Actual source code: ex4.c

petsc-3.3-p7 2013-05-11
```  2: /* Program usage:  mpiexec -n <procs> ex4 [-help] [all PETSc options] */

4: static char help[] ="Solves a simple time-dependent linear PDE (the heat equation).\n\
5: Input parameters include:\n\
6:   -m <points>, where <points> = number of grid points\n\
7:   -time_dependent_rhs : Treat the problem as having a time-dependent right-hand side\n\
8:   -debug              : Activate debugging printouts\n\
9:   -nox                : Deactivate x-window graphics\n\n";

11: /*
12:    Concepts: TS^time-dependent linear problems
13:    Concepts: TS^heat equation
14:    Concepts: TS^diffusion equation
15:    Processors: n
16: */

18: /* ------------------------------------------------------------------------

20:    This program solves the one-dimensional heat equation (also called the
21:    diffusion equation),
22:        u_t = u_xx,
23:    on the domain 0 <= x <= 1, with the boundary conditions
24:        u(t,0) = 0, u(t,1) = 0,
25:    and the initial condition
26:        u(0,x) = sin(6*pi*x) + 3*sin(2*pi*x).
27:    This is a linear, second-order, parabolic equation.

29:    We discretize the right-hand side using finite differences with
30:    uniform grid spacing h:
31:        u_xx = (u_{i+1} - 2u_{i} + u_{i-1})/(h^2)
32:    We then demonstrate time evolution using the various TS methods by
33:    running the program via
34:        mpiexec -n <procs> ex3 -ts_type <timestepping solver>

36:    We compare the approximate solution with the exact solution, given by
37:        u_exact(x,t) = exp(-36*pi*pi*t) * sin(6*pi*x) +
38:                       3*exp(-4*pi*pi*t) * sin(2*pi*x)

40:    Notes:
41:    This code demonstrates the TS solver interface to two variants of
42:    linear problems, u_t = f(u,t), namely
43:      - time-dependent f:   f(u,t) is a function of t
44:      - time-independent f: f(u,t) is simply f(u)

46:     The uniprocessor version of this code is ts/examples/tutorials/ex3.c

48:   ------------------------------------------------------------------------- */

50: /*
51:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs) to manage
52:    the parallel grid.  Include "petscts.h" so that we can use TS solvers.
53:    Note that this file automatically includes:
54:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
55:      petscmat.h  - matrices
56:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
57:      petscviewer.h - viewers               petscpc.h   - preconditioners
58:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
59: */

61: #include <petscdmda.h>
62: #include <petscts.h>

64: /*
65:    User-defined application context - contains data needed by the
66:    application-provided call-back routines.
67: */
68: typedef struct {
69:   MPI_Comm    comm;              /* communicator */
70:   DM          da;                /* distributed array data structure */
71:   Vec         localwork;         /* local ghosted work vector */
72:   Vec         u_local;           /* local ghosted approximate solution vector */
73:   Vec         solution;          /* global exact solution vector */
74:   PetscInt    m;                 /* total number of grid points */
75:   PetscReal   h;                 /* mesh width h = 1/(m-1) */
76:   PetscBool   debug;             /* flag (1 indicates activation of debugging printouts) */
77:   PetscViewer viewer1,viewer2;  /* viewers for the solution and error */
78:   PetscReal   norm_2,norm_max;  /* error norms */
79: } AppCtx;

81: /*
82:    User-defined routines
83: */
84: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
85: extern PetscErrorCode RHSMatrixHeat(TS,PetscReal,Vec,Mat*,Mat*,MatStructure*,void*);
86: extern PetscErrorCode RHSFunctionHeat(TS,PetscReal,Vec,Vec,void*);
87: extern PetscErrorCode Monitor(TS,PetscInt,PetscReal,Vec,void*);
88: extern PetscErrorCode ExactSolution(PetscReal,Vec,AppCtx*);

92: int main(int argc,char **argv)
93: {
94:   AppCtx         appctx;                 /* user-defined application context */
95:   TS             ts;                     /* timestepping context */
96:   Mat            A;                      /* matrix data structure */
97:   Vec            u;                      /* approximate solution vector */
98:   PetscReal      time_total_max = 1.0;   /* default max total time */
99:   PetscInt       time_steps_max = 100;   /* default max timesteps */
100:   PetscDraw      draw;                   /* drawing context */
102:   PetscInt       steps,m;
103:   PetscMPIInt    size;
104:   PetscReal      dt,ftime;
105:   PetscBool      flg;
106:   TSProblemType  tsproblem = TS_LINEAR;

108:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
109:      Initialize program and set problem parameters
110:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
111:
112:   PetscInitialize(&argc,&argv,(char*)0,help);
113:   appctx.comm = PETSC_COMM_WORLD;

115:   m    = 60;
116:   PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
117:   PetscOptionsHasName(PETSC_NULL,"-debug",&appctx.debug);
118:   appctx.m        = m;
119:   appctx.h        = 1.0/(m-1.0);
120:   appctx.norm_2   = 0.0;
121:   appctx.norm_max = 0.0;
122:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
123:   PetscPrintf(PETSC_COMM_WORLD,"Solving a linear TS problem, number of processors = %d\n",size);

125:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
126:      Create vector data structures
127:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
128:   /*
129:      Create distributed array (DMDA) to manage parallel grid and vectors
130:      and to set up the ghost point communication pattern.  There are M
131:      total grid values spread equally among all the processors.
132:   */

134:   DMDACreate1d(PETSC_COMM_WORLD,DMDA_BOUNDARY_NONE,m,1,1,PETSC_NULL,&appctx.da);

136:   /*
137:      Extract global and local vectors from DMDA; we use these to store the
138:      approximate solution.  Then duplicate these for remaining vectors that
139:      have the same types.
140:   */
141:   DMCreateGlobalVector(appctx.da,&u);
142:   DMCreateLocalVector(appctx.da,&appctx.u_local);

144:   /*
145:      Create local work vector for use in evaluating right-hand-side function;
146:      create global work vector for storing exact solution.
147:   */
148:   VecDuplicate(appctx.u_local,&appctx.localwork);
149:   VecDuplicate(u,&appctx.solution);

151:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152:      Set up displays to show graphs of the solution and error
153:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

155:   PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,380,400,160,&appctx.viewer1);
156:   PetscViewerDrawGetDraw(appctx.viewer1,0,&draw);
157:   PetscDrawSetDoubleBuffer(draw);
158:   PetscViewerDrawOpen(PETSC_COMM_WORLD,0,"",80,0,400,160,&appctx.viewer2);
159:   PetscViewerDrawGetDraw(appctx.viewer2,0,&draw);
160:   PetscDrawSetDoubleBuffer(draw);

162:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
163:      Create timestepping solver context
164:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

166:   TSCreate(PETSC_COMM_WORLD,&ts);

168:   flg = PETSC_FALSE;
169:   PetscOptionsGetBool(PETSC_NULL,"-nonlinear",&flg,PETSC_NULL);
170:   TSSetProblemType(ts,flg ? TS_NONLINEAR : TS_LINEAR);

172:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
173:      Set optional user-defined monitoring routine
174:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
175:   TSMonitorSet(ts,Monitor,&appctx,PETSC_NULL);

177:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

179:      Create matrix data structure; set matrix evaluation routine.
180:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

182:   MatCreate(PETSC_COMM_WORLD,&A);
183:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m,m);
184:   MatSetFromOptions(A);
185:   MatSetUp(A);

187:   flg = PETSC_FALSE;
188:   PetscOptionsGetBool(PETSC_NULL,"-time_dependent_rhs",&flg,PETSC_NULL);
189:   if (flg) {
190:     /*
191:        For linear problems with a time-dependent f(u,t) in the equation
192:        u_t = f(u,t), the user provides the discretized right-hand-side
193:        as a time-dependent matrix.
194:     */
195:     TSSetRHSFunction(ts,PETSC_NULL,TSComputeRHSFunctionLinear,&appctx);
196:     TSSetRHSJacobian(ts,A,A,RHSMatrixHeat,&appctx);
197:   } else {
198:     /*
199:        For linear problems with a time-independent f(u) in the equation
200:        u_t = f(u), the user provides the discretized right-hand-side
201:        as a matrix only once, and then sets a null matrix evaluation
202:        routine.
203:     */
204:     MatStructure A_structure;
205:     RHSMatrixHeat(ts,0.0,u,&A,&A,&A_structure,&appctx);
206:     TSSetRHSFunction(ts,PETSC_NULL,TSComputeRHSFunctionLinear,&appctx);
207:     TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,&appctx);
208:   }
209:
210:   if (tsproblem == TS_NONLINEAR) {
211:     SNES snes;
212:     TSSetRHSFunction(ts,PETSC_NULL,RHSFunctionHeat,&appctx);
213:     TSGetSNES(ts,&snes);
214:     SNESSetJacobian(snes,PETSC_NULL,PETSC_NULL,SNESDefaultComputeJacobian,PETSC_NULL);
215:   }

217:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
218:      Set solution vector and initial timestep
219:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

221:   dt = appctx.h*appctx.h/2.0;
222:   TSSetInitialTimeStep(ts,0.0,dt);
223:   TSSetSolution(ts,u);

225:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
226:      Customize timestepping solver:
227:        - Set the solution method to be the Backward Euler method.
228:        - Set timestepping duration info
229:      Then set runtime options, which can override these defaults.
230:      For example,
231:           -ts_max_steps <maxsteps> -ts_final_time <maxtime>
232:      to override the defaults set by TSSetDuration().
233:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

235:   TSSetDuration(ts,time_steps_max,time_total_max);
236:   TSSetFromOptions(ts);

238:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
239:      Solve the problem
240:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

242:   /*
243:      Evaluate initial conditions
244:   */
245:   InitialConditions(u,&appctx);

247:   /*
248:      Run the timestepping solver
249:   */
250:   TSSolve(ts,u,&ftime);
251:   TSGetTimeStepNumber(ts,&steps);

253:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
254:      View timestepping solver info
255:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
256:   PetscPrintf(PETSC_COMM_WORLD,"Total timesteps %D, Final time %G\n",steps,ftime);
257:   PetscPrintf(PETSC_COMM_WORLD,"Avg. error (2 norm) = %G Avg. error (max norm) = %G\n",appctx.norm_2/steps,appctx.norm_max/steps);

259:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
260:      Free work space.  All PETSc objects should be destroyed when they
261:      are no longer needed.
262:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

264:   TSDestroy(&ts);
265:   MatDestroy(&A);
266:   VecDestroy(&u);
267:   PetscViewerDestroy(&appctx.viewer1);
268:   PetscViewerDestroy(&appctx.viewer2);
269:   VecDestroy(&appctx.localwork);
270:   VecDestroy(&appctx.solution);
271:   VecDestroy(&appctx.u_local);
272:   DMDestroy(&appctx.da);

274:   /*
275:      Always call PetscFinalize() before exiting a program.  This routine
276:        - finalizes the PETSc libraries as well as MPI
277:        - provides summary and diagnostic information if certain runtime
278:          options are chosen (e.g., -log_summary).
279:   */
280:   PetscFinalize();
281:   return 0;
282: }
283: /* --------------------------------------------------------------------- */
286: /*
287:    InitialConditions - Computes the solution at the initial time.

289:    Input Parameter:
290:    u - uninitialized solution vector (global)
291:    appctx - user-defined application context

293:    Output Parameter:
294:    u - vector with solution at initial time (global)
295: */
296: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
297: {
298:   PetscScalar    *u_localptr,h = appctx->h;
299:   PetscInt       i,mybase,myend;

302:   /*
303:      Determine starting point of each processor's range of
304:      grid values.
305:   */
306:   VecGetOwnershipRange(u,&mybase,&myend);

308:   /*
309:     Get a pointer to vector data.
310:     - For default PETSc vectors, VecGetArray() returns a pointer to
311:       the data array.  Otherwise, the routine is implementation dependent.
312:     - You MUST call VecRestoreArray() when you no longer need access to
313:       the array.
314:     - Note that the Fortran interface to VecGetArray() differs from the
315:       C version.  See the users manual for details.
316:   */
317:   VecGetArray(u,&u_localptr);

319:   /*
320:      We initialize the solution array by simply writing the solution
321:      directly into the array locations.  Alternatively, we could use
322:      VecSetValues() or VecSetValuesLocal().
323:   */
324:   for (i=mybase; i<myend; i++) {
325:     u_localptr[i-mybase] = PetscSinScalar(PETSC_PI*i*6.*h) + 3.*PetscSinScalar(PETSC_PI*i*2.*h);
326:   }

328:   /*
329:      Restore vector
330:   */
331:   VecRestoreArray(u,&u_localptr);

333:   /*
334:      Print debugging information if desired
335:   */
336:   if (appctx->debug) {
337:      PetscPrintf(appctx->comm,"initial guess vector\n");
338:      VecView(u,PETSC_VIEWER_STDOUT_WORLD);
339:   }

341:   return 0;
342: }
343: /* --------------------------------------------------------------------- */
346: /*
347:    ExactSolution - Computes the exact solution at a given time.

349:    Input Parameters:
350:    t - current time
351:    solution - vector in which exact solution will be computed
352:    appctx - user-defined application context

354:    Output Parameter:
355:    solution - vector with the newly computed exact solution
356: */
357: PetscErrorCode ExactSolution(PetscReal t,Vec solution,AppCtx *appctx)
358: {
359:   PetscScalar    *s_localptr,h = appctx->h,ex1,ex2,sc1,sc2;
360:   PetscInt       i,mybase,myend;

363:   /*
364:      Determine starting and ending points of each processor's
365:      range of grid values
366:   */
367:   VecGetOwnershipRange(solution,&mybase,&myend);

369:   /*
370:      Get a pointer to vector data.
371:   */
372:   VecGetArray(solution,&s_localptr);

374:   /*
375:      Simply write the solution directly into the array locations.
376:      Alternatively, we culd use VecSetValues() or VecSetValuesLocal().
377:   */
378:   ex1 = exp(-36.*PETSC_PI*PETSC_PI*t); ex2 = exp(-4.*PETSC_PI*PETSC_PI*t);
379:   sc1 = PETSC_PI*6.*h;                 sc2 = PETSC_PI*2.*h;
380:   for (i=mybase; i<myend; i++) {
381:     s_localptr[i-mybase] = PetscSinScalar(sc1*(PetscReal)i)*ex1 + 3.*PetscSinScalar(sc2*(PetscReal)i)*ex2;
382:   }

384:   /*
385:      Restore vector
386:   */
387:   VecRestoreArray(solution,&s_localptr);
388:   return 0;
389: }
390: /* --------------------------------------------------------------------- */
393: /*
394:    Monitor - User-provided routine to monitor the solution computed at
395:    each timestep.  This example plots the solution and computes the
396:    error in two different norms.

398:    Input Parameters:
399:    ts     - the timestep context
400:    step   - the count of the current step (with 0 meaning the
401:              initial condition)
402:    time   - the current time
403:    u      - the solution at this timestep
404:    ctx    - the user-provided context for this monitoring routine.
405:             In this case we use the application context which contains
406:             information about the problem size, workspace and the exact
407:             solution.
408: */
409: PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal time,Vec u,void *ctx)
410: {
411:   AppCtx         *appctx = (AppCtx*) ctx;   /* user-defined application context */
413:   PetscReal      norm_2,norm_max;

415:   /*
416:      View a graph of the current iterate
417:   */
418:   VecView(u,appctx->viewer2);

420:   /*
421:      Compute the exact solution
422:   */
423:   ExactSolution(time,appctx->solution,appctx);

425:   /*
426:      Print debugging information if desired
427:   */
428:   if (appctx->debug) {
429:      PetscPrintf(appctx->comm,"Computed solution vector\n");
430:      VecView(u,PETSC_VIEWER_STDOUT_WORLD);
431:      PetscPrintf(appctx->comm,"Exact solution vector\n");
432:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
433:   }

435:   /*
436:      Compute the 2-norm and max-norm of the error
437:   */
438:   VecAXPY(appctx->solution,-1.0,u);
439:   VecNorm(appctx->solution,NORM_2,&norm_2);
440:   norm_2 = PetscSqrtReal(appctx->h)*norm_2;
441:   VecNorm(appctx->solution,NORM_MAX,&norm_max);

443:   /*
444:      PetscPrintf() causes only the first processor in this
445:      communicator to print the timestep information.
446:   */
447:   PetscPrintf(appctx->comm,"Timestep %D: time = %G 2-norm error = %G max norm error = %G\n",
448:               step,time,norm_2,norm_max);
449:   appctx->norm_2   += norm_2;
450:   appctx->norm_max += norm_max;

452:   /*
453:      View a graph of the error
454:   */
455:   VecView(appctx->solution,appctx->viewer1);

457:   /*
458:      Print debugging information if desired
459:   */
460:   if (appctx->debug) {
461:      PetscPrintf(appctx->comm,"Error vector\n");
462:      VecView(appctx->solution,PETSC_VIEWER_STDOUT_WORLD);
463:   }

465:   return 0;
466: }

468: /* --------------------------------------------------------------------- */
471: /*
472:    RHSMatrixHeat - User-provided routine to compute the right-hand-side
473:    matrix for the heat equation.

475:    Input Parameters:
476:    ts - the TS context
477:    t - current time
478:    global_in - global input vector
479:    dummy - optional user-defined context, as set by TSetRHSJacobian()

481:    Output Parameters:
482:    AA - Jacobian matrix
483:    BB - optionally different preconditioning matrix
484:    str - flag indicating matrix structure

486:   Notes:
487:   RHSMatrixHeat computes entries for the locally owned part of the system.
488:    - Currently, all PETSc parallel matrix formats are partitioned by
489:      contiguous chunks of rows across the processors.
490:    - Each processor needs to insert only elements that it owns
491:      locally (but any non-local elements will be sent to the
492:      appropriate processor during matrix assembly).
493:    - Always specify global row and columns of matrix entries when
494:      using MatSetValues(); we could alternatively use MatSetValuesLocal().
495:    - Here, we set all entries for a particular row at once.
496:    - Note that MatSetValues() uses 0-based row and column numbers
497:      in Fortran as well as in C.
498: */
499: PetscErrorCode RHSMatrixHeat(TS ts,PetscReal t,Vec X,Mat *AA,Mat *BB,MatStructure *str,void *ctx)
500: {
501:   Mat            A = *AA;                      /* Jacobian matrix */
502:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined application context */
504:   PetscInt       i,mstart,mend,idx[3];
505:   PetscScalar    v[3],stwo = -2./(appctx->h*appctx->h),sone = -.5*stwo;

507:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
508:      Compute entries for the locally owned part of the matrix
509:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

511:   MatGetOwnershipRange(A,&mstart,&mend);

513:   /*
514:      Set matrix rows corresponding to boundary data
515:   */

517:   if (mstart == 0) {  /* first processor only */
518:     v[0] = 1.0;
519:     MatSetValues(A,1,&mstart,1,&mstart,v,INSERT_VALUES);
520:     mstart++;
521:   }

523:   if (mend == appctx->m) { /* last processor only */
524:     mend--;
525:     v[0] = 1.0;
526:     MatSetValues(A,1,&mend,1,&mend,v,INSERT_VALUES);
527:   }

529:   /*
530:      Set matrix rows corresponding to interior data.  We construct the
531:      matrix one row at a time.
532:   */
533:   v[0] = sone; v[1] = stwo; v[2] = sone;
534:   for (i=mstart; i<mend; i++) {
535:     idx[0] = i-1; idx[1] = i; idx[2] = i+1;
536:     MatSetValues(A,1,&i,3,idx,v,INSERT_VALUES);
537:   }

539:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
540:      Complete the matrix assembly process and set some options
541:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
542:   /*
543:      Assemble matrix, using the 2-step process:
544:        MatAssemblyBegin(), MatAssemblyEnd()
545:      Computations can be done while messages are in transition
546:      by placing code between these two statements.
547:   */
548:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
549:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

551:   /*
552:      Set flag to indicate that the Jacobian matrix retains an identical
553:      nonzero structure throughout all timestepping iterations (although the
554:      values of the entries change). Thus, we can save some work in setting
555:      up the preconditioner (e.g., no need to redo symbolic factorization for
556:      ILU/ICC preconditioners).
557:       - If the nonzero structure of the matrix is different during
558:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
559:         must be used instead.  If you are unsure whether the matrix
560:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
561:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
562:         believes your assertion and does not check the structure
563:         of the matrix.  If you erroneously claim that the structure
564:         is the same when it actually is not, the new preconditioner
565:         will not function correctly.  Thus, use this optimization
566:         feature with caution!
567:   */
568:   *str = SAME_NONZERO_PATTERN;

570:   /*
571:      Set and option to indicate that we will never add a new nonzero location
572:      to the matrix. If we do, it will generate an error.
573:   */
574:   MatSetOption(A,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);

576:   return 0;
577: }

581: PetscErrorCode RHSFunctionHeat(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
582: {
584:   Mat            A;
585:   MatStructure   A_structure;

588:   TSGetRHSJacobian(ts,&A,PETSC_NULL,PETSC_NULL,&ctx);
589:   RHSMatrixHeat(ts,t,globalin,&A,PETSC_NULL,&A_structure,ctx);
590:   /* MatView(A,PETSC_VIEWER_STDOUT_WORLD); */
591:   MatMult(A,globalin,globalout);
592:   return(0);
593: }

```