Actual source code: dt.c

petsc-master 2015-07-31
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  1: /* Discretization tools */

  3: #include <petscconf.h>
  4: #if defined(PETSC_HAVE_MATHIMF_H)
  5: #include <mathimf.h>           /* this needs to be included before math.h */
  6: #endif
  7: #ifdef PETSC_HAVE_MPFR
  8: #include <mpfr.h>
  9: #endif

 11: #include <petscdt.h>            /*I "petscdt.h" I*/
 12: #include <petscblaslapack.h>
 13: #include <petsc/private/petscimpl.h>
 14: #include <petsc/private/dtimpl.h>
 15: #include <petscviewer.h>
 16: #include <petscdmplex.h>
 17: #include <petscdmshell.h>

 19: static PetscBool GaussCite       = PETSC_FALSE;
 20: const char       GaussCitation[] = "@article{GolubWelsch1969,\n"
 21:                                    "  author  = {Golub and Welsch},\n"
 22:                                    "  title   = {Calculation of Quadrature Rules},\n"
 23:                                    "  journal = {Math. Comp.},\n"
 24:                                    "  volume  = {23},\n"
 25:                                    "  number  = {106},\n"
 26:                                    "  pages   = {221--230},\n"
 27:                                    "  year    = {1969}\n}\n";

 31: /*@
 32:   PetscQuadratureCreate - Create a PetscQuadrature object

 34:   Collective on MPI_Comm

 36:   Input Parameter:
 37: . comm - The communicator for the PetscQuadrature object

 39:   Output Parameter:
 40: . q  - The PetscQuadrature object

 42:   Level: beginner

 44: .keywords: PetscQuadrature, quadrature, create
 45: .seealso: PetscQuadratureDestroy(), PetscQuadratureGetData()
 46: @*/
 47: PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
 48: {

 53:   DMInitializePackage();
 54:   PetscHeaderCreate(*q,PETSC_OBJECT_CLASSID,"PetscQuadrature","Quadrature","DT",comm,PetscQuadratureDestroy,PetscQuadratureView);
 55:   (*q)->dim       = -1;
 56:   (*q)->order     = -1;
 57:   (*q)->numPoints = 0;
 58:   (*q)->points    = NULL;
 59:   (*q)->weights   = NULL;
 60:   return(0);
 61: }

 65: /*@
 66:   PetscQuadratureDuplicate - Create a deep copy of the PetscQuadrature object

 68:   Collective on PetscQuadrature

 70:   Input Parameter:
 71: . q  - The PetscQuadrature object

 73:   Output Parameter:
 74: . r  - The new PetscQuadrature object

 76:   Level: beginner

 78: .keywords: PetscQuadrature, quadrature, clone
 79: .seealso: PetscQuadratureCreate(), PetscQuadratureDestroy(), PetscQuadratureGetData()
 80: @*/
 81: PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
 82: {
 83:   PetscInt         order, dim, Nq;
 84:   const PetscReal *points, *weights;
 85:   PetscReal       *p, *w;
 86:   PetscErrorCode   ierr;

 90:   PetscQuadratureCreate(PetscObjectComm((PetscObject) q), r);
 91:   PetscQuadratureGetOrder(q, &order);
 92:   PetscQuadratureSetOrder(*r, order);
 93:   PetscQuadratureGetData(q, &dim, &Nq, &points, &weights);
 94:   PetscMalloc1(Nq*dim, &p);
 95:   PetscMalloc1(Nq, &w);
 96:   PetscMemcpy(p, points, Nq*dim * sizeof(PetscReal));
 97:   PetscMemcpy(w, weights, Nq * sizeof(PetscReal));
 98:   PetscQuadratureSetData(*r, dim, Nq, p, w);
 99:   return(0);
100: }

104: /*@
105:   PetscQuadratureDestroy - Destroys a PetscQuadrature object

107:   Collective on PetscQuadrature

109:   Input Parameter:
110: . q  - The PetscQuadrature object

112:   Level: beginner

114: .keywords: PetscQuadrature, quadrature, destroy
115: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
116: @*/
117: PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
118: {

122:   if (!*q) return(0);
124:   if (--((PetscObject)(*q))->refct > 0) {
125:     *q = NULL;
126:     return(0);
127:   }
128:   PetscFree((*q)->points);
129:   PetscFree((*q)->weights);
130:   PetscHeaderDestroy(q);
131:   return(0);
132: }

136: /*@
137:   PetscQuadratureGetOrder - Return the quadrature information

139:   Not collective

141:   Input Parameter:
142: . q - The PetscQuadrature object

144:   Output Parameter:
145: . order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated

147:   Output Parameter:

149:   Level: intermediate

151: .seealso: PetscQuadratureSetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
152: @*/
153: PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature q, PetscInt *order)
154: {
158:   *order = q->order;
159:   return(0);
160: }

164: /*@
165:   PetscQuadratureSetOrder - Return the quadrature information

167:   Not collective

169:   Input Parameters:
170: + q - The PetscQuadrature object
171: - order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated

173:   Level: intermediate

175: .seealso: PetscQuadratureGetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
176: @*/
177: PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature q, PetscInt order)
178: {
181:   q->order = order;
182:   return(0);
183: }

187: /*@C
188:   PetscQuadratureGetData - Returns the data defining the quadrature

190:   Not collective

192:   Input Parameter:
193: . q  - The PetscQuadrature object

195:   Output Parameters:
196: + dim - The spatial dimension
197: . npoints - The number of quadrature points
198: . points - The coordinates of each quadrature point
199: - weights - The weight of each quadrature point

201:   Level: intermediate

203: .keywords: PetscQuadrature, quadrature
204: .seealso: PetscQuadratureCreate(), PetscQuadratureSetData()
205: @*/
206: PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
207: {
210:   if (dim) {
212:     *dim = q->dim;
213:   }
214:   if (npoints) {
216:     *npoints = q->numPoints;
217:   }
218:   if (points) {
220:     *points = q->points;
221:   }
222:   if (weights) {
224:     *weights = q->weights;
225:   }
226:   return(0);
227: }

231: /*@C
232:   PetscQuadratureSetData - Sets the data defining the quadrature

234:   Not collective

236:   Input Parameters:
237: + q  - The PetscQuadrature object
238: . dim - The spatial dimension
239: . npoints - The number of quadrature points
240: . points - The coordinates of each quadrature point
241: - weights - The weight of each quadrature point

243:   Level: intermediate

245: .keywords: PetscQuadrature, quadrature
246: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
247: @*/
248: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
249: {
252:   if (dim >= 0)     q->dim       = dim;
253:   if (npoints >= 0) q->numPoints = npoints;
254:   if (points) {
256:     q->points = points;
257:   }
258:   if (weights) {
260:     q->weights = weights;
261:   }
262:   return(0);
263: }

267: /*@C
268:   PetscQuadratureView - Views a PetscQuadrature object

270:   Collective on PetscQuadrature

272:   Input Parameters:
273: + q  - The PetscQuadrature object
274: - viewer - The PetscViewer object

276:   Level: beginner

278: .keywords: PetscQuadrature, quadrature, view
279: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
280: @*/
281: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
282: {
283:   PetscInt       q, d;

287:   PetscObjectPrintClassNamePrefixType((PetscObject)quad,viewer);
288:   PetscViewerASCIIPrintf(viewer, "Quadrature on %d points\n  (", quad->numPoints);
289:   for (q = 0; q < quad->numPoints; ++q) {
290:     for (d = 0; d < quad->dim; ++d) {
291:       if (d) PetscViewerASCIIPrintf(viewer, ", ");
292:       PetscViewerASCIIPrintf(viewer, "%g\n", (double)quad->points[q*quad->dim+d]);
293:     }
294:     PetscViewerASCIIPrintf(viewer, ") %g\n", (double)quad->weights[q]);
295:   }
296:   return(0);
297: }

301: /*@C
302:   PetscQuadratureExpandComposite - Return a quadrature over the composite element, which has the original quadrature in each subelement

304:   Not collective

306:   Input Parameter:
307: + q - The original PetscQuadrature
308: . numSubelements - The number of subelements the original element is divided into
309: . v0 - An array of the initial points for each subelement
310: - jac - An array of the Jacobian mappings from the reference to each subelement

312:   Output Parameters:
313: . dim - The dimension

315:   Note: Together v0 and jac define an affine mapping from the original reference element to each subelement

317:   Level: intermediate

319: .seealso: PetscFECreate(), PetscSpaceGetDimension(), PetscDualSpaceGetDimension()
320: @*/
321: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
322: {
323:   const PetscReal *points,    *weights;
324:   PetscReal       *pointsRef, *weightsRef;
325:   PetscInt         dim, order, npoints, npointsRef, c, p, d, e;
326:   PetscErrorCode   ierr;

333:   PetscQuadratureCreate(PETSC_COMM_SELF, qref);
334:   PetscQuadratureGetOrder(q, &order);
335:   PetscQuadratureGetData(q, &dim, &npoints, &points, &weights);
336:   npointsRef = npoints*numSubelements;
337:   PetscMalloc1(npointsRef*dim,&pointsRef);
338:   PetscMalloc1(npointsRef,&weightsRef);
339:   for (c = 0; c < numSubelements; ++c) {
340:     for (p = 0; p < npoints; ++p) {
341:       for (d = 0; d < dim; ++d) {
342:         pointsRef[(c*npoints + p)*dim+d] = v0[c*dim+d];
343:         for (e = 0; e < dim; ++e) {
344:           pointsRef[(c*npoints + p)*dim+d] += jac[(c*dim + d)*dim+e]*(points[p*dim+e] + 1.0);
345:         }
346:       }
347:       /* Could also use detJ here */
348:       weightsRef[c*npoints+p] = weights[p]/numSubelements;
349:     }
350:   }
351:   PetscQuadratureSetOrder(*qref, order);
352:   PetscQuadratureSetData(*qref, dim, npointsRef, pointsRef, weightsRef);
353:   return(0);
354: }

358: /*@
359:    PetscDTLegendreEval - evaluate Legendre polynomial at points

361:    Not Collective

363:    Input Arguments:
364: +  npoints - number of spatial points to evaluate at
365: .  points - array of locations to evaluate at
366: .  ndegree - number of basis degrees to evaluate
367: -  degrees - sorted array of degrees to evaluate

369:    Output Arguments:
370: +  B - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension npoints*ndegrees, allocated by caller) (or NULL)
371: .  D - row-oriented derivative evaluation matrix (or NULL)
372: -  D2 - row-oriented second derivative evaluation matrix (or NULL)

374:    Level: intermediate

376: .seealso: PetscDTGaussQuadrature()
377: @*/
378: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
379: {
380:   PetscInt i,maxdegree;

383:   if (!npoints || !ndegree) return(0);
384:   maxdegree = degrees[ndegree-1];
385:   for (i=0; i<npoints; i++) {
386:     PetscReal pm1,pm2,pd1,pd2,pdd1,pdd2,x;
387:     PetscInt  j,k;
388:     x    = points[i];
389:     pm2  = 0;
390:     pm1  = 1;
391:     pd2  = 0;
392:     pd1  = 0;
393:     pdd2 = 0;
394:     pdd1 = 0;
395:     k    = 0;
396:     if (degrees[k] == 0) {
397:       if (B) B[i*ndegree+k] = pm1;
398:       if (D) D[i*ndegree+k] = pd1;
399:       if (D2) D2[i*ndegree+k] = pdd1;
400:       k++;
401:     }
402:     for (j=1; j<=maxdegree; j++,k++) {
403:       PetscReal p,d,dd;
404:       p    = ((2*j-1)*x*pm1 - (j-1)*pm2)/j;
405:       d    = pd2 + (2*j-1)*pm1;
406:       dd   = pdd2 + (2*j-1)*pd1;
407:       pm2  = pm1;
408:       pm1  = p;
409:       pd2  = pd1;
410:       pd1  = d;
411:       pdd2 = pdd1;
412:       pdd1 = dd;
413:       if (degrees[k] == j) {
414:         if (B) B[i*ndegree+k] = p;
415:         if (D) D[i*ndegree+k] = d;
416:         if (D2) D2[i*ndegree+k] = dd;
417:       }
418:     }
419:   }
420:   return(0);
421: }

425: /*@
426:    PetscDTGaussQuadrature - create Gauss quadrature

428:    Not Collective

430:    Input Arguments:
431: +  npoints - number of points
432: .  a - left end of interval (often-1)
433: -  b - right end of interval (often +1)

435:    Output Arguments:
436: +  x - quadrature points
437: -  w - quadrature weights

439:    Level: intermediate

441:    References:
442:    Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 221--230, 1969.

444: .seealso: PetscDTLegendreEval()
445: @*/
446: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
447: {
449:   PetscInt       i;
450:   PetscReal      *work;
451:   PetscScalar    *Z;
452:   PetscBLASInt   N,LDZ,info;

455:   PetscCitationsRegister(GaussCitation, &GaussCite);
456:   /* Set up the Golub-Welsch system */
457:   for (i=0; i<npoints; i++) {
458:     x[i] = 0;                   /* diagonal is 0 */
459:     if (i) w[i-1] = 0.5 / PetscSqrtReal(1 - 1./PetscSqr(2*i));
460:   }
461:   PetscMalloc2(npoints*npoints,&Z,PetscMax(1,2*npoints-2),&work);
462:   PetscBLASIntCast(npoints,&N);
463:   LDZ  = N;
464:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
465:   PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&N,x,w,Z,&LDZ,work,&info));
466:   PetscFPTrapPop();
467:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");

469:   for (i=0; i<(npoints+1)/2; i++) {
470:     PetscReal y = 0.5 * (-x[i] + x[npoints-i-1]); /* enforces symmetry */
471:     x[i]           = (a+b)/2 - y*(b-a)/2;
472:     x[npoints-i-1] = (a+b)/2 + y*(b-a)/2;

474:     w[i] = w[npoints-1-i] = 0.5*(b-a)*(PetscSqr(PetscAbsScalar(Z[i*npoints])) + PetscSqr(PetscAbsScalar(Z[(npoints-i-1)*npoints])));
475:   }
476:   PetscFree2(Z,work);
477:   return(0);
478: }

482: /*@
483:   PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

485:   Not Collective

487:   Input Arguments:
488: + dim     - The spatial dimension
489: . npoints - number of points in one dimension
490: . a       - left end of interval (often-1)
491: - b       - right end of interval (often +1)

493:   Output Argument:
494: . q - A PetscQuadrature object

496:   Level: intermediate

498: .seealso: PetscDTGaussQuadrature(), PetscDTLegendreEval()
499: @*/
500: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
501: {
502:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k;
503:   PetscReal     *x, *w, *xw, *ww;

507:   PetscMalloc1(totpoints*dim,&x);
508:   PetscMalloc1(totpoints,&w);
509:   /* Set up the Golub-Welsch system */
510:   switch (dim) {
511:   case 0:
512:     PetscFree(x);
513:     PetscFree(w);
514:     PetscMalloc1(1, &x);
515:     PetscMalloc1(1, &w);
516:     x[0] = 0.0;
517:     w[0] = 1.0;
518:     break;
519:   case 1:
520:     PetscDTGaussQuadrature(npoints, a, b, x, w);
521:     break;
522:   case 2:
523:     PetscMalloc2(npoints,&xw,npoints,&ww);
524:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
525:     for (i = 0; i < npoints; ++i) {
526:       for (j = 0; j < npoints; ++j) {
527:         x[(i*npoints+j)*dim+0] = xw[i];
528:         x[(i*npoints+j)*dim+1] = xw[j];
529:         w[i*npoints+j]         = ww[i] * ww[j];
530:       }
531:     }
532:     PetscFree2(xw,ww);
533:     break;
534:   case 3:
535:     PetscMalloc2(npoints,&xw,npoints,&ww);
536:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
537:     for (i = 0; i < npoints; ++i) {
538:       for (j = 0; j < npoints; ++j) {
539:         for (k = 0; k < npoints; ++k) {
540:           x[((i*npoints+j)*npoints+k)*dim+0] = xw[i];
541:           x[((i*npoints+j)*npoints+k)*dim+1] = xw[j];
542:           x[((i*npoints+j)*npoints+k)*dim+2] = xw[k];
543:           w[(i*npoints+j)*npoints+k]         = ww[i] * ww[j] * ww[k];
544:         }
545:       }
546:     }
547:     PetscFree2(xw,ww);
548:     break;
549:   default:
550:     SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
551:   }
552:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
553:   PetscQuadratureSetOrder(*q, npoints);
554:   PetscQuadratureSetData(*q, dim, totpoints, x, w);
555:   return(0);
556: }

560: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
561:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
562: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorial_Internal(PetscInt n, PetscReal *factorial)
563: {
564:   PetscReal f = 1.0;
565:   PetscInt  i;

568:   for (i = 1; i < n+1; ++i) f *= i;
569:   *factorial = f;
570:   return(0);
571: }

575: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
576:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
577: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
578: {
579:   PetscReal apb, pn1, pn2;
580:   PetscInt  k;

583:   if (!n) {*P = 1.0; return(0);}
584:   if (n == 1) {*P = 0.5 * (a - b + (a + b + 2.0) * x); return(0);}
585:   apb = a + b;
586:   pn2 = 1.0;
587:   pn1 = 0.5 * (a - b + (apb + 2.0) * x);
588:   *P  = 0.0;
589:   for (k = 2; k < n+1; ++k) {
590:     PetscReal a1 = 2.0 * k * (k + apb) * (2.0*k + apb - 2.0);
591:     PetscReal a2 = (2.0 * k + apb - 1.0) * (a*a - b*b);
592:     PetscReal a3 = (2.0 * k + apb - 2.0) * (2.0 * k + apb - 1.0) * (2.0 * k + apb);
593:     PetscReal a4 = 2.0 * (k + a - 1.0) * (k + b - 1.0) * (2.0 * k + apb);

595:     a2  = a2 / a1;
596:     a3  = a3 / a1;
597:     a4  = a4 / a1;
598:     *P  = (a2 + a3 * x) * pn1 - a4 * pn2;
599:     pn2 = pn1;
600:     pn1 = *P;
601:   }
602:   return(0);
603: }

607: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
608: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
609: {
610:   PetscReal      nP;

614:   if (!n) {*P = 0.0; return(0);}
615:   PetscDTComputeJacobi(a+1, b+1, n-1, x, &nP);
616:   *P   = 0.5 * (a + b + n + 1) * nP;
617:   return(0);
618: }

622: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
623: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapSquareToTriangle_Internal(PetscReal x, PetscReal y, PetscReal *xi, PetscReal *eta)
624: {
626:   *xi  = 0.5 * (1.0 + x) * (1.0 - y) - 1.0;
627:   *eta = y;
628:   return(0);
629: }

633: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
634: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapCubeToTetrahedron_Internal(PetscReal x, PetscReal y, PetscReal z, PetscReal *xi, PetscReal *eta, PetscReal *zeta)
635: {
637:   *xi   = 0.25 * (1.0 + x) * (1.0 - y) * (1.0 - z) - 1.0;
638:   *eta  = 0.5  * (1.0 + y) * (1.0 - z) - 1.0;
639:   *zeta = z;
640:   return(0);
641: }

645: static PetscErrorCode PetscDTGaussJacobiQuadrature1D_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
646: {
647:   PetscInt       maxIter = 100;
648:   PetscReal      eps     = 1.0e-8;
649:   PetscReal      a1, a2, a3, a4, a5, a6;
650:   PetscInt       k;


655:   a1      = PetscPowReal(2.0, a+b+1);
656: #if defined(PETSC_HAVE_TGAMMA)
657:   a2      = PetscTGamma(a + npoints + 1);
658:   a3      = PetscTGamma(b + npoints + 1);
659:   a4      = PetscTGamma(a + b + npoints + 1);
660: #else
661:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
662: #endif

664:   PetscDTFactorial_Internal(npoints, &a5);
665:   a6   = a1 * a2 * a3 / a4 / a5;
666:   /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
667:    Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
668:   for (k = 0; k < npoints; ++k) {
669:     PetscReal r = -PetscCosReal((2.0*k + 1.0) * PETSC_PI / (2.0 * npoints)), dP;
670:     PetscInt  j;

672:     if (k > 0) r = 0.5 * (r + x[k-1]);
673:     for (j = 0; j < maxIter; ++j) {
674:       PetscReal s = 0.0, delta, f, fp;
675:       PetscInt  i;

677:       for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
678:       PetscDTComputeJacobi(a, b, npoints, r, &f);
679:       PetscDTComputeJacobiDerivative(a, b, npoints, r, &fp);
680:       delta = f / (fp - f * s);
681:       r     = r - delta;
682:       if (PetscAbsReal(delta) < eps) break;
683:     }
684:     x[k] = r;
685:     PetscDTComputeJacobiDerivative(a, b, npoints, x[k], &dP);
686:     w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
687:   }
688:   return(0);
689: }

693: /*@C
694:   PetscDTGaussJacobiQuadrature - create Gauss-Jacobi quadrature for a simplex

696:   Not Collective

698:   Input Arguments:
699: + dim   - The simplex dimension
700: . order - The number of points in one dimension
701: . a     - left end of interval (often-1)
702: - b     - right end of interval (often +1)

704:   Output Argument:
705: . q - A PetscQuadrature object

707:   Level: intermediate

709:   References:
710:   Karniadakis and Sherwin.
711:   FIAT

713: .seealso: PetscDTGaussTensorQuadrature(), PetscDTGaussQuadrature()
714: @*/
715: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt dim, PetscInt order, PetscReal a, PetscReal b, PetscQuadrature *q)
716: {
717:   PetscInt       npoints = dim > 1 ? dim > 2 ? order*PetscSqr(order) : PetscSqr(order) : order;
718:   PetscReal     *px, *wx, *py, *wy, *pz, *wz, *x, *w;
719:   PetscInt       i, j, k;

723:   if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
724:   PetscMalloc1(npoints*dim, &x);
725:   PetscMalloc1(npoints, &w);
726:   switch (dim) {
727:   case 0:
728:     PetscFree(x);
729:     PetscFree(w);
730:     PetscMalloc1(1, &x);
731:     PetscMalloc1(1, &w);
732:     x[0] = 0.0;
733:     w[0] = 1.0;
734:     break;
735:   case 1:
736:     PetscDTGaussJacobiQuadrature1D_Internal(order, 0.0, 0.0, x, w);
737:     break;
738:   case 2:
739:     PetscMalloc4(order,&px,order,&wx,order,&py,order,&wy);
740:     PetscDTGaussJacobiQuadrature1D_Internal(order, 0.0, 0.0, px, wx);
741:     PetscDTGaussJacobiQuadrature1D_Internal(order, 1.0, 0.0, py, wy);
742:     for (i = 0; i < order; ++i) {
743:       for (j = 0; j < order; ++j) {
744:         PetscDTMapSquareToTriangle_Internal(px[i], py[j], &x[(i*order+j)*2+0], &x[(i*order+j)*2+1]);
745:         w[i*order+j] = 0.5 * wx[i] * wy[j];
746:       }
747:     }
748:     PetscFree4(px,wx,py,wy);
749:     break;
750:   case 3:
751:     PetscMalloc6(order,&px,order,&wx,order,&py,order,&wy,order,&pz,order,&wz);
752:     PetscDTGaussJacobiQuadrature1D_Internal(order, 0.0, 0.0, px, wx);
753:     PetscDTGaussJacobiQuadrature1D_Internal(order, 1.0, 0.0, py, wy);
754:     PetscDTGaussJacobiQuadrature1D_Internal(order, 2.0, 0.0, pz, wz);
755:     for (i = 0; i < order; ++i) {
756:       for (j = 0; j < order; ++j) {
757:         for (k = 0; k < order; ++k) {
758:           PetscDTMapCubeToTetrahedron_Internal(px[i], py[j], pz[k], &x[((i*order+j)*order+k)*3+0], &x[((i*order+j)*order+k)*3+1], &x[((i*order+j)*order+k)*3+2]);
759:           w[(i*order+j)*order+k] = 0.125 * wx[i] * wy[j] * wz[k];
760:         }
761:       }
762:     }
763:     PetscFree6(px,wx,py,wy,pz,wz);
764:     break;
765:   default:
766:     SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
767:   }
768:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
769:   PetscQuadratureSetOrder(*q, order);
770:   PetscQuadratureSetData(*q, dim, npoints, x, w);
771:   return(0);
772: }

776: /*@C
777:   PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell

779:   Not Collective

781:   Input Arguments:
782: + dim   - The cell dimension
783: . level - The number of points in one dimension, 2^l
784: . a     - left end of interval (often-1)
785: - b     - right end of interval (often +1)

787:   Output Argument:
788: . q - A PetscQuadrature object

790:   Level: intermediate

792: .seealso: PetscDTGaussTensorQuadrature()
793: @*/
794: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
795: {
796:   const PetscInt  p     = 16;                        /* Digits of precision in the evaluation */
797:   const PetscReal alpha = (b-a)/2.;                  /* Half-width of the integration interval */
798:   const PetscReal beta  = (b+a)/2.;                  /* Center of the integration interval */
799:   const PetscReal h     = PetscPowReal(2.0, -level); /* Step size, length between x_k */
800:   PetscReal       xk;                                /* Quadrature point x_k on reference domain [-1, 1] */
801:   PetscReal       wk    = 0.5*PETSC_PI;              /* Quadrature weight at x_k */
802:   PetscReal      *x, *w;
803:   PetscInt        K, k, npoints;
804:   PetscErrorCode  ierr;

807:   if (dim > 1) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %d not yet implemented", dim);
808:   if (!level) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
809:   /* Find K such that the weights are < 32 digits of precision */
810:   for (K = 1; PetscAbsReal(PetscLog10Real(wk)) < 2*p; ++K) {
811:     wk = 0.5*h*PETSC_PI*PetscCoshReal(K*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(K*h)));
812:   }
813:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
814:   PetscQuadratureSetOrder(*q, 2*K+1);
815:   npoints = 2*K-1;
816:   PetscMalloc1(npoints*dim, &x);
817:   PetscMalloc1(npoints, &w);
818:   /* Center term */
819:   x[0] = beta;
820:   w[0] = 0.5*alpha*PETSC_PI;
821:   for (k = 1; k < K; ++k) {
822:     wk = 0.5*alpha*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
823:     xk = tanh(0.5*PETSC_PI*PetscSinhReal(k*h));
824:     x[2*k-1] = -alpha*xk+beta;
825:     w[2*k-1] = wk;
826:     x[2*k+0] =  alpha*xk+beta;
827:     w[2*k+0] = wk;
828:   }
829:   PetscQuadratureSetData(*q, dim, npoints, x, w);
830:   return(0);
831: }

835: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
836: {
837:   const PetscInt  p     = 16;        /* Digits of precision in the evaluation */
838:   const PetscReal alpha = (b-a)/2.;  /* Half-width of the integration interval */
839:   const PetscReal beta  = (b+a)/2.;  /* Center of the integration interval */
840:   PetscReal       h     = 1.0;       /* Step size, length between x_k */
841:   PetscInt        l     = 0;         /* Level of refinement, h = 2^{-l} */
842:   PetscReal       osum  = 0.0;       /* Integral on last level */
843:   PetscReal       psum  = 0.0;       /* Integral on the level before the last level */
844:   PetscReal       sum;               /* Integral on current level */
845:   PetscReal       yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
846:   PetscReal       lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
847:   PetscReal       wk;                /* Quadrature weight at x_k */
848:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
849:   PetscInt        d;                 /* Digits of precision in the integral */

852:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
853:   /* Center term */
854:   func(beta, &lval);
855:   sum = 0.5*alpha*PETSC_PI*lval;
856:   /* */
857:   do {
858:     PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
859:     PetscInt  k = 1;

861:     ++l;
862:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
863:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
864:     psum = osum;
865:     osum = sum;
866:     h   *= 0.5;
867:     sum *= 0.5;
868:     do {
869:       wk = 0.5*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
870:       yk = 1.0/(PetscExpReal(0.5*PETSC_PI*PetscSinhReal(k*h)) * PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
871:       lx = -alpha*(1.0 - yk)+beta;
872:       rx =  alpha*(1.0 - yk)+beta;
873:       func(lx, &lval);
874:       func(rx, &rval);
875:       lterm   = alpha*wk*lval;
876:       maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
877:       sum    += lterm;
878:       rterm   = alpha*wk*rval;
879:       maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
880:       sum    += rterm;
881:       /* if (l == 1) printf("k is %d and sum is %15.15f and wk is %15.15f\n", k, sum, wk); */
882:       ++k;
883:       /* Only need to evaluate every other point on refined levels */
884:       if (l != 1) ++k;
885:     } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */

887:     d1 = PetscLog10Real(PetscAbsReal(sum - osum));
888:     d2 = PetscLog10Real(PetscAbsReal(sum - psum));
889:     d3 = PetscLog10Real(maxTerm) - p;
890:     d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
891:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
892:   } while (d < digits && l < 12);
893:   *sol = sum;

895:   return(0);
896: }

898: #ifdef PETSC_HAVE_MPFR
901: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
902: {
903:   const PetscInt  safetyFactor = 2;  /* Calculate abcissa until 2*p digits */
904:   PetscInt        l            = 0;  /* Level of refinement, h = 2^{-l} */
905:   mpfr_t          alpha;             /* Half-width of the integration interval */
906:   mpfr_t          beta;              /* Center of the integration interval */
907:   mpfr_t          h;                 /* Step size, length between x_k */
908:   mpfr_t          osum;              /* Integral on last level */
909:   mpfr_t          psum;              /* Integral on the level before the last level */
910:   mpfr_t          sum;               /* Integral on current level */
911:   mpfr_t          yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
912:   mpfr_t          lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
913:   mpfr_t          wk;                /* Quadrature weight at x_k */
914:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
915:   PetscInt        d;                 /* Digits of precision in the integral */
916:   mpfr_t          pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;

919:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
920:   /* Create high precision storage */
921:   mpfr_inits2(PetscCeilReal(safetyFactor*digits*PetscLogReal(10.)/PetscLogReal(2.)), alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
922:   /* Initialization */
923:   mpfr_set_d(alpha, 0.5*(b-a), MPFR_RNDN);
924:   mpfr_set_d(beta,  0.5*(b+a), MPFR_RNDN);
925:   mpfr_set_d(osum,  0.0,       MPFR_RNDN);
926:   mpfr_set_d(psum,  0.0,       MPFR_RNDN);
927:   mpfr_set_d(h,     1.0,       MPFR_RNDN);
928:   mpfr_const_pi(pi2, MPFR_RNDN);
929:   mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
930:   /* Center term */
931:   func(0.5*(b+a), &lval);
932:   mpfr_set(sum, pi2, MPFR_RNDN);
933:   mpfr_mul(sum, sum, alpha, MPFR_RNDN);
934:   mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
935:   /* */
936:   do {
937:     PetscReal d1, d2, d3, d4;
938:     PetscInt  k = 1;

940:     ++l;
941:     mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
942:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
943:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
944:     mpfr_set(psum, osum, MPFR_RNDN);
945:     mpfr_set(osum,  sum, MPFR_RNDN);
946:     mpfr_mul_d(h,   h,   0.5, MPFR_RNDN);
947:     mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
948:     do {
949:       mpfr_set_si(kh, k, MPFR_RNDN);
950:       mpfr_mul(kh, kh, h, MPFR_RNDN);
951:       /* Weight */
952:       mpfr_set(wk, h, MPFR_RNDN);
953:       mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
954:       mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
955:       mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
956:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
957:       mpfr_sqr(tmp, tmp, MPFR_RNDN);
958:       mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
959:       mpfr_div(wk, wk, tmp, MPFR_RNDN);
960:       /* Abscissa */
961:       mpfr_set_d(yk, 1.0, MPFR_RNDZ);
962:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
963:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
964:       mpfr_exp(tmp, msinh, MPFR_RNDN);
965:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
966:       /* Quadrature points */
967:       mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
968:       mpfr_mul(lx, lx, alpha, MPFR_RNDU);
969:       mpfr_add(lx, lx, beta, MPFR_RNDU);
970:       mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
971:       mpfr_mul(rx, rx, alpha, MPFR_RNDD);
972:       mpfr_add(rx, rx, beta, MPFR_RNDD);
973:       /* Evaluation */
974:       func(mpfr_get_d(lx, MPFR_RNDU), &lval);
975:       func(mpfr_get_d(rx, MPFR_RNDD), &rval);
976:       /* Update */
977:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
978:       mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
979:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
980:       mpfr_abs(tmp, tmp, MPFR_RNDN);
981:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
982:       mpfr_set(curTerm, tmp, MPFR_RNDN);
983:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
984:       mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
985:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
986:       mpfr_abs(tmp, tmp, MPFR_RNDN);
987:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
988:       mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
989:       /* if (l == 1) printf("k is %d and sum is %15.15f and wk is %15.15f\n", k, sum, wk); */
990:       ++k;
991:       /* Only need to evaluate every other point on refined levels */
992:       if (l != 1) ++k;
993:       mpfr_log10(tmp, wk, MPFR_RNDN);
994:       mpfr_abs(tmp, tmp, MPFR_RNDN);
995:     } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor*digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
996:     mpfr_sub(tmp, sum, osum, MPFR_RNDN);
997:     mpfr_abs(tmp, tmp, MPFR_RNDN);
998:     mpfr_log10(tmp, tmp, MPFR_RNDN);
999:     d1 = mpfr_get_d(tmp, MPFR_RNDN);
1000:     mpfr_sub(tmp, sum, psum, MPFR_RNDN);
1001:     mpfr_abs(tmp, tmp, MPFR_RNDN);
1002:     mpfr_log10(tmp, tmp, MPFR_RNDN);
1003:     d2 = mpfr_get_d(tmp, MPFR_RNDN);
1004:     mpfr_log10(tmp, maxTerm, MPFR_RNDN);
1005:     d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
1006:     mpfr_log10(tmp, curTerm, MPFR_RNDN);
1007:     d4 = mpfr_get_d(tmp, MPFR_RNDN);
1008:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
1009:   } while (d < digits && l < 8);
1010:   *sol = mpfr_get_d(sum, MPFR_RNDN);
1011:   /* Cleanup */
1012:   mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
1013:   return(0);
1014: }
1015: #else
1018: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
1019: {
1020:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
1021: }
1022: #endif

1026: /* Overwrites A. Can only handle full-rank problems with m>=n
1027:  * A in column-major format
1028:  * Ainv in row-major format
1029:  * tau has length m
1030:  * worksize must be >= max(1,n)
1031:  */
1032: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
1033: {
1035:   PetscBLASInt   M,N,K,lda,ldb,ldwork,info;
1036:   PetscScalar    *A,*Ainv,*R,*Q,Alpha;

1039: #if defined(PETSC_USE_COMPLEX)
1040:   {
1041:     PetscInt i,j;
1042:     PetscMalloc2(m*n,&A,m*n,&Ainv);
1043:     for (j=0; j<n; j++) {
1044:       for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
1045:     }
1046:     mstride = m;
1047:   }
1048: #else
1049:   A = A_in;
1050:   Ainv = Ainv_out;
1051: #endif

1053:   PetscBLASIntCast(m,&M);
1054:   PetscBLASIntCast(n,&N);
1055:   PetscBLASIntCast(mstride,&lda);
1056:   PetscBLASIntCast(worksize,&ldwork);
1057:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
1058:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info));
1059:   PetscFPTrapPop();
1060:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
1061:   R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

1063:   /* Extract an explicit representation of Q */
1064:   Q = Ainv;
1065:   PetscMemcpy(Q,A,mstride*n*sizeof(PetscScalar));
1066:   K = N;                        /* full rank */
1067:   PetscStackCallBLAS("LAPACKungqr",LAPACKungqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info));
1068:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");

1070:   /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
1071:   Alpha = 1.0;
1072:   ldb = lda;
1073:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb));
1074:   /* Ainv is Q, overwritten with inverse */

1076: #if defined(PETSC_USE_COMPLEX)
1077:   {
1078:     PetscInt i;
1079:     for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
1080:     PetscFree2(A,Ainv);
1081:   }
1082: #endif
1083:   return(0);
1084: }

1088: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
1089: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
1090: {
1092:   PetscReal      *Bv;
1093:   PetscInt       i,j;

1096:   PetscMalloc1((ninterval+1)*ndegree,&Bv);
1097:   /* Point evaluation of L_p on all the source vertices */
1098:   PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
1099:   /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
1100:   for (i=0; i<ninterval; i++) {
1101:     for (j=0; j<ndegree; j++) {
1102:       if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
1103:       else           B[i*ndegree+j]   = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
1104:     }
1105:   }
1106:   PetscFree(Bv);
1107:   return(0);
1108: }

1112: /*@
1113:    PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

1115:    Not Collective

1117:    Input Arguments:
1118: +  degree - degree of reconstruction polynomial
1119: .  nsource - number of source intervals
1120: .  sourcex - sorted coordinates of source cell boundaries (length nsource+1)
1121: .  ntarget - number of target intervals
1122: -  targetx - sorted coordinates of target cell boundaries (length ntarget+1)

1124:    Output Arguments:
1125: .  R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]

1127:    Level: advanced

1129: .seealso: PetscDTLegendreEval()
1130: @*/
1131: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
1132: {
1134:   PetscInt       i,j,k,*bdegrees,worksize;
1135:   PetscReal      xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
1136:   PetscScalar    *tau,*work;

1142:   if (degree >= nsource) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Reconstruction degree %D must be less than number of source intervals %D",degree,nsource);
1143: #if defined(PETSC_USE_DEBUG)
1144:   for (i=0; i<nsource; i++) {
1145:     if (sourcex[i] >= sourcex[i+1]) SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_ARG_CORRUPT,"Source interval %D has negative orientation (%g,%g)",i,(double)sourcex[i],(double)sourcex[i+1]);
1146:   }
1147:   for (i=0; i<ntarget; i++) {
1148:     if (targetx[i] >= targetx[i+1]) SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_ARG_CORRUPT,"Target interval %D has negative orientation (%g,%g)",i,(double)targetx[i],(double)targetx[i+1]);
1149:   }
1150: #endif
1151:   xmin = PetscMin(sourcex[0],targetx[0]);
1152:   xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
1153:   center = (xmin + xmax)/2;
1154:   hscale = (xmax - xmin)/2;
1155:   worksize = nsource;
1156:   PetscMalloc4(degree+1,&bdegrees,nsource+1,&sourcey,nsource*(degree+1),&Bsource,worksize,&work);
1157:   PetscMalloc4(nsource,&tau,nsource*(degree+1),&Bsinv,ntarget+1,&targety,ntarget*(degree+1),&Btarget);
1158:   for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
1159:   for (i=0; i<=degree; i++) bdegrees[i] = i+1;
1160:   PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
1161:   PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
1162:   for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
1163:   PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
1164:   for (i=0; i<ntarget; i++) {
1165:     PetscReal rowsum = 0;
1166:     for (j=0; j<nsource; j++) {
1167:       PetscReal sum = 0;
1168:       for (k=0; k<degree+1; k++) {
1169:         sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
1170:       }
1171:       R[i*nsource+j] = sum;
1172:       rowsum += sum;
1173:     }
1174:     for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
1175:   }
1176:   PetscFree4(bdegrees,sourcey,Bsource,work);
1177:   PetscFree4(tau,Bsinv,targety,Btarget);
1178:   return(0);
1179: }