as a function of
 540                                 <literal>(neq, t, y, ydot, h, cj, rewt, savr, x1, x2, ...)</literal>.
 541                             </para>
 542                         </listitem>
 543                         <listitem>
 544                             <para>A character string.</para>
 545                             <para>It must refer to the name of a C function or a Fortran subroutine linked with Scilab
 546                             </para>
 547                             <para>In C, the syntax must be:</para>
 548                             <programlisting role="no-scilab-exec"><![CDATA[
 549 void pjac (double*res, int*ires, int*neq, double*t, double*y, double*ydot, double*rewt, double*savr,
 550 double*wk, double*h, double*cj, double*wp, int*iwp, int*ier, double*rpar, int*ipar)
 551  ]]></programlisting>
 552                             <para>In Fortran, it must be:</para>
 553                             <programlisting role="no-scilab-exec"><![CDATA[
 554 subroutine pjac (res, ires, neq, t, y, ydot, rewt, savr,
 555   wk, h, cj, wp, iwp, ier, rpar, ipar)
 556 double precision res(*), t, y(*), ydot(*), rewt(*), savr(*),
 557   wk(*), h, cj, wp(*), rpar(*)
 558 integer ires, neq, iwp(*), ier, ipar(*)
 559  ]]></programlisting>
 560                         </listitem>
 561                     </itemizedlist>
 562                 </listitem>
 563             </varlistentry>
 564             <varlistentry>
 565                 <term>hd</term>
 566                 <listitem>
 567                     <para>
 568                         real vector which allows to store the <literal>daskr</literal>
 569                         context and to resume integration.
 570                     </para>
 571                 </listitem>
 572             </varlistentry>
 573             <varlistentry>
 574                 <term>r</term>
 575                 <listitem>
 576                     <para>
 577                         real matrix. Each column is the vector <literal>[t; x(t); xdot(t)]</literal> where
 578                         <literal>t</literal> is the time index for which the solution has been computed,
 579                         <literal>x(t)</literal> is the value of the computed solution,
 580                         <literal>xdot(t)</literal> is the derivative of the computed solution.
 581                     </para>
 582                 </listitem>
 583             </varlistentry>
 584             <varlistentry>
 585                 <term>nn</term>
 586                 <listitem>
 587                     <para>
 588                         a vector with two entries <literal>[times num]</literal>,
 589                         <literal>times</literal> is the value of the time at which the surface is crossed,
 590                         <literal>num</literal> is the number of the crossed surface.
 591                     </para>
 592                 </listitem>
 593             </varlistentry>
 594         </variablelist>
 595     </refsection>
 596     <refsection>
 597         <title>Description</title>
 598         <para>Solution of the implicit differential equation:</para>
 599         <programlisting role="no-scilab-exec"><![CDATA[
 600 g(t, y, ydot) = 0
 601 y(t0) = y0  and  ydot(t0) = ydot0
 602  ]]></programlisting>
 603         <para>Returns the surface crossing instants and the number of the surface
 604             reached in <literal>nn</literal>.
 605         </para>
 606         <para>Detailed examples can be found in SCI/modules/differential_equations/tests/unit_tests/daskr.tst</para>
 607     </refsection>
 608     <refsection>
 609         <title>Examples</title>
 610         <programlisting role="example"><![CDATA[
 611 // dy/dt = ((2*log(y)+8)/t-5)*y,  y(1) = 1,  1 <= t <= 6
 612 // g1 = ((2*log(y)+8)/t-5)*y
 613 // g2 = log(y) - 2.2491
 614 y0 = 1; t = 2:6; t0 = 1; y0d = 3;
 615 atol = 1.d-6; rtol = 0; ng = 2;
 616
 617 deff('[delta, ires] = res1(t, y, ydot)', 'ires = 0; delta = ydot-((2*log(y)+8)/t-5)*y')
 618 deff('[rts] = gr1(t, y)', 'rts = [((2*log(y)+8)/t-5)*y; log(y)-2.2491]')
 619
 620 [yy, nn] = daskr([y0, y0d], t0, t, atol, rtol, res1, ng, gr1);
 621 nn
 622
 623 // Should return nn = [2.4698972 2]
 624  ]]></programlisting>
 625     </refsection>
 626     <refsection role="see also">
 627         <title>See also</title>
 628         <simplelist type="inline">
 629             <member>
 630                 <link linkend="ode">ode</link>
 631             </member>
 632             <member>
 633                 <link linkend="dasrt">dasrt</link>
 634             </member>
 635             <member>
 636                 <link linkend="dassl">dassl</link>
 637             </member>
 638             <member>
 639                 <link linkend="impl">impl</link>
 640             </member>
 641             <member>
 642                 <link linkend="call">call</link>
 643             </member>
 644             <member>
 645                 <link linkend="link">link</link>
 646             </member>
 647             <member>
 648                 <link linkend="external">external</link>
 649             </member>
 650         </simplelist>
 651     </refsection>
 652     <refsection>
 653         <title>History</title>
 654         <revhistory>
 655             <revision>
 656                 <revnumber>5.5.0</revnumber>
 657                 <revdescription>daskr solver added</revdescription>
 658             </revision>
 659         </revhistory>
 660     </refsection>
 661 </refentry>