Actual source code: dt.c

petsc-3.6.1 2015-07-22
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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

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

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

 28: /*@
 29:   PetscQuadratureCreate - Create a PetscQuadrature object

 31:   Collective on MPI_Comm

 33:   Input Parameter:
 34: . comm - The communicator for the PetscQuadrature object

 36:   Output Parameter:
 37: . q  - The PetscQuadrature object

 39:   Level: beginner

 41: .keywords: PetscQuadrature, quadrature, create
 42: .seealso: PetscQuadratureDestroy(), PetscQuadratureGetData()
 43: @*/
 44: PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
 45: {

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

 62: /*@
 63:   PetscQuadratureDuplicate - Create a deep copy of the PetscQuadrature object

 65:   Collective on PetscQuadrature

 67:   Input Parameter:
 68: . q  - The PetscQuadrature object

 70:   Output Parameter:
 71: . r  - The new PetscQuadrature object

 73:   Level: beginner

 75: .keywords: PetscQuadrature, quadrature, clone
 76: .seealso: PetscQuadratureCreate(), PetscQuadratureDestroy(), PetscQuadratureGetData()
 77: @*/
 78: PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
 79: {
 80:   PetscInt         order, dim, Nq;
 81:   const PetscReal *points, *weights;
 82:   PetscReal       *p, *w;
 83:   PetscErrorCode   ierr;

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

101: /*@
102:   PetscQuadratureDestroy - Destroys a PetscQuadrature object

104:   Collective on PetscQuadrature

106:   Input Parameter:
107: . q  - The PetscQuadrature object

109:   Level: beginner

111: .keywords: PetscQuadrature, quadrature, destroy
112: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
113: @*/
114: PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
115: {

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

133: /*@
134:   PetscQuadratureGetOrder - Return the quadrature information

136:   Not collective

138:   Input Parameter:
139: . q - The PetscQuadrature object

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

144:   Output Parameter:

146:   Level: intermediate

148: .seealso: PetscQuadratureSetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
149: @*/
150: PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature q, PetscInt *order)
151: {
155:   *order = q->order;
156:   return(0);
157: }

161: /*@
162:   PetscQuadratureSetOrder - Return the quadrature information

164:   Not collective

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

170:   Level: intermediate

172: .seealso: PetscQuadratureGetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
173: @*/
174: PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature q, PetscInt order)
175: {
178:   q->order = order;
179:   return(0);
180: }

184: /*@C
185:   PetscQuadratureGetData - Returns the data defining the quadrature

187:   Not collective

189:   Input Parameter:
190: . q  - The PetscQuadrature object

192:   Output Parameters:
193: + dim - The spatial dimension
194: . npoints - The number of quadrature points
195: . points - The coordinates of each quadrature point
196: - weights - The weight of each quadrature point

198:   Level: intermediate

200: .keywords: PetscQuadrature, quadrature
201: .seealso: PetscQuadratureCreate(), PetscQuadratureSetData()
202: @*/
203: PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
204: {
207:   if (dim) {
209:     *dim = q->dim;
210:   }
211:   if (npoints) {
213:     *npoints = q->numPoints;
214:   }
215:   if (points) {
217:     *points = q->points;
218:   }
219:   if (weights) {
221:     *weights = q->weights;
222:   }
223:   return(0);
224: }

228: /*@C
229:   PetscQuadratureSetData - Sets the data defining the quadrature

231:   Not collective

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

240:   Level: intermediate

242: .keywords: PetscQuadrature, quadrature
243: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
244: @*/
245: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
246: {
249:   if (dim >= 0)     q->dim       = dim;
250:   if (npoints >= 0) q->numPoints = npoints;
251:   if (points) {
253:     q->points = points;
254:   }
255:   if (weights) {
257:     q->weights = weights;
258:   }
259:   return(0);
260: }

264: /*@C
265:   PetscQuadratureView - Views a PetscQuadrature object

267:   Collective on PetscQuadrature

269:   Input Parameters:
270: + q  - The PetscQuadrature object
271: - viewer - The PetscViewer object

273:   Level: beginner

275: .keywords: PetscQuadrature, quadrature, view
276: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
277: @*/
278: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
279: {
280:   PetscInt       q, d;

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

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

301:   Not collective

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

309:   Output Parameters:
310: . dim - The dimension

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

314:   Level: intermediate

316: .seealso: PetscFECreate(), PetscSpaceGetDimension(), PetscDualSpaceGetDimension()
317: @*/
318: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
319: {
320:   const PetscReal *points,    *weights;
321:   PetscReal       *pointsRef, *weightsRef;
322:   PetscInt         dim, order, npoints, npointsRef, c, p, d, e;
323:   PetscErrorCode   ierr;

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

355: /*@
356:    PetscDTLegendreEval - evaluate Legendre polynomial at points

358:    Not Collective

360:    Input Arguments:
361: +  npoints - number of spatial points to evaluate at
362: .  points - array of locations to evaluate at
363: .  ndegree - number of basis degrees to evaluate
364: -  degrees - sorted array of degrees to evaluate

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

371:    Level: intermediate

373: .seealso: PetscDTGaussQuadrature()
374: @*/
375: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
376: {
377:   PetscInt i,maxdegree;

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

422: /*@
423:    PetscDTGaussQuadrature - create Gauss quadrature

425:    Not Collective

427:    Input Arguments:
428: +  npoints - number of points
429: .  a - left end of interval (often-1)
430: -  b - right end of interval (often +1)

432:    Output Arguments:
433: +  x - quadrature points
434: -  w - quadrature weights

436:    Level: intermediate

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

441: .seealso: PetscDTLegendreEval()
442: @*/
443: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
444: {
446:   PetscInt       i;
447:   PetscReal      *work;
448:   PetscScalar    *Z;
449:   PetscBLASInt   N,LDZ,info;

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

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

471:     w[i] = w[npoints-1-i] = 0.5*(b-a)*(PetscSqr(PetscAbsScalar(Z[i*npoints])) + PetscSqr(PetscAbsScalar(Z[(npoints-i-1)*npoints])));
472:   }
473:   PetscFree2(Z,work);
474:   return(0);
475: }

479: /*@
480:   PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

482:   Not Collective

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

490:   Output Argument:
491: . q - A PetscQuadrature object

493:   Level: intermediate

495: .seealso: PetscDTGaussQuadrature(), PetscDTLegendreEval()
496: @*/
497: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
498: {
499:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k;
500:   PetscReal     *x, *w, *xw, *ww;

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

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

565:   for (i = 1; i < n+1; ++i) f *= i;
566:   *factorial = f;
567:   return(0);
568: }

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

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

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

604: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
605: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
606: {
607:   PetscReal      nP;

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

619: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
620: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapSquareToTriangle_Internal(PetscReal x, PetscReal y, PetscReal *xi, PetscReal *eta)
621: {
623:   *xi  = 0.5 * (1.0 + x) * (1.0 - y) - 1.0;
624:   *eta = y;
625:   return(0);
626: }

630: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
631: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapCubeToTetrahedron_Internal(PetscReal x, PetscReal y, PetscReal z, PetscReal *xi, PetscReal *eta, PetscReal *zeta)
632: {
634:   *xi   = 0.25 * (1.0 + x) * (1.0 - y) * (1.0 - z) - 1.0;
635:   *eta  = 0.5  * (1.0 + y) * (1.0 - z) - 1.0;
636:   *zeta = z;
637:   return(0);
638: }

642: static PetscErrorCode PetscDTGaussJacobiQuadrature1D_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
643: {
644:   PetscInt       maxIter = 100;
645:   PetscReal      eps     = 1.0e-8;
646:   PetscReal      a1, a2, a3, a4, a5, a6;
647:   PetscInt       k;


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

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

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

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

690: /*@C
691:   PetscDTGaussJacobiQuadrature - create Gauss-Jacobi quadrature for a simplex

693:   Not Collective

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

701:   Output Argument:
702: . q - A PetscQuadrature object

704:   Level: intermediate

706:   References:
707:   Karniadakis and Sherwin.
708:   FIAT

710: .seealso: PetscDTGaussTensorQuadrature(), PetscDTGaussQuadrature()
711: @*/
712: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt dim, PetscInt order, PetscReal a, PetscReal b, PetscQuadrature *q)
713: {
714:   PetscInt       npoints = dim > 1 ? dim > 2 ? order*PetscSqr(order) : PetscSqr(order) : order;
715:   PetscReal     *px, *wx, *py, *wy, *pz, *wz, *x, *w;
716:   PetscInt       i, j, k;

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

773: /* Overwrites A. Can only handle full-rank problems with m>=n
774:  * A in column-major format
775:  * Ainv in row-major format
776:  * tau has length m
777:  * worksize must be >= max(1,n)
778:  */
779: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
780: {
782:   PetscBLASInt   M,N,K,lda,ldb,ldwork,info;
783:   PetscScalar    *A,*Ainv,*R,*Q,Alpha;

786: #if defined(PETSC_USE_COMPLEX)
787:   {
788:     PetscInt i,j;
789:     PetscMalloc2(m*n,&A,m*n,&Ainv);
790:     for (j=0; j<n; j++) {
791:       for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
792:     }
793:     mstride = m;
794:   }
795: #else
796:   A = A_in;
797:   Ainv = Ainv_out;
798: #endif

800:   PetscBLASIntCast(m,&M);
801:   PetscBLASIntCast(n,&N);
802:   PetscBLASIntCast(mstride,&lda);
803:   PetscBLASIntCast(worksize,&ldwork);
804:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
805:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info));
806:   PetscFPTrapPop();
807:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
808:   R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

810:   /* Extract an explicit representation of Q */
811:   Q = Ainv;
812:   PetscMemcpy(Q,A,mstride*n*sizeof(PetscScalar));
813:   K = N;                        /* full rank */
814:   PetscStackCallBLAS("LAPACKungqr",LAPACKungqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info));
815:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");

817:   /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
818:   Alpha = 1.0;
819:   ldb = lda;
820:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb));
821:   /* Ainv is Q, overwritten with inverse */

823: #if defined(PETSC_USE_COMPLEX)
824:   {
825:     PetscInt i;
826:     for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
827:     PetscFree2(A,Ainv);
828:   }
829: #endif
830:   return(0);
831: }

835: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
836: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
837: {
839:   PetscReal      *Bv;
840:   PetscInt       i,j;

843:   PetscMalloc1((ninterval+1)*ndegree,&Bv);
844:   /* Point evaluation of L_p on all the source vertices */
845:   PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
846:   /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
847:   for (i=0; i<ninterval; i++) {
848:     for (j=0; j<ndegree; j++) {
849:       if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
850:       else           B[i*ndegree+j]   = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
851:     }
852:   }
853:   PetscFree(Bv);
854:   return(0);
855: }

859: /*@
860:    PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

862:    Not Collective

864:    Input Arguments:
865: +  degree - degree of reconstruction polynomial
866: .  nsource - number of source intervals
867: .  sourcex - sorted coordinates of source cell boundaries (length nsource+1)
868: .  ntarget - number of target intervals
869: -  targetx - sorted coordinates of target cell boundaries (length ntarget+1)

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

874:    Level: advanced

876: .seealso: PetscDTLegendreEval()
877: @*/
878: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
879: {
881:   PetscInt       i,j,k,*bdegrees,worksize;
882:   PetscReal      xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
883:   PetscScalar    *tau,*work;

889:   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);
890: #if defined(PETSC_USE_DEBUG)
891:   for (i=0; i<nsource; i++) {
892:     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]);
893:   }
894:   for (i=0; i<ntarget; i++) {
895:     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]);
896:   }
897: #endif
898:   xmin = PetscMin(sourcex[0],targetx[0]);
899:   xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
900:   center = (xmin + xmax)/2;
901:   hscale = (xmax - xmin)/2;
902:   worksize = nsource;
903:   PetscMalloc4(degree+1,&bdegrees,nsource+1,&sourcey,nsource*(degree+1),&Bsource,worksize,&work);
904:   PetscMalloc4(nsource,&tau,nsource*(degree+1),&Bsinv,ntarget+1,&targety,ntarget*(degree+1),&Btarget);
905:   for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
906:   for (i=0; i<=degree; i++) bdegrees[i] = i+1;
907:   PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
908:   PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
909:   for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
910:   PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
911:   for (i=0; i<ntarget; i++) {
912:     PetscReal rowsum = 0;
913:     for (j=0; j<nsource; j++) {
914:       PetscReal sum = 0;
915:       for (k=0; k<degree+1; k++) {
916:         sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
917:       }
918:       R[i*nsource+j] = sum;
919:       rowsum += sum;
920:     }
921:     for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
922:   }
923:   PetscFree4(bdegrees,sourcey,Bsource,work);
924:   PetscFree4(tau,Bsinv,targety,Btarget);
925:   return(0);
926: }