Actual source code: dt.c

petsc-master 2020-09-18
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  1: /* Discretization tools */

  3: #include <petscdt.h>
  4: #include <petscblaslapack.h>
  5: #include <petsc/private/petscimpl.h>
  6: #include <petsc/private/dtimpl.h>
  7: #include <petscviewer.h>
  8: #include <petscdmplex.h>
  9: #include <petscdmshell.h>

 11: #if defined(PETSC_HAVE_MPFR)
 12: #include <mpfr.h>
 13: #endif

 15: const char *const PetscDTNodeTypes[] = {"gaussjacobi", "equispaced", "tanhsinh", "PETSCDTNODES_", NULL};

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

 27: /* Numerical tests in src/dm/dt/tests/ex1.c show that when computing the nodes and weights of Gauss-Jacobi
 28:    quadrature rules:

 30:    - in double precision, Newton's method and Golub & Welsch both work for moderate degrees (< 100),
 31:    - in single precision, Newton's method starts producing incorrect roots around n = 15, but
 32:      the weights from Golub & Welsch become a problem before then: they produces errors
 33:      in computing the Jacobi-polynomial Gram matrix around n = 6.

 35:    So we default to Newton's method (required fewer dependencies) */
 36: PetscBool PetscDTGaussQuadratureNewton_Internal = PETSC_TRUE;

 38: PetscClassId PETSCQUADRATURE_CLASSID = 0;

 40: /*@
 41:   PetscQuadratureCreate - Create a PetscQuadrature object

 43:   Collective

 45:   Input Parameter:
 46: . comm - The communicator for the PetscQuadrature object

 48:   Output Parameter:
 49: . q  - The PetscQuadrature object

 51:   Level: beginner

 53: .seealso: PetscQuadratureDestroy(), PetscQuadratureGetData()
 54: @*/
 55: PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
 56: {

 61:   DMInitializePackage();
 62:   PetscHeaderCreate(*q,PETSCQUADRATURE_CLASSID,"PetscQuadrature","Quadrature","DT",comm,PetscQuadratureDestroy,PetscQuadratureView);
 63:   (*q)->dim       = -1;
 64:   (*q)->Nc        =  1;
 65:   (*q)->order     = -1;
 66:   (*q)->numPoints = 0;
 67:   (*q)->points    = NULL;
 68:   (*q)->weights   = NULL;
 69:   return(0);
 70: }

 72: /*@
 73:   PetscQuadratureDuplicate - Create a deep copy of the PetscQuadrature object

 75:   Collective on q

 77:   Input Parameter:
 78: . q  - The PetscQuadrature object

 80:   Output Parameter:
 81: . r  - The new PetscQuadrature object

 83:   Level: beginner

 85: .seealso: PetscQuadratureCreate(), PetscQuadratureDestroy(), PetscQuadratureGetData()
 86: @*/
 87: PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
 88: {
 89:   PetscInt         order, dim, Nc, Nq;
 90:   const PetscReal *points, *weights;
 91:   PetscReal       *p, *w;
 92:   PetscErrorCode   ierr;

 96:   PetscQuadratureCreate(PetscObjectComm((PetscObject) q), r);
 97:   PetscQuadratureGetOrder(q, &order);
 98:   PetscQuadratureSetOrder(*r, order);
 99:   PetscQuadratureGetData(q, &dim, &Nc, &Nq, &points, &weights);
100:   PetscMalloc1(Nq*dim, &p);
101:   PetscMalloc1(Nq*Nc, &w);
102:   PetscArraycpy(p, points, Nq*dim);
103:   PetscArraycpy(w, weights, Nc * Nq);
104:   PetscQuadratureSetData(*r, dim, Nc, Nq, p, w);
105:   return(0);
106: }

108: /*@
109:   PetscQuadratureDestroy - Destroys a PetscQuadrature object

111:   Collective on q

113:   Input Parameter:
114: . q  - The PetscQuadrature object

116:   Level: beginner

118: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
119: @*/
120: PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
121: {

125:   if (!*q) return(0);
127:   if (--((PetscObject)(*q))->refct > 0) {
128:     *q = NULL;
129:     return(0);
130:   }
131:   PetscFree((*q)->points);
132:   PetscFree((*q)->weights);
133:   PetscHeaderDestroy(q);
134:   return(0);
135: }

137: /*@
138:   PetscQuadratureGetOrder - Return the order of the method

140:   Not collective

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

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

148:   Level: intermediate

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

161: /*@
162:   PetscQuadratureSetOrder - Return the order of the method

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: }

182: /*@
183:   PetscQuadratureGetNumComponents - Return the number of components for functions to be integrated

185:   Not collective

187:   Input Parameter:
188: . q - The PetscQuadrature object

190:   Output Parameter:
191: . Nc - The number of components

193:   Note: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.

195:   Level: intermediate

197: .seealso: PetscQuadratureSetNumComponents(), PetscQuadratureGetData(), PetscQuadratureSetData()
198: @*/
199: PetscErrorCode PetscQuadratureGetNumComponents(PetscQuadrature q, PetscInt *Nc)
200: {
204:   *Nc = q->Nc;
205:   return(0);
206: }

208: /*@
209:   PetscQuadratureSetNumComponents - Return the number of components for functions to be integrated

211:   Not collective

213:   Input Parameters:
214: + q  - The PetscQuadrature object
215: - Nc - The number of components

217:   Note: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.

219:   Level: intermediate

221: .seealso: PetscQuadratureGetNumComponents(), PetscQuadratureGetData(), PetscQuadratureSetData()
222: @*/
223: PetscErrorCode PetscQuadratureSetNumComponents(PetscQuadrature q, PetscInt Nc)
224: {
227:   q->Nc = Nc;
228:   return(0);
229: }

231: /*@C
232:   PetscQuadratureGetData - Returns the data defining the quadrature

234:   Not collective

236:   Input Parameter:
237: . q  - The PetscQuadrature object

239:   Output Parameters:
240: + dim - The spatial dimension
241: . Nc - The number of components
242: . npoints - The number of quadrature points
243: . points - The coordinates of each quadrature point
244: - weights - The weight of each quadrature point

246:   Level: intermediate

248:   Fortran Notes:
249:     From Fortran you must call PetscQuadratureRestoreData() when you are done with the data

251: .seealso: PetscQuadratureCreate(), PetscQuadratureSetData()
252: @*/
253: PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *Nc, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
254: {
257:   if (dim) {
259:     *dim = q->dim;
260:   }
261:   if (Nc) {
263:     *Nc = q->Nc;
264:   }
265:   if (npoints) {
267:     *npoints = q->numPoints;
268:   }
269:   if (points) {
271:     *points = q->points;
272:   }
273:   if (weights) {
275:     *weights = q->weights;
276:   }
277:   return(0);
278: }

280: static PetscErrorCode PetscDTJacobianInverse_Internal(PetscInt m, PetscInt n, const PetscReal J[], PetscReal Jinv[])
281: {
282:   PetscScalar    *Js, *Jinvs;
283:   PetscInt       i, j, k;
284:   PetscBLASInt   bm, bn, info;

288:   if (!m || !n) return(0);
289:   PetscBLASIntCast(m, &bm);
290:   PetscBLASIntCast(n, &bn);
291: #if defined(PETSC_USE_COMPLEX)
292:   PetscMalloc2(m*n, &Js, m*n, &Jinvs);
293:   for (i = 0; i < m*n; i++) Js[i] = J[i];
294: #else
295:   Js = (PetscReal *) J;
296:   Jinvs = Jinv;
297: #endif
298:   if (m == n) {
299:     PetscBLASInt *pivots;
300:     PetscScalar *W;

302:     PetscMalloc2(m, &pivots, m, &W);

304:     PetscArraycpy(Jinvs, Js, m * m);
305:     PetscStackCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bm, &bm, Jinvs, &bm, pivots, &info));
306:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetrf %D",(PetscInt)info);
307:     PetscStackCallBLAS("LAPACKgetri", LAPACKgetri_(&bm, Jinvs, &bm, pivots, W, &bm, &info));
308:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetri %D",(PetscInt)info);
309:     PetscFree2(pivots, W);
310:   } else if (m < n) {
311:     PetscScalar *JJT;
312:     PetscBLASInt *pivots;
313:     PetscScalar *W;

315:     PetscMalloc1(m*m, &JJT);
316:     PetscMalloc2(m, &pivots, m, &W);
317:     for (i = 0; i < m; i++) {
318:       for (j = 0; j < m; j++) {
319:         PetscScalar val = 0.;

321:         for (k = 0; k < n; k++) val += Js[i * n + k] * Js[j * n + k];
322:         JJT[i * m + j] = val;
323:       }
324:     }

326:     PetscStackCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bm, &bm, JJT, &bm, pivots, &info));
327:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetrf %D",(PetscInt)info);
328:     PetscStackCallBLAS("LAPACKgetri", LAPACKgetri_(&bm, JJT, &bm, pivots, W, &bm, &info));
329:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetri %D",(PetscInt)info);
330:     for (i = 0; i < n; i++) {
331:       for (j = 0; j < m; j++) {
332:         PetscScalar val = 0.;

334:         for (k = 0; k < m; k++) val += Js[k * n + i] * JJT[k * m + j];
335:         Jinvs[i * m + j] = val;
336:       }
337:     }
338:     PetscFree2(pivots, W);
339:     PetscFree(JJT);
340:   } else {
341:     PetscScalar *JTJ;
342:     PetscBLASInt *pivots;
343:     PetscScalar *W;

345:     PetscMalloc1(n*n, &JTJ);
346:     PetscMalloc2(n, &pivots, n, &W);
347:     for (i = 0; i < n; i++) {
348:       for (j = 0; j < n; j++) {
349:         PetscScalar val = 0.;

351:         for (k = 0; k < m; k++) val += Js[k * n + i] * Js[k * n + j];
352:         JTJ[i * n + j] = val;
353:       }
354:     }

356:     PetscStackCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bn, &bn, JTJ, &bn, pivots, &info));
357:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetrf %D",(PetscInt)info);
358:     PetscStackCallBLAS("LAPACKgetri", LAPACKgetri_(&bn, JTJ, &bn, pivots, W, &bn, &info));
359:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetri %D",(PetscInt)info);
360:     for (i = 0; i < n; i++) {
361:       for (j = 0; j < m; j++) {
362:         PetscScalar val = 0.;

364:         for (k = 0; k < n; k++) val += JTJ[i * n + k] * Js[j * n + k];
365:         Jinvs[i * m + j] = val;
366:       }
367:     }
368:     PetscFree2(pivots, W);
369:     PetscFree(JTJ);
370:   }
371: #if defined(PETSC_USE_COMPLEX)
372:   for (i = 0; i < m*n; i++) Jinv[i] = PetscRealPart(Jinvs[i]);
373:   PetscFree2(Js, Jinvs);
374: #endif
375:   return(0);
376: }

378: /*@
379:    PetscQuadraturePushForward - Push forward a quadrature functional under an affine transformation.

381:    Collecive on PetscQuadrature

383:    Input Arguments:
384: +  q - the quadrature functional
385: .  imageDim - the dimension of the image of the transformation
386: .  origin - a point in the original space
387: .  originImage - the image of the origin under the transformation
388: .  J - the Jacobian of the image: an [imageDim x dim] matrix in row major order
389: -  formDegree - transform the quadrature weights as k-forms of this form degree (if the number of components is a multiple of (dim choose formDegree), it is assumed that they represent multiple k-forms) [see PetscDTAltVPullback() for interpretation of formDegree]

391:    Output Arguments:
392: .  Jinvstarq - a quadrature rule where each point is the image of a point in the original quadrature rule, and where the k-form weights have been pulled-back by the pseudoinverse of J to the k-form weights in the image space.

394:    Note: the new quadrature rule will have a different number of components if spaces have different dimensions.  For example, pushing a 2-form forward from a two dimensional space to a three dimensional space changes the number of components from 1 to 3.

396:    Level: intermediate

398: .seealso: PetscDTAltVPullback(), PetscDTAltVPullbackMatrix()
399: @*/
400: PetscErrorCode PetscQuadraturePushForward(PetscQuadrature q, PetscInt imageDim, const PetscReal origin[], const PetscReal originImage[], const PetscReal J[], PetscInt formDegree, PetscQuadrature *Jinvstarq)
401: {
402:   PetscInt         dim, Nc, imageNc, formSize, Ncopies, imageFormSize, Npoints, pt, i, j, c;
403:   const PetscReal *points;
404:   const PetscReal *weights;
405:   PetscReal       *imagePoints, *imageWeights;
406:   PetscReal       *Jinv;
407:   PetscReal       *Jinvstar;
408:   PetscErrorCode   ierr;

412:   if (imageDim < PetscAbsInt(formDegree)) SETERRQ2(PetscObjectComm((PetscObject)q), PETSC_ERR_ARG_INCOMP, "Cannot represent a %D-form in %D dimensions", PetscAbsInt(formDegree), imageDim);
413:   PetscQuadratureGetData(q, &dim, &Nc, &Npoints, &points, &weights);
414:   PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &formSize);
415:   if (Nc % formSize) SETERRQ2(PetscObjectComm((PetscObject)q), PETSC_ERR_ARG_INCOMP, "Number of components %D is not a multiple of formSize %D\n", Nc, formSize);
416:   Ncopies = Nc / formSize;
417:   PetscDTBinomialInt(imageDim, PetscAbsInt(formDegree), &imageFormSize);
418:   imageNc = Ncopies * imageFormSize;
419:   PetscMalloc1(Npoints * imageDim, &imagePoints);
420:   PetscMalloc1(Npoints * imageNc, &imageWeights);
421:   PetscMalloc2(imageDim * dim, &Jinv, formSize * imageFormSize, &Jinvstar);
422:   PetscDTJacobianInverse_Internal(imageDim, dim, J, Jinv);
423:   PetscDTAltVPullbackMatrix(imageDim, dim, Jinv, formDegree, Jinvstar);
424:   for (pt = 0; pt < Npoints; pt++) {
425:     const PetscReal *point = &points[pt * dim];
426:     PetscReal       *imagePoint = &imagePoints[pt * imageDim];

428:     for (i = 0; i < imageDim; i++) {
429:       PetscReal val = originImage[i];

431:       for (j = 0; j < dim; j++) val += J[i * dim + j] * (point[j] - origin[j]);
432:       imagePoint[i] = val;
433:     }
434:     for (c = 0; c < Ncopies; c++) {
435:       const PetscReal *form = &weights[pt * Nc + c * formSize];
436:       PetscReal       *imageForm = &imageWeights[pt * imageNc + c * imageFormSize];

438:       for (i = 0; i < imageFormSize; i++) {
439:         PetscReal val = 0.;

441:         for (j = 0; j < formSize; j++) val += Jinvstar[i * formSize + j] * form[j];
442:         imageForm[i] = val;
443:       }
444:     }
445:   }
446:   PetscQuadratureCreate(PetscObjectComm((PetscObject)q), Jinvstarq);
447:   PetscQuadratureSetData(*Jinvstarq, imageDim, imageNc, Npoints, imagePoints, imageWeights);
448:   PetscFree2(Jinv, Jinvstar);
449:   return(0);
450: }

452: /*@C
453:   PetscQuadratureSetData - Sets the data defining the quadrature

455:   Not collective

457:   Input Parameters:
458: + q  - The PetscQuadrature object
459: . dim - The spatial dimension
460: . Nc - The number of components
461: . npoints - The number of quadrature points
462: . points - The coordinates of each quadrature point
463: - weights - The weight of each quadrature point

465:   Note: This routine owns the references to points and weights, so they must be allocated using PetscMalloc() and the user should not free them.

467:   Level: intermediate

469: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
470: @*/
471: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt Nc, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
472: {
475:   if (dim >= 0)     q->dim       = dim;
476:   if (Nc >= 0)      q->Nc        = Nc;
477:   if (npoints >= 0) q->numPoints = npoints;
478:   if (points) {
480:     q->points = points;
481:   }
482:   if (weights) {
484:     q->weights = weights;
485:   }
486:   return(0);
487: }

489: static PetscErrorCode PetscQuadratureView_Ascii(PetscQuadrature quad, PetscViewer v)
490: {
491:   PetscInt          q, d, c;
492:   PetscViewerFormat format;
493:   PetscErrorCode    ierr;

496:   if (quad->Nc > 1) {PetscViewerASCIIPrintf(v, "Quadrature of order %D on %D points (dim %D) with %D components\n", quad->order, quad->numPoints, quad->dim, quad->Nc);}
497:   else              {PetscViewerASCIIPrintf(v, "Quadrature of order %D on %D points (dim %D)\n", quad->order, quad->numPoints, quad->dim);}
498:   PetscViewerGetFormat(v, &format);
499:   if (format != PETSC_VIEWER_ASCII_INFO_DETAIL) return(0);
500:   for (q = 0; q < quad->numPoints; ++q) {
501:     PetscViewerASCIIPrintf(v, "p%D (", q);
502:     PetscViewerASCIIUseTabs(v, PETSC_FALSE);
503:     for (d = 0; d < quad->dim; ++d) {
504:       if (d) {PetscViewerASCIIPrintf(v, ", ");}
505:       PetscViewerASCIIPrintf(v, "%+g", (double)quad->points[q*quad->dim+d]);
506:     }
507:     PetscViewerASCIIPrintf(v, ") ");
508:     if (quad->Nc > 1) {PetscViewerASCIIPrintf(v, "w%D (", q);}
509:     for (c = 0; c < quad->Nc; ++c) {
510:       if (c) {PetscViewerASCIIPrintf(v, ", ");}
511:       PetscViewerASCIIPrintf(v, "%+g", (double)quad->weights[q*quad->Nc+c]);
512:     }
513:     if (quad->Nc > 1) {PetscViewerASCIIPrintf(v, ")");}
514:     PetscViewerASCIIPrintf(v, "\n");
515:     PetscViewerASCIIUseTabs(v, PETSC_TRUE);
516:   }
517:   return(0);
518: }

520: /*@C
521:   PetscQuadratureView - Views a PetscQuadrature object

523:   Collective on quad

525:   Input Parameters:
526: + quad  - The PetscQuadrature object
527: - viewer - The PetscViewer object

529:   Level: beginner

531: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
532: @*/
533: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
534: {
535:   PetscBool      iascii;

541:   if (!viewer) {PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject) quad), &viewer);}
542:   PetscObjectTypeCompare((PetscObject) viewer, PETSCVIEWERASCII, &iascii);
543:   PetscViewerASCIIPushTab(viewer);
544:   if (iascii) {PetscQuadratureView_Ascii(quad, viewer);}
545:   PetscViewerASCIIPopTab(viewer);
546:   return(0);
547: }

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

552:   Not collective

554:   Input Parameter:
555: + q - The original PetscQuadrature
556: . numSubelements - The number of subelements the original element is divided into
557: . v0 - An array of the initial points for each subelement
558: - jac - An array of the Jacobian mappings from the reference to each subelement

560:   Output Parameters:
561: . dim - The dimension

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

565:  Not available from Fortran

567:   Level: intermediate

569: .seealso: PetscFECreate(), PetscSpaceGetDimension(), PetscDualSpaceGetDimension()
570: @*/
571: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
572: {
573:   const PetscReal *points,    *weights;
574:   PetscReal       *pointsRef, *weightsRef;
575:   PetscInt         dim, Nc, order, npoints, npointsRef, c, p, cp, d, e;
576:   PetscErrorCode   ierr;

583:   PetscQuadratureCreate(PETSC_COMM_SELF, qref);
584:   PetscQuadratureGetOrder(q, &order);
585:   PetscQuadratureGetData(q, &dim, &Nc, &npoints, &points, &weights);
586:   npointsRef = npoints*numSubelements;
587:   PetscMalloc1(npointsRef*dim,&pointsRef);
588:   PetscMalloc1(npointsRef*Nc, &weightsRef);
589:   for (c = 0; c < numSubelements; ++c) {
590:     for (p = 0; p < npoints; ++p) {
591:       for (d = 0; d < dim; ++d) {
592:         pointsRef[(c*npoints + p)*dim+d] = v0[c*dim+d];
593:         for (e = 0; e < dim; ++e) {
594:           pointsRef[(c*npoints + p)*dim+d] += jac[(c*dim + d)*dim+e]*(points[p*dim+e] + 1.0);
595:         }
596:       }
597:       /* Could also use detJ here */
598:       for (cp = 0; cp < Nc; ++cp) weightsRef[(c*npoints+p)*Nc+cp] = weights[p*Nc+cp]/numSubelements;
599:     }
600:   }
601:   PetscQuadratureSetOrder(*qref, order);
602:   PetscQuadratureSetData(*qref, dim, Nc, npointsRef, pointsRef, weightsRef);
603:   return(0);
604: }

606: /* Compute the coefficients for the Jacobi polynomial recurrence,
607:  *
608:  * J^{a,b}_n(x) = (cnm1 + cnm1x * x) * J^{a,b}_{n-1}(x) - cnm2 * J^{a,b}_{n-2}(x).
609:  */
610: #define PetscDTJacobiRecurrence_Internal(n,a,b,cnm1,cnm1x,cnm2) \
611: do {                                                            \
612:   PetscReal _a = (a);                                           \
613:   PetscReal _b = (b);                                           \
614:   PetscReal _n = (n);                                           \
615:   if (n == 1) {                                                 \
616:     (cnm1) = (_a-_b) * 0.5;                                     \
617:     (cnm1x) = (_a+_b+2.)*0.5;                                   \
618:     (cnm2) = 0.;                                                \
619:   } else {                                                      \
620:     PetscReal _2n = _n+_n;                                      \
621:     PetscReal _d = (_2n*(_n+_a+_b)*(_2n+_a+_b-2));              \
622:     PetscReal _n1 = (_2n+_a+_b-1.)*(_a*_a-_b*_b);               \
623:     PetscReal _n1x = (_2n+_a+_b-1.)*(_2n+_a+_b)*(_2n+_a+_b-2);  \
624:     PetscReal _n2 = 2.*((_n+_a-1.)*(_n+_b-1.)*(_2n+_a+_b));     \
625:     (cnm1) = _n1 / _d;                                          \
626:     (cnm1x) = _n1x / _d;                                        \
627:     (cnm2) = _n2 / _d;                                          \
628:   }                                                             \
629: } while (0)

631: /*@
632:   PetscDTJacobiNorm - Compute the weighted L2 norm of a Jacobi polynomial.

634:   $\| P^{\alpha,\beta}_n \|_{\alpha,\beta}^2 = \int_{-1}^1 (1 + x)^{\alpha} (1 - x)^{\beta} P^{\alpha,\beta}_n (x)^2 dx.$

636:   Input Arguments:
637: - alpha - the left exponent > -1
638: . beta - the right exponent > -1
639: + n - the polynomial degree

641:   Output Arguments:
642: . norm - the weighted L2 norm

644:   Level: beginner

646: .seealso: PetscDTJacobiEval()
647: @*/
648: PetscErrorCode PetscDTJacobiNorm(PetscReal alpha, PetscReal beta, PetscInt n, PetscReal *norm)
649: {
650:   PetscReal twoab1;
651:   PetscReal gr;

654:   if (alpha <= -1.) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Exponent alpha %g <= -1. invalid\n", (double) alpha);
655:   if (beta <= -1.) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Exponent beta %g <= -1. invalid\n", (double) beta);
656:   if (n < 0) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "n %D < 0 invalid\n", n);
657:   twoab1 = PetscPowReal(2., alpha + beta + 1.);
658: #if defined(PETSC_HAVE_LGAMMA)
659:   if (!n) {
660:     gr = PetscExpReal(PetscLGamma(alpha+1.) + PetscLGamma(beta+1.) - PetscLGamma(alpha+beta+2.));
661:   } else {
662:     gr = PetscExpReal(PetscLGamma(n+alpha+1.) + PetscLGamma(n+beta+1.) - (PetscLGamma(n+1.) + PetscLGamma(n+alpha+beta+1.))) / (n+n+alpha+beta+1.);
663:   }
664: #else
665:   {
666:     PetscInt alphai = (PetscInt) alpha;
667:     PetscInt betai = (PetscInt) beta;
668:     PetscInt i;

670:     gr = n ? (1. / (n+n+alpha+beta+1.)) : 1.;
671:     if ((PetscReal) alphai == alpha) {
672:       if (!n) {
673:         for (i = 0; i < alphai; i++) gr *= (i+1.) / (beta+i+1.);
674:         gr /= (alpha+beta+1.);
675:       } else {
676:         for (i = 0; i < alphai; i++) gr *= (n+i+1.) / (n+beta+i+1.);
677:       }
678:     } else if ((PetscReal) betai == beta) {
679:       if (!n) {
680:         for (i = 0; i < betai; i++) gr *= (i+1.) / (alpha+i+2.);
681:         gr /= (alpha+beta+1.);
682:       } else {
683:         for (i = 0; i < betai; i++) gr *= (n+i+1.) / (n+alpha+i+1.);
684:       }
685:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"lgamma() - math routine is unavailable.");
686:   }
687: #endif
688:   *norm = PetscSqrtReal(twoab1 * gr);
689:   return(0);
690: }

692: static PetscErrorCode PetscDTJacobiEval_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscInt k, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *p)
693: {
694:   PetscReal ak, bk;
695:   PetscReal abk1;
696:   PetscInt i,l,maxdegree;

699:   maxdegree = degrees[ndegree-1] - k;
700:   ak = a + k;
701:   bk = b + k;
702:   abk1 = a + b + k + 1.;
703:   if (maxdegree < 0) {
704:     for (i = 0; i < npoints; i++) for (l = 0; l < ndegree; l++) p[i*ndegree+l] = 0.;
705:     return(0);
706:   }
707:   for (i=0; i<npoints; i++) {
708:     PetscReal pm1,pm2,x;
709:     PetscReal cnm1, cnm1x, cnm2;
710:     PetscInt  j,m;

712:     x    = points[i];
713:     pm2  = 1.;
714:     PetscDTJacobiRecurrence_Internal(1,ak,bk,cnm1,cnm1x,cnm2);
715:     pm1 = (cnm1 + cnm1x*x);
716:     l    = 0;
717:     while (l < ndegree && degrees[l] - k < 0) {
718:       p[l++] = 0.;
719:     }
720:     while (l < ndegree && degrees[l] - k == 0) {
721:       p[l] = pm2;
722:       for (m = 0; m < k; m++) p[l] *= (abk1 + m) * 0.5;
723:       l++;
724:     }
725:     while (l < ndegree && degrees[l] - k == 1) {
726:       p[l] = pm1;
727:       for (m = 0; m < k; m++) p[l] *= (abk1 + 1 + m) * 0.5;
728:       l++;
729:     }
730:     for (j=2; j<=maxdegree; j++) {
731:       PetscReal pp;

733:       PetscDTJacobiRecurrence_Internal(j,ak,bk,cnm1,cnm1x,cnm2);
734:       pp   = (cnm1 + cnm1x*x)*pm1 - cnm2*pm2;
735:       pm2  = pm1;
736:       pm1  = pp;
737:       while (l < ndegree && degrees[l] - k == j) {
738:         p[l] = pp;
739:         for (m = 0; m < k; m++) p[l] *= (abk1 + j + m) * 0.5;
740:         l++;
741:       }
742:     }
743:     p += ndegree;
744:   }
745:   return(0);
746: }

748: /*@
749:   PetscDTJacobiEvalJet - Evaluate the jet (function and derivatives) of the Jacobi polynomials basis up to a given degree.  The Jacobi polynomials with indices $\alpha$ and $\beta$ are orthogonal with respect to the weighted inner product $\langle f, g \rangle = \int_{-1}^1 (1+x)^{\alpha} (1-x)^{\beta) f(x) g(x) dx$.

751:   Input Arguments:
752: + alpha - the left exponent of the weight
753: . beta - the right exponetn of the weight
754: . npoints - the number of points to evaluate the polynomials at
755: . points - [npoints] array of point coordinates
756: . degree - the maximm degree polynomial space to evaluate, (degree + 1) will be evaluated total.
757: - k - the maximum derivative to evaluate in the jet, (k + 1) will be evaluated total.

759:   Output Argments:
760: - p - an array containing the evaluations of the Jacobi polynomials's jets on the points.  the size is (degree + 1) x
761:   (k + 1) x npoints, which also describes the order of the dimensions of this three-dimensional array: the first
762:   (slowest varying) dimension is polynomial degree; the second dimension is derivative order; the third (fastest
763:   varying) dimension is the index of the evaluation point.

765:   Level: advanced

767: .seealso: PetscDTJacobiEval(), PetscDTPKDEvalJet()
768: @*/
769: PetscErrorCode PetscDTJacobiEvalJet(PetscReal alpha, PetscReal beta, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
770: {
771:   PetscInt        i, j, l;
772:   PetscInt       *degrees;
773:   PetscReal      *psingle;
774:   PetscErrorCode  ierr;

777:   if (degree == 0) {
778:     PetscInt zero = 0;

780:     for (i = 0; i <= k; i++) {
781:       PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, 1, &zero, &p[i*npoints]);
782:     }
783:     return(0);
784:   }
785:   PetscMalloc1(degree + 1, &degrees);
786:   PetscMalloc1((degree + 1) * npoints, &psingle);
787:   for (i = 0; i <= degree; i++) degrees[i] = i;
788:   for (i = 0; i <= k; i++) {
789:     PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, degree + 1, degrees, psingle);
790:     for (j = 0; j <= degree; j++) {
791:       for (l = 0; l < npoints; l++) {
792:         p[(j * (k + 1) + i) * npoints + l] = psingle[l * (degree + 1) + j];
793:       }
794:     }
795:   }
796:   PetscFree(psingle);
797:   PetscFree(degrees);
798:   return(0);
799: }

801: /*@
802:    PetscDTJacobiEval - evaluate Jacobi polynomials for the weight function $(1.+x)^{\alpha} (1.-x)^{\beta}$
803:                        at points

805:    Not Collective

807:    Input Arguments:
808: +  npoints - number of spatial points to evaluate at
809: .  alpha - the left exponent > -1
810: .  beta - the right exponent > -1
811: .  points - array of locations to evaluate at
812: .  ndegree - number of basis degrees to evaluate
813: -  degrees - sorted array of degrees to evaluate

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

820:    Level: intermediate

822: .seealso: PetscDTGaussQuadrature()
823: @*/
824: PetscErrorCode PetscDTJacobiEval(PetscInt npoints,PetscReal alpha, PetscReal beta, const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
825: {

829:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
830:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");
831:   if (!npoints || !ndegree) return(0);
832:   if (B)  {PetscDTJacobiEval_Internal(npoints, alpha, beta, 0, points, ndegree, degrees, B);}
833:   if (D)  {PetscDTJacobiEval_Internal(npoints, alpha, beta, 1, points, ndegree, degrees, D);}
834:   if (D2) {PetscDTJacobiEval_Internal(npoints, alpha, beta, 2, points, ndegree, degrees, D2);}
835:   return(0);
836: }

838: /*@
839:    PetscDTLegendreEval - evaluate Legendre polynomials at points

841:    Not Collective

843:    Input Arguments:
844: +  npoints - number of spatial points to evaluate at
845: .  points - array of locations to evaluate at
846: .  ndegree - number of basis degrees to evaluate
847: -  degrees - sorted array of degrees to evaluate

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

854:    Level: intermediate

856: .seealso: PetscDTGaussQuadrature()
857: @*/
858: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
859: {

863:   PetscDTJacobiEval(npoints, 0., 0., points, ndegree, degrees, B, D, D2);
864:   return(0);
865: }

867: /*@
868:   PetscDTIndexToGradedOrder - convert an index into a tuple of monomial degrees in a graded order (that is, if the degree sum of tuple x is less than the degree sum of tuple y, then the index of x is smaller than the index of y)

870:   Input Parameters:
871: + len - the desired length of the degree tuple
872: - index - the index to convert: should be >= 0

874:   Output Parameter:
875: . degtup - will be filled with a tuple of degrees

877:   Level: beginner

879:   Note: for two tuples x and y with the same degree sum, partial degree sums over the final elements of the tuples
880:   acts as a tiebreaker.  For example, (2, 1, 1) and (1, 2, 1) have the same degree sum, but the degree sum over the
881:   last two elements is smaller for the former, so (2, 1, 1) < (1, 2, 1).

883: .seealso: PetscDTGradedOrderToIndex()
884: @*/
885: PetscErrorCode PetscDTIndexToGradedOrder(PetscInt len, PetscInt index, PetscInt degtup[])
886: {
887:   PetscInt i, total;
888:   PetscInt sum;

891:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
892:   if (index < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
893:   total = 1;
894:   sum = 0;
895:   while (index >= total) {
896:     index -= total;
897:     total = (total * (len + sum)) / (sum + 1);
898:     sum++;
899:   }
900:   for (i = 0; i < len; i++) {
901:     PetscInt c;

903:     degtup[i] = sum;
904:     for (c = 0, total = 1; c < sum; c++) {
905:       /* going into the loop, total is the number of way to have a tuple of sum exactly c with length len - 1 - i */
906:       if (index < total) break;
907:       index -= total;
908:       total = (total * (len - 1 - i + c)) / (c + 1);
909:       degtup[i]--;
910:     }
911:     sum -= degtup[i];
912:   }
913:   return(0);
914: }

916: /*@
917:   PetscDTGradedOrderToIndex - convert a tuple into an index in a graded order, the inverse of PetscDTIndexToGradedOrder().

919:   Input Parameters:
920: + len - the length of the degree tuple
921: - degtup - tuple with this length

923:   Output Parameter:
924: . index - index in graded order: >= 0

926:   Level: Beginner

928:   Note: for two tuples x and y with the same degree sum, partial degree sums over the final elements of the tuples
929:   acts as a tiebreaker.  For example, (2, 1, 1) and (1, 2, 1) have the same degree sum, but the degree sum over the
930:   last two elements is smaller for the former, so (2, 1, 1) < (1, 2, 1).

932: .seealso: PetscDTIndexToGradedOrder()
933: @*/
934: PetscErrorCode PetscDTGradedOrderToIndex(PetscInt len, const PetscInt degtup[], PetscInt *index)
935: {
936:   PetscInt i, idx, sum, total;

939:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
940:   for (i = 0, sum = 0; i < len; i++) sum += degtup[i];
941:   idx = 0;
942:   total = 1;
943:   for (i = 0; i < sum; i++) {
944:     idx += total;
945:     total = (total * (len + i)) / (i + 1);
946:   }
947:   for (i = 0; i < len - 1; i++) {
948:     PetscInt c;

950:     total = 1;
951:     sum -= degtup[i];
952:     for (c = 0; c < sum; c++) {
953:       idx += total;
954:       total = (total * (len - 1 - i + c)) / (c + 1);
955:     }
956:   }
957:   *index = idx;
958:   return(0);
959: }

961: static PetscBool PKDCite = PETSC_FALSE;
962: const char       PKDCitation[] = "@article{Kirby2010,\n"
963:                                  "  title={Singularity-free evaluation of collapsed-coordinate orthogonal polynomials},\n"
964:                                  "  author={Kirby, Robert C},\n"
965:                                  "  journal={ACM Transactions on Mathematical Software (TOMS)},\n"
966:                                  "  volume={37},\n"
967:                                  "  number={1},\n"
968:                                  "  pages={1--16},\n"
969:                                  "  year={2010},\n"
970:                                  "  publisher={ACM New York, NY, USA}\n}\n";

972: /*@
973:   PetscDTPKDEvalJet - Evaluate the jet (function and derivatives) of the Prioriol-Koornwinder-Dubiner (PKD) basis for
974:   the space of polynomials up to a given degree.  The PKD basis is L2-orthonormal on the biunit simplex (which is used
975:   as the reference element for finite elements in PETSc), which makes it a stable basis to use for evaluating
976:   polynomials in that domain.

978:   Input Arguments:
979: + dim - the number of variables in the multivariate polynomials
980: . npoints - the number of points to evaluate the polynomials at
981: . points - [npoints x dim] array of point coordinates
982: . degree - the degree (sum of degrees on the variables in a monomial) of the polynomial space to evaluate.  There are ((dim + degree) choose dim) polynomials in this space.
983: - k - the maximum order partial derivative to evaluate in the jet.  There are (dim + k choose dim) partial derivatives
984:   in the jet.  Choosing k = 0 means to evaluate just the function and no derivatives

986:   Output Argments:
987: - p - an array containing the evaluations of the PKD polynomials' jets on the points.  The size is ((dim + degree)
988:   choose dim) x ((dim + k) choose dim) x npoints, which also describes the order of the dimensions of this
989:   three-dimensional array: the first (slowest varying) dimension is basis function index; the second dimension is jet
990:   index; the third (fastest varying) dimension is the index of the evaluation point.

992:   Level: advanced

994:   Note: The ordering of the basis functions, and the ordering of the derivatives in the jet, both follow the graded
995:   ordering of PetscDTIndexToGradedOrder() and PetscDTGradedOrderToIndex().  For example, in 3D, the polynomial with
996:   leading monomial x^3,y^1,z^2, which as degree tuple (2,0,1), which by PetscDTGradedOrderToIndex() has index 12 (it is the 13th basis function in the space);
997:   the partial derivative $\partial_x \partial_z$ has order tuple (1,0,1), appears at index 6 in the jet (it is the 7th partial derivative in the jet).

999:   The implementation uses Kirby's singularity-free evaluation algorithm, https://doi.org/10.1145/1644001.1644006.

1001: .seealso: PetscDTGradedOrderToIndex(), PetscDTIndexToGradedOrder(), PetscDTJacobiEvalJet()
1002: @*/
1003: PetscErrorCode PetscDTPKDEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
1004: {
1005:   PetscInt        degidx, kidx, d, pt;
1006:   PetscInt        Nk, Ndeg;
1007:   PetscInt       *ktup, *degtup;
1008:   PetscReal      *scales, initscale, scaleexp;
1009:   PetscErrorCode  ierr;

1012:   PetscCitationsRegister(PKDCitation, &PKDCite);
1013:   PetscDTBinomialInt(dim + k, k, &Nk);
1014:   PetscDTBinomialInt(degree + dim, degree, &Ndeg);
1015:   PetscMalloc2(dim, &degtup, dim, &ktup);
1016:   PetscMalloc1(Ndeg, &scales);
1017:   initscale = 1.;
1018:   if (dim > 1) {
1019:     PetscDTBinomial(dim,2,&scaleexp);
1020:     initscale = PetscPowReal(2.,scaleexp*0.5);
1021:   }
1022:   for (degidx = 0; degidx < Ndeg; degidx++) {
1023:     PetscInt e, i;
1024:     PetscInt m1idx = -1, m2idx = -1;
1025:     PetscInt n;
1026:     PetscInt degsum;
1027:     PetscReal alpha;
1028:     PetscReal cnm1, cnm1x, cnm2;
1029:     PetscReal norm;

1031:     PetscDTIndexToGradedOrder(dim, degidx, degtup);
1032:     for (d = dim - 1; d >= 0; d--) if (degtup[d]) break;
1033:     if (d < 0) { /* constant is 1 everywhere, all derivatives are zero */
1034:       scales[degidx] = initscale;
1035:       for (e = 0; e < dim; e++) {
1036:         PetscDTJacobiNorm(e,0.,0,&norm);
1037:         scales[degidx] /= norm;
1038:       }
1039:       for (i = 0; i < npoints; i++) p[degidx * Nk * npoints + i] = 1.;
1040:       for (i = 0; i < (Nk - 1) * npoints; i++) p[(degidx * Nk + 1) * npoints + i] = 0.;
1041:       continue;
1042:     }
1043:     n = degtup[d];
1044:     degtup[d]--;
1045:     PetscDTGradedOrderToIndex(dim, degtup, &m1idx);
1046:     if (degtup[d] > 0) {
1047:       degtup[d]--;
1048:       PetscDTGradedOrderToIndex(dim, degtup, &m2idx);
1049:       degtup[d]++;
1050:     }
1051:     degtup[d]++;
1052:     for (e = 0, degsum = 0; e < d; e++) degsum += degtup[e];
1053:     alpha = 2 * degsum + d;
1054:     PetscDTJacobiRecurrence_Internal(n,alpha,0.,cnm1,cnm1x,cnm2);


1057:     scales[degidx] = initscale;
1058:     for (e = 0, degsum = 0; e < dim; e++) {
1059:       PetscInt  f;
1060:       PetscReal ealpha;
1061:       PetscReal enorm;

1063:       ealpha = 2 * degsum + e;
1064:       for (f = 0; f < degsum; f++) scales[degidx] *= 2.;
1065:       PetscDTJacobiNorm(ealpha,0.,degtup[e],&enorm);
1066:       scales[degidx] /= enorm;
1067:       degsum += degtup[e];
1068:     }

1070:     for (pt = 0; pt < npoints; pt++) {
1071:       /* compute the multipliers */
1072:       PetscReal thetanm1, thetanm1x, thetanm2;

1074:       thetanm1x = dim - (d+1) + 2.*points[pt * dim + d];
1075:       for (e = d+1; e < dim; e++) thetanm1x += points[pt * dim + e];
1076:       thetanm1x *= 0.5;
1077:       thetanm1 = (2. - (dim-(d+1)));
1078:       for (e = d+1; e < dim; e++) thetanm1 -= points[pt * dim + e];
1079:       thetanm1 *= 0.5;
1080:       thetanm2 = thetanm1 * thetanm1;

1082:       for (kidx = 0; kidx < Nk; kidx++) {
1083:         PetscInt f;

1085:         PetscDTIndexToGradedOrder(dim, kidx, ktup);
1086:         /* first sum in the same derivative terms */
1087:         p[(degidx * Nk + kidx) * npoints + pt] = (cnm1 * thetanm1 + cnm1x * thetanm1x) * p[(m1idx * Nk + kidx) * npoints + pt];
1088:         if (m2idx >= 0) {
1089:           p[(degidx * Nk + kidx) * npoints + pt] -= cnm2 * thetanm2 * p[(m2idx * Nk + kidx) * npoints + pt];
1090:         }

1092:         for (f = d; f < dim; f++) {
1093:           PetscInt km1idx, mplty = ktup[f];

1095:           if (!mplty) continue;
1096:           ktup[f]--;
1097:           PetscDTGradedOrderToIndex(dim, ktup, &km1idx);

1099:           /* the derivative of  cnm1x * thetanm1x  wrt x variable f is 0.5 * cnm1x if f > d otherwise it is cnm1x */
1100:           /* the derivative of  cnm1  * thetanm1   wrt x variable f is 0 if f == d, otherwise it is -0.5 * cnm1 */
1101:           /* the derivative of -cnm2  * thetanm2   wrt x variable f is 0 if f == d, otherwise it is cnm2 * thetanm1 */
1102:           if (f > d) {
1103:             PetscInt f2;

1105:             p[(degidx * Nk + kidx) * npoints + pt] += mplty * 0.5 * (cnm1x - cnm1) * p[(m1idx * Nk + km1idx) * npoints + pt];
1106:             if (m2idx >= 0) {
1107:               p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm2 * thetanm1 * p[(m2idx * Nk + km1idx) * npoints + pt];
1108:               /* second derivatives of -cnm2  * thetanm2   wrt x variable f,f2 is like - 0.5 * cnm2 */
1109:               for (f2 = f; f2 < dim; f2++) {
1110:                 PetscInt km2idx, mplty2 = ktup[f2];
1111:                 PetscInt factor;

1113:                 if (!mplty2) continue;
1114:                 ktup[f2]--;
1115:                 PetscDTGradedOrderToIndex(dim, ktup, &km2idx);

1117:                 factor = mplty * mplty2;
1118:                 if (f == f2) factor /= 2;
1119:                 p[(degidx * Nk + kidx) * npoints + pt] -= 0.5 * factor * cnm2 * p[(m2idx * Nk + km2idx) * npoints + pt];
1120:                 ktup[f2]++;
1121:               }
1122:             }
1123:           } else {
1124:             p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm1x * p[(m1idx * Nk + km1idx) * npoints + pt];
1125:           }
1126:           ktup[f]++;
1127:         }
1128:       }
1129:     }
1130:   }
1131:   for (degidx = 0; degidx < Ndeg; degidx++) {
1132:     PetscReal scale = scales[degidx];
1133:     PetscInt i;

1135:     for (i = 0; i < Nk * npoints; i++) p[degidx*Nk*npoints + i] *= scale;
1136:   }
1137:   PetscFree(scales);
1138:   PetscFree2(degtup, ktup);
1139:   return(0);
1140: }

1142: /* solve the symmetric tridiagonal eigenvalue system, writing the eigenvalues into eigs and the eigenvectors into V
1143:  * with lds n; diag and subdiag are overwritten */
1144: static PetscErrorCode PetscDTSymmetricTridiagonalEigensolve(PetscInt n, PetscReal diag[], PetscReal subdiag[],
1145:                                                             PetscReal eigs[], PetscScalar V[])
1146: {
1147:   char jobz = 'V'; /* eigenvalues and eigenvectors */
1148:   char range = 'A'; /* all eigenvalues will be found */
1149:   PetscReal VL = 0.; /* ignored because range is 'A' */
1150:   PetscReal VU = 0.; /* ignored because range is 'A' */
1151:   PetscBLASInt IL = 0; /* ignored because range is 'A' */
1152:   PetscBLASInt IU = 0; /* ignored because range is 'A' */
1153:   PetscReal abstol = 0.; /* unused */
1154:   PetscBLASInt bn, bm, ldz; /* bm will equal bn on exit */
1155:   PetscBLASInt *isuppz;
1156:   PetscBLASInt lwork, liwork;
1157:   PetscReal workquery;
1158:   PetscBLASInt  iworkquery;
1159:   PetscBLASInt *iwork;
1160:   PetscBLASInt info;
1161:   PetscReal *work = NULL;

1165: #if !defined(PETSCDTGAUSSIANQUADRATURE_EIG)
1166:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
1167: #endif
1168:   PetscBLASIntCast(n, &bn);
1169:   PetscBLASIntCast(n, &ldz);
1170: #if !defined(PETSC_MISSING_LAPACK_STEGR)
1171:   PetscMalloc1(2 * n, &isuppz);
1172:   lwork = -1;
1173:   liwork = -1;
1174:   PetscStackCallBLAS("LAPACKstegr",LAPACKstegr_(&jobz,&range,&bn,diag,subdiag,&VL,&VU,&IL,&IU,&abstol,&bm,eigs,V,&ldz,isuppz,&workquery,&lwork,&iworkquery,&liwork,&info));
1175:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEGR error");
1176:   lwork = (PetscBLASInt) workquery;
1177:   liwork = (PetscBLASInt) iworkquery;
1178:   PetscMalloc2(lwork, &work, liwork, &iwork);
1179:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
1180:   PetscStackCallBLAS("LAPACKstegr",LAPACKstegr_(&jobz,&range,&bn,diag,subdiag,&VL,&VU,&IL,&IU,&abstol,&bm,eigs,V,&ldz,isuppz,work,&lwork,iwork,&liwork,&info));
1181:   PetscFPTrapPop();
1182:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEGR error");
1183:   PetscFree2(work, iwork);
1184:   PetscFree(isuppz);
1185: #elif !defined(PETSC_MISSING_LAPACK_STEQR)
1186:   jobz = 'I'; /* Compute eigenvalues and eigenvectors of the
1187:                  tridiagonal matrix.  Z is initialized to the identity
1188:                  matrix. */
1189:   PetscMalloc1(PetscMax(1,2*n-2),&work);
1190:   PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&bn,diag,subdiag,V,&ldz,work,&info));
1191:   PetscFPTrapPop();
1192:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");
1193:   PetscFree(work);
1194:   PetscArraycpy(eigs,diag,n);
1195: #endif
1196:   return(0);
1197: }

1199: /* Formula for the weights at the endpoints (-1 and 1) of Gauss-Lobatto-Jacobi
1200:  * quadrature rules on the interval [-1, 1] */
1201: static PetscErrorCode PetscDTGaussLobattoJacobiEndweights_Internal(PetscInt n, PetscReal alpha, PetscReal beta, PetscReal *leftw, PetscReal *rightw)
1202: {
1203:   PetscReal twoab1;
1204:   PetscInt  m = n - 2;
1205:   PetscReal a = alpha + 1.;
1206:   PetscReal b = beta + 1.;
1207:   PetscReal gra, grb;

1210:   twoab1 = PetscPowReal(2., a + b - 1.);
1211: #if defined(PETSC_HAVE_LGAMMA)
1212:   grb = PetscExpReal(2. * PetscLGamma(b+1.) + PetscLGamma(m+1.) + PetscLGamma(m+a+1.) -
1213:                      (PetscLGamma(m+b+1) + PetscLGamma(m+a+b+1.)));
1214:   gra = PetscExpReal(2. * PetscLGamma(a+1.) + PetscLGamma(m+1.) + PetscLGamma(m+b+1.) -
1215:                      (PetscLGamma(m+a+1) + PetscLGamma(m+a+b+1.)));
1216: #else
1217:   {
1218:     PetscInt alphai = (PetscInt) alpha;
1219:     PetscInt betai = (PetscInt) beta;

1222:     if ((PetscReal) alphai == alpha && (PetscReal) betai == beta) {
1223:       PetscReal binom1, binom2;

1225:       PetscDTBinomial(m+b, b, &binom1);
1226:       PetscDTBinomial(m+a+b, b, &binom2);
1227:       grb = 1./ (binom1 * binom2);
1228:       PetscDTBinomial(m+a, a, &binom1);
1229:       PetscDTBinomial(m+a+b, a, &binom2);
1230:       gra = 1./ (binom1 * binom2);
1231:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"lgamma() - math routine is unavailable.");
1232:   }
1233: #endif
1234:   *leftw = twoab1 * grb / b;
1235:   *rightw = twoab1 * gra / a;
1236:   return(0);
1237: }

1239: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
1240:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
1241: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
1242: {
1243:   PetscReal pn1, pn2;
1244:   PetscReal cnm1, cnm1x, cnm2;
1245:   PetscInt  k;

1248:   if (!n) {*P = 1.0; return(0);}
1249:   PetscDTJacobiRecurrence_Internal(1,a,b,cnm1,cnm1x,cnm2);
1250:   pn2 = 1.;
1251:   pn1 = cnm1 + cnm1x*x;
1252:   if (n == 1) {*P = pn1; return(0);}
1253:   *P  = 0.0;
1254:   for (k = 2; k < n+1; ++k) {
1255:     PetscDTJacobiRecurrence_Internal(k,a,b,cnm1,cnm1x,cnm2);

1257:     *P  = (cnm1 + cnm1x*x)*pn1 - cnm2*pn2;
1258:     pn2 = pn1;
1259:     pn1 = *P;
1260:   }
1261:   return(0);
1262: }

1264: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
1265: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscInt k, PetscReal *P)
1266: {
1267:   PetscReal      nP;
1268:   PetscInt       i;

1272:   *P = 0.0;
1273:   if (k > n) return(0);
1274:   PetscDTComputeJacobi(a+k, b+k, n-k, x, &nP);
1275:   for (i = 0; i < k; i++) nP *= (a + b + n + 1. + i) * 0.5;
1276:   *P = nP;
1277:   return(0);
1278: }

1280: static PetscErrorCode PetscDTGaussJacobiQuadrature_Newton_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal x[], PetscReal w[])
1281: {
1282:   PetscInt       maxIter = 100;
1283:   PetscReal      eps     = PetscExpReal(0.75 * PetscLogReal(PETSC_MACHINE_EPSILON));
1284:   PetscReal      a1, a6, gf;
1285:   PetscInt       k;


1290:   a1 = PetscPowReal(2.0, a+b+1);
1291: #if defined(PETSC_HAVE_LGAMMA)
1292:   {
1293:     PetscReal a2, a3, a4, a5;
1294:     a2 = PetscLGamma(a + npoints + 1);
1295:     a3 = PetscLGamma(b + npoints + 1);
1296:     a4 = PetscLGamma(a + b + npoints + 1);
1297:     a5 = PetscLGamma(npoints + 1);
1298:     gf = PetscExpReal(a2 + a3 - (a4 + a5));
1299:   }
1300: #else
1301:   {
1302:     PetscInt ia, ib;

1304:     ia = (PetscInt) a;
1305:     ib = (PetscInt) b;
1306:     gf = 1.;
1307:     if (ia == a && ia >= 0) { /* compute ratio of rising factorals wrt a */
1308:       for (k = 0; k < ia; k++) gf *= (npoints + 1. + k) / (npoints + b + 1. + k);
1309:     } else if (b == b && ib >= 0) { /* compute ratio of rising factorials wrt b */
1310:       for (k = 0; k < ib; k++) gf *= (npoints + 1. + k) / (npoints + a + 1. + k);
1311:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"lgamma() - math routine is unavailable.");
1312:   }
1313: #endif

1315:   a6   = a1 * gf;
1316:   /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
1317:    Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
1318:   for (k = 0; k < npoints; ++k) {
1319:     PetscReal r = PetscCosReal(PETSC_PI * (1. - (4.*k + 3. + 2.*b) / (4.*npoints + 2.*(a + b + 1.)))), dP;
1320:     PetscInt  j;

1322:     if (k > 0) r = 0.5 * (r + x[k-1]);
1323:     for (j = 0; j < maxIter; ++j) {
1324:       PetscReal s = 0.0, delta, f, fp;
1325:       PetscInt  i;

1327:       for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
1328:       PetscDTComputeJacobi(a, b, npoints, r, &f);
1329:       PetscDTComputeJacobiDerivative(a, b, npoints, r, 1, &fp);
1330:       delta = f / (fp - f * s);
1331:       r     = r - delta;
1332:       if (PetscAbsReal(delta) < eps) break;
1333:     }
1334:     x[k] = r;
1335:     PetscDTComputeJacobiDerivative(a, b, npoints, x[k], 1, &dP);
1336:     w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
1337:   }
1338:   return(0);
1339: }

1341: /* Compute the diagonals of the Jacobi matrix used in Golub & Welsch algorithms for Gauss-Jacobi
1342:  * quadrature weight calculations on [-1,1] for exponents (1. + x)^a (1.-x)^b */
1343: static PetscErrorCode PetscDTJacobiMatrix_Internal(PetscInt nPoints, PetscReal a, PetscReal b, PetscReal *d, PetscReal *s)
1344: {
1345:   PetscInt       i;

1348:   for (i = 0; i < nPoints; i++) {
1349:     PetscReal A, B, C;

1351:     PetscDTJacobiRecurrence_Internal(i+1,a,b,A,B,C);
1352:     d[i] = -A / B;
1353:     if (i) s[i-1] *= C / B;
1354:     if (i < nPoints - 1) s[i] = 1. / B;
1355:   }
1356:   return(0);
1357: }

1359: static PetscErrorCode PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
1360: {
1361:   PetscReal mu0;
1362:   PetscReal ga, gb, gab;
1363:   PetscInt i;

1367:   PetscCitationsRegister(GolubWelschCitation, &GolubWelschCite);

1369: #if defined(PETSC_HAVE_TGAMMA)
1370:   ga  = PetscTGamma(a + 1);
1371:   gb  = PetscTGamma(b + 1);
1372:   gab = PetscTGamma(a + b + 2);
1373: #else
1374:   {
1375:     PetscInt ia, ib;

1377:     ia = (PetscInt) a;
1378:     ib = (PetscInt) b;
1379:     if (ia == a && ib == b && ia + 1 > 0 && ib + 1 > 0 && ia + ib + 2 > 0) { /* All gamma(x) terms are (x-1)! terms */
1380:       PetscDTFactorial(ia, &ga);
1381:       PetscDTFactorial(ib, &gb);
1382:       PetscDTFactorial(ia + ib + 1, &gb);
1383:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
1384:   }
1385: #endif
1386:   mu0 = PetscPowReal(2.,a + b + 1.) * ga * gb / gab;

1388: #if defined(PETSCDTGAUSSIANQUADRATURE_EIG)
1389:   {
1390:     PetscReal *diag, *subdiag;
1391:     PetscScalar *V;

1393:     PetscMalloc2(npoints, &diag, npoints, &subdiag);
1394:     PetscMalloc1(npoints*npoints, &V);
1395:     PetscDTJacobiMatrix_Internal(npoints, a, b, diag, subdiag);
1396:     for (i = 0; i < npoints - 1; i++) subdiag[i] = PetscSqrtReal(subdiag[i]);
1397:     PetscDTSymmetricTridiagonalEigensolve(npoints, diag, subdiag, x, V);
1398:     for (i = 0; i < npoints; i++) w[i] = PetscSqr(PetscRealPart(V[i * npoints])) * mu0;
1399:     PetscFree(V);
1400:     PetscFree2(diag, subdiag);
1401:   }
1402: #else
1403:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
1404: #endif
1405:   { /* As of March 2, 2020, The Sun Performance Library breaks the LAPACK contract for xstegr and xsteqr: the
1406:        eigenvalues are not guaranteed to be in ascending order.  So we heave a passive aggressive sigh and check that
1407:        the eigenvalues are sorted */
1408:     PetscBool sorted;

1410:     PetscSortedReal(npoints, x, &sorted);
1411:     if (!sorted) {
1412:       PetscInt *order, i;
1413:       PetscReal *tmp;

1415:       PetscMalloc2(npoints, &order, npoints, &tmp);
1416:       for (i = 0; i < npoints; i++) order[i] = i;
1417:       PetscSortRealWithPermutation(npoints, x, order);
1418:       PetscArraycpy(tmp, x, npoints);
1419:       for (i = 0; i < npoints; i++) x[i] = tmp[order[i]];
1420:       PetscArraycpy(tmp, w, npoints);
1421:       for (i = 0; i < npoints; i++) w[i] = tmp[order[i]];
1422:       PetscFree2(order, tmp);
1423:     }
1424:   }
1425:   return(0);
1426: }

1428: static PetscErrorCode PetscDTGaussJacobiQuadrature_Internal(PetscInt npoints,PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
1429: {

1433:   if (npoints < 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of points must be positive");
1434:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
1435:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
1436:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");

1438:   if (newton) {
1439:     PetscDTGaussJacobiQuadrature_Newton_Internal(npoints, alpha, beta, x, w);
1440:   } else {
1441:     PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(npoints, alpha, beta, x, w);
1442:   }
1443:   if (alpha == beta) { /* symmetrize */
1444:     PetscInt i;
1445:     for (i = 0; i < (npoints + 1) / 2; i++) {
1446:       PetscInt  j  = npoints - 1 - i;
1447:       PetscReal xi = x[i];
1448:       PetscReal xj = x[j];
1449:       PetscReal wi = w[i];
1450:       PetscReal wj = w[j];

1452:       x[i] = (xi - xj) / 2.;
1453:       x[j] = (xj - xi) / 2.;
1454:       w[i] = w[j] = (wi + wj) / 2.;
1455:     }
1456:   }
1457:   return(0);
1458: }

1460: /*@
1461:   PetscDTGaussJacobiQuadrature - quadrature for the interval [a, b] with the weight function
1462:   $(x-a)^\alpha (x-b)^\beta$.

1464:   Not collective

1466:   Input Parameters:
1467: + npoints - the number of points in the quadrature rule
1468: . a - the left endpoint of the interval
1469: . b - the right endpoint of the interval
1470: . alpha - the left exponent
1471: - beta - the right exponent

1473:   Output Parameters:
1474: + x - array of length npoints, the locations of the quadrature points
1475: - w - array of length npoints, the weights of the quadrature points

1477:   Level: intermediate

1479:   Note: this quadrature rule is exact for polynomials up to degree 2*npoints - 1.
1480: @*/
1481: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt npoints,PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
1482: {
1483:   PetscInt       i;

1487:   PetscDTGaussJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal);
1488:   if (a != -1. || b != 1.) { /* shift */
1489:     for (i = 0; i < npoints; i++) {
1490:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1491:       w[i] *= (b - a) / 2.;
1492:     }
1493:   }
1494:   return(0);
1495: }

1497: static PetscErrorCode PetscDTGaussLobattoJacobiQuadrature_Internal(PetscInt npoints,PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
1498: {
1499:   PetscInt       i;

1503:   if (npoints < 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of points must be positive");
1504:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
1505:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
1506:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");

1508:   x[0] = -1.;
1509:   x[npoints-1] = 1.;
1510:   if (npoints > 2) {
1511:     PetscDTGaussJacobiQuadrature_Internal(npoints-2, alpha+1., beta+1., &x[1], &w[1], newton);
1512:   }
1513:   for (i = 1; i < npoints - 1; i++) {
1514:     w[i] /= (1. - x[i]*x[i]);
1515:   }
1516:   PetscDTGaussLobattoJacobiEndweights_Internal(npoints, alpha, beta, &w[0], &w[npoints-1]);
1517:   return(0);
1518: }

1520: /*@
1521:   PetscDTGaussLobattoJacobiQuadrature - quadrature for the interval [a, b] with the weight function
1522:   $(x-a)^\alpha (x-b)^\beta$, with endpoints a and b included as quadrature points.

1524:   Not collective

1526:   Input Parameters:
1527: + npoints - the number of points in the quadrature rule
1528: . a - the left endpoint of the interval
1529: . b - the right endpoint of the interval
1530: . alpha - the left exponent
1531: - beta - the right exponent

1533:   Output Parameters:
1534: + x - array of length npoints, the locations of the quadrature points
1535: - w - array of length npoints, the weights of the quadrature points

1537:   Level: intermediate

1539:   Note: this quadrature rule is exact for polynomials up to degree 2*npoints - 3.
1540: @*/
1541: PetscErrorCode PetscDTGaussLobattoJacobiQuadrature(PetscInt npoints,PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
1542: {
1543:   PetscInt       i;

1547:   PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal);
1548:   if (a != -1. || b != 1.) { /* shift */
1549:     for (i = 0; i < npoints; i++) {
1550:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1551:       w[i] *= (b - a) / 2.;
1552:     }
1553:   }
1554:   return(0);
1555: }

1557: /*@
1558:    PetscDTGaussQuadrature - create Gauss-Legendre quadrature

1560:    Not Collective

1562:    Input Arguments:
1563: +  npoints - number of points
1564: .  a - left end of interval (often-1)
1565: -  b - right end of interval (often +1)

1567:    Output Arguments:
1568: +  x - quadrature points
1569: -  w - quadrature weights

1571:    Level: intermediate

1573:    References:
1574: .   1. - Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 1969.

1576: .seealso: PetscDTLegendreEval()
1577: @*/
1578: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
1579: {
1580:   PetscInt       i;

1584:   PetscDTGaussJacobiQuadrature_Internal(npoints, 0., 0., x, w, PetscDTGaussQuadratureNewton_Internal);
1585:   if (a != -1. || b != 1.) { /* shift */
1586:     for (i = 0; i < npoints; i++) {
1587:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1588:       w[i] *= (b - a) / 2.;
1589:     }
1590:   }
1591:   return(0);
1592: }

1594: /*@C
1595:    PetscDTGaussLobattoLegendreQuadrature - creates a set of the locations and weights of the Gauss-Lobatto-Legendre
1596:                       nodes of a given size on the domain [-1,1]

1598:    Not Collective

1600:    Input Parameter:
1601: +  n - number of grid nodes
1602: -  type - PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA or PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON

1604:    Output Arguments:
1605: +  x - quadrature points
1606: -  w - quadrature weights

1608:    Notes:
1609:     For n > 30  the Newton approach computes duplicate (incorrect) values for some nodes because the initial guess is apparently not
1610:           close enough to the desired solution

1612:    These are useful for implementing spectral methods based on Gauss-Lobatto-Legendre (GLL) nodes

1614:    See  https://epubs.siam.org/doi/abs/10.1137/110855442  https://epubs.siam.org/doi/abs/10.1137/120889873 for better ways to compute GLL nodes

1616:    Level: intermediate

1618: .seealso: PetscDTGaussQuadrature()

1620: @*/
1621: PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt npoints,PetscGaussLobattoLegendreCreateType type,PetscReal *x,PetscReal *w)
1622: {
1623:   PetscBool      newton;

1627:   if (npoints < 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Must provide at least 2 grid points per element");
1628:   newton = (PetscBool) (type == PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON);
1629:   PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, 0., 0., x, w, newton);
1630:   return(0);
1631: }

1633: /*@
1634:   PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

1636:   Not Collective

1638:   Input Arguments:
1639: + dim     - The spatial dimension
1640: . Nc      - The number of components
1641: . npoints - number of points in one dimension
1642: . a       - left end of interval (often-1)
1643: - b       - right end of interval (often +1)

1645:   Output Argument:
1646: . q - A PetscQuadrature object

1648:   Level: intermediate

1650: .seealso: PetscDTGaussQuadrature(), PetscDTLegendreEval()
1651: @*/
1652: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
1653: {
1654:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k, c;
1655:   PetscReal     *x, *w, *xw, *ww;

1659:   PetscMalloc1(totpoints*dim,&x);
1660:   PetscMalloc1(totpoints*Nc,&w);
1661:   /* Set up the Golub-Welsch system */
1662:   switch (dim) {
1663:   case 0:
1664:     PetscFree(x);
1665:     PetscFree(w);
1666:     PetscMalloc1(1, &x);
1667:     PetscMalloc1(Nc, &w);
1668:     x[0] = 0.0;
1669:     for (c = 0; c < Nc; ++c) w[c] = 1.0;
1670:     break;
1671:   case 1:
1672:     PetscMalloc1(npoints,&ww);
1673:     PetscDTGaussQuadrature(npoints, a, b, x, ww);
1674:     for (i = 0; i < npoints; ++i) for (c = 0; c < Nc; ++c) w[i*Nc+c] = ww[i];
1675:     PetscFree(ww);
1676:     break;
1677:   case 2:
1678:     PetscMalloc2(npoints,&xw,npoints,&ww);
1679:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
1680:     for (i = 0; i < npoints; ++i) {
1681:       for (j = 0; j < npoints; ++j) {
1682:         x[(i*npoints+j)*dim+0] = xw[i];
1683:         x[(i*npoints+j)*dim+1] = xw[j];
1684:         for (c = 0; c < Nc; ++c) w[(i*npoints+j)*Nc+c] = ww[i] * ww[j];
1685:       }
1686:     }
1687:     PetscFree2(xw,ww);
1688:     break;
1689:   case 3:
1690:     PetscMalloc2(npoints,&xw,npoints,&ww);
1691:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
1692:     for (i = 0; i < npoints; ++i) {
1693:       for (j = 0; j < npoints; ++j) {
1694:         for (k = 0; k < npoints; ++k) {
1695:           x[((i*npoints+j)*npoints+k)*dim+0] = xw[i];
1696:           x[((i*npoints+j)*npoints+k)*dim+1] = xw[j];
1697:           x[((i*npoints+j)*npoints+k)*dim+2] = xw[k];
1698:           for (c = 0; c < Nc; ++c) w[((i*npoints+j)*npoints+k)*Nc+c] = ww[i] * ww[j] * ww[k];
1699:         }
1700:       }
1701:     }
1702:     PetscFree2(xw,ww);
1703:     break;
1704:   default:
1705:     SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
1706:   }
1707:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
1708:   PetscQuadratureSetOrder(*q, 2*npoints-1);
1709:   PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);
1710:   PetscObjectChangeTypeName((PetscObject)*q,"GaussTensor");
1711:   return(0);
1712: }

1714: /*@
1715:   PetscDTStroudConicalQuadrature - create Stroud conical quadrature for a simplex

1717:   Not Collective

1719:   Input Arguments:
1720: + dim     - The simplex dimension
1721: . Nc      - The number of components
1722: . npoints - The number of points in one dimension
1723: . a       - left end of interval (often-1)
1724: - b       - right end of interval (often +1)

1726:   Output Argument:
1727: . q - A PetscQuadrature object

1729:   Level: intermediate

1731:   References:
1732: .  1. - Karniadakis and Sherwin.  FIAT

1734:   Note: For dim == 1, this is Gauss-Legendre quadrature

1736: .seealso: PetscDTGaussTensorQuadrature(), PetscDTGaussQuadrature()
1737: @*/
1738: PetscErrorCode PetscDTStroudConicalQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
1739: {
1740:   PetscInt       totprev, totrem;
1741:   PetscInt       totpoints;
1742:   PetscReal     *p1, *w1;
1743:   PetscReal     *x, *w;
1744:   PetscInt       i, j, k, l, m, pt, c;

1748:   if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
1749:   totpoints = 1;
1750:   for (i = 0, totpoints = 1; i < dim; i++) totpoints *= npoints;
1751:   PetscMalloc1(totpoints*dim, &x);
1752:   PetscMalloc1(totpoints*Nc, &w);
1753:   PetscMalloc2(npoints, &p1, npoints, &w1);
1754:   for (i = 0; i < totpoints*Nc; i++) w[i] = 1.;
1755:   for (i = 0, totprev = 1, totrem = totpoints / npoints; i < dim; i++) {
1756:     PetscReal mul;

1758:     mul = PetscPowReal(2.,-i);
1759:     PetscDTGaussJacobiQuadrature(npoints, -1., 1., i, 0.0, p1, w1);
1760:     for (pt = 0, l = 0; l < totprev; l++) {
1761:       for (j = 0; j < npoints; j++) {
1762:         for (m = 0; m < totrem; m++, pt++) {
1763:           for (k = 0; k < i; k++) x[pt*dim+k] = (x[pt*dim+k]+1.)*(1.-p1[j])*0.5 - 1.;
1764:           x[pt * dim + i] = p1[j];
1765:           for (c = 0; c < Nc; c++) w[pt*Nc + c] *= mul * w1[j];
1766:         }
1767:       }
1768:     }
1769:     totprev *= npoints;
1770:     totrem /= npoints;
1771:   }
1772:   PetscFree2(p1, w1);
1773:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
1774:   PetscQuadratureSetOrder(*q, 2*npoints-1);
1775:   PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);
1776:   PetscObjectChangeTypeName((PetscObject)*q,"StroudConical");
1777:   return(0);
1778: }

1780: /*@
1781:   PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell

1783:   Not Collective

1785:   Input Arguments:
1786: + dim   - The cell dimension
1787: . level - The number of points in one dimension, 2^l
1788: . a     - left end of interval (often-1)
1789: - b     - right end of interval (often +1)

1791:   Output Argument:
1792: . q - A PetscQuadrature object

1794:   Level: intermediate

1796: .seealso: PetscDTGaussTensorQuadrature()
1797: @*/
1798: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
1799: {
1800:   const PetscInt  p     = 16;                        /* Digits of precision in the evaluation */
1801:   const PetscReal alpha = (b-a)/2.;                  /* Half-width of the integration interval */
1802:   const PetscReal beta  = (b+a)/2.;                  /* Center of the integration interval */
1803:   const PetscReal h     = PetscPowReal(2.0, -level); /* Step size, length between x_k */
1804:   PetscReal       xk;                                /* Quadrature point x_k on reference domain [-1, 1] */
1805:   PetscReal       wk    = 0.5*PETSC_PI;              /* Quadrature weight at x_k */
1806:   PetscReal      *x, *w;
1807:   PetscInt        K, k, npoints;
1808:   PetscErrorCode  ierr;

1811:   if (dim > 1) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %d not yet implemented", dim);
1812:   if (!level) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
1813:   /* Find K such that the weights are < 32 digits of precision */
1814:   for (K = 1; PetscAbsReal(PetscLog10Real(wk)) < 2*p; ++K) {
1815:     wk = 0.5*h*PETSC_PI*PetscCoshReal(K*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(K*h)));
1816:   }
1817:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
1818:   PetscQuadratureSetOrder(*q, 2*K+1);
1819:   npoints = 2*K-1;
1820:   PetscMalloc1(npoints*dim, &x);
1821:   PetscMalloc1(npoints, &w);
1822:   /* Center term */
1823:   x[0] = beta;
1824:   w[0] = 0.5*alpha*PETSC_PI;
1825:   for (k = 1; k < K; ++k) {
1826:     wk = 0.5*alpha*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
1827:     xk = PetscTanhReal(0.5*PETSC_PI*PetscSinhReal(k*h));
1828:     x[2*k-1] = -alpha*xk+beta;
1829:     w[2*k-1] = wk;
1830:     x[2*k+0] =  alpha*xk+beta;
1831:     w[2*k+0] = wk;
1832:   }
1833:   PetscQuadratureSetData(*q, dim, 1, npoints, x, w);
1834:   return(0);
1835: }

1837: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
1838: {
1839:   const PetscInt  p     = 16;        /* Digits of precision in the evaluation */
1840:   const PetscReal alpha = (b-a)/2.;  /* Half-width of the integration interval */
1841:   const PetscReal beta  = (b+a)/2.;  /* Center of the integration interval */
1842:   PetscReal       h     = 1.0;       /* Step size, length between x_k */
1843:   PetscInt        l     = 0;         /* Level of refinement, h = 2^{-l} */
1844:   PetscReal       osum  = 0.0;       /* Integral on last level */
1845:   PetscReal       psum  = 0.0;       /* Integral on the level before the last level */
1846:   PetscReal       sum;               /* Integral on current level */
1847:   PetscReal       yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
1848:   PetscReal       lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
1849:   PetscReal       wk;                /* Quadrature weight at x_k */
1850:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
1851:   PetscInt        d;                 /* Digits of precision in the integral */

1854:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
1855:   /* Center term */
1856:   func(beta, &lval);
1857:   sum = 0.5*alpha*PETSC_PI*lval;
1858:   /* */
1859:   do {
1860:     PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
1861:     PetscInt  k = 1;

1863:     ++l;
1864:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
1865:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
1866:     psum = osum;
1867:     osum = sum;
1868:     h   *= 0.5;
1869:     sum *= 0.5;
1870:     do {
1871:       wk = 0.5*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
1872:       yk = 1.0/(PetscExpReal(0.5*PETSC_PI*PetscSinhReal(k*h)) * PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
1873:       lx = -alpha*(1.0 - yk)+beta;
1874:       rx =  alpha*(1.0 - yk)+beta;
1875:       func(lx, &lval);
1876:       func(rx, &rval);
1877:       lterm   = alpha*wk*lval;
1878:       maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
1879:       sum    += lterm;
1880:       rterm   = alpha*wk*rval;
1881:       maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
1882:       sum    += rterm;
1883:       ++k;
1884:       /* Only need to evaluate every other point on refined levels */
1885:       if (l != 1) ++k;
1886:     } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */

1888:     d1 = PetscLog10Real(PetscAbsReal(sum - osum));
1889:     d2 = PetscLog10Real(PetscAbsReal(sum - psum));
1890:     d3 = PetscLog10Real(maxTerm) - p;
1891:     if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
1892:     else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
1893:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
1894:   } while (d < digits && l < 12);
1895:   *sol = sum;

1897:   return(0);
1898: }

1900: #if defined(PETSC_HAVE_MPFR)
1901: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
1902: {
1903:   const PetscInt  safetyFactor = 2;  /* Calculate abcissa until 2*p digits */
1904:   PetscInt        l            = 0;  /* Level of refinement, h = 2^{-l} */
1905:   mpfr_t          alpha;             /* Half-width of the integration interval */
1906:   mpfr_t          beta;              /* Center of the integration interval */
1907:   mpfr_t          h;                 /* Step size, length between x_k */
1908:   mpfr_t          osum;              /* Integral on last level */
1909:   mpfr_t          psum;              /* Integral on the level before the last level */
1910:   mpfr_t          sum;               /* Integral on current level */
1911:   mpfr_t          yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
1912:   mpfr_t          lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
1913:   mpfr_t          wk;                /* Quadrature weight at x_k */
1914:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
1915:   PetscInt        d;                 /* Digits of precision in the integral */
1916:   mpfr_t          pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;

1919:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
1920:   /* Create high precision storage */
1921:   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);
1922:   /* Initialization */
1923:   mpfr_set_d(alpha, 0.5*(b-a), MPFR_RNDN);
1924:   mpfr_set_d(beta,  0.5*(b+a), MPFR_RNDN);
1925:   mpfr_set_d(osum,  0.0,       MPFR_RNDN);
1926:   mpfr_set_d(psum,  0.0,       MPFR_RNDN);
1927:   mpfr_set_d(h,     1.0,       MPFR_RNDN);
1928:   mpfr_const_pi(pi2, MPFR_RNDN);
1929:   mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
1930:   /* Center term */
1931:   func(0.5*(b+a), &lval);
1932:   mpfr_set(sum, pi2, MPFR_RNDN);
1933:   mpfr_mul(sum, sum, alpha, MPFR_RNDN);
1934:   mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
1935:   /* */
1936:   do {
1937:     PetscReal d1, d2, d3, d4;
1938:     PetscInt  k = 1;

1940:     ++l;
1941:     mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
1942:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
1943:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
1944:     mpfr_set(psum, osum, MPFR_RNDN);
1945:     mpfr_set(osum,  sum, MPFR_RNDN);
1946:     mpfr_mul_d(h,   h,   0.5, MPFR_RNDN);
1947:     mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
1948:     do {
1949:       mpfr_set_si(kh, k, MPFR_RNDN);
1950:       mpfr_mul(kh, kh, h, MPFR_RNDN);
1951:       /* Weight */
1952:       mpfr_set(wk, h, MPFR_RNDN);
1953:       mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
1954:       mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
1955:       mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
1956:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
1957:       mpfr_sqr(tmp, tmp, MPFR_RNDN);
1958:       mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
1959:       mpfr_div(wk, wk, tmp, MPFR_RNDN);
1960:       /* Abscissa */
1961:       mpfr_set_d(yk, 1.0, MPFR_RNDZ);
1962:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
1963:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
1964:       mpfr_exp(tmp, msinh, MPFR_RNDN);
1965:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
1966:       /* Quadrature points */
1967:       mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
1968:       mpfr_mul(lx, lx, alpha, MPFR_RNDU);
1969:       mpfr_add(lx, lx, beta, MPFR_RNDU);
1970:       mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
1971:       mpfr_mul(rx, rx, alpha, MPFR_RNDD);
1972:       mpfr_add(rx, rx, beta, MPFR_RNDD);
1973:       /* Evaluation */
1974:       func(mpfr_get_d(lx, MPFR_RNDU), &lval);
1975:       func(mpfr_get_d(rx, MPFR_RNDD), &rval);
1976:       /* Update */
1977:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
1978:       mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
1979:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
1980:       mpfr_abs(tmp, tmp, MPFR_RNDN);
1981:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
1982:       mpfr_set(curTerm, tmp, MPFR_RNDN);
1983:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
1984:       mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
1985:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
1986:       mpfr_abs(tmp, tmp, MPFR_RNDN);
1987:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
1988:       mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
1989:       ++k;
1990:       /* Only need to evaluate every other point on refined levels */
1991:       if (l != 1) ++k;
1992:       mpfr_log10(tmp, wk, MPFR_RNDN);
1993:       mpfr_abs(tmp, tmp, MPFR_RNDN);
1994:     } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor*digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
1995:     mpfr_sub(tmp, sum, osum, MPFR_RNDN);
1996:     mpfr_abs(tmp, tmp, MPFR_RNDN);
1997:     mpfr_log10(tmp, tmp, MPFR_RNDN);
1998:     d1 = mpfr_get_d(tmp, MPFR_RNDN);
1999:     mpfr_sub(tmp, sum, psum, MPFR_RNDN);
2000:     mpfr_abs(tmp, tmp, MPFR_RNDN);
2001:     mpfr_log10(tmp, tmp, MPFR_RNDN);
2002:     d2 = mpfr_get_d(tmp, MPFR_RNDN);
2003:     mpfr_log10(tmp, maxTerm, MPFR_RNDN);
2004:     d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
2005:     mpfr_log10(tmp, curTerm, MPFR_RNDN);
2006:     d4 = mpfr_get_d(tmp, MPFR_RNDN);
2007:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
2008:   } while (d < digits && l < 8);
2009:   *sol = mpfr_get_d(sum, MPFR_RNDN);
2010:   /* Cleanup */
2011:   mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
2012:   return(0);
2013: }
2014: #else

2016: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
2017: {
2018:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
2019: }
2020: #endif

2022: /* Overwrites A. Can only handle full-rank problems with m>=n
2023:  * A in column-major format
2024:  * Ainv in row-major format
2025:  * tau has length m
2026:  * worksize must be >= max(1,n)
2027:  */
2028: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
2029: {
2031:   PetscBLASInt   M,N,K,lda,ldb,ldwork,info;
2032:   PetscScalar    *A,*Ainv,*R,*Q,Alpha;

2035: #if defined(PETSC_USE_COMPLEX)
2036:   {
2037:     PetscInt i,j;
2038:     PetscMalloc2(m*n,&A,m*n,&Ainv);
2039:     for (j=0; j<n; j++) {
2040:       for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
2041:     }
2042:     mstride = m;
2043:   }
2044: #else
2045:   A = A_in;
2046:   Ainv = Ainv_out;
2047: #endif

2049:   PetscBLASIntCast(m,&M);
2050:   PetscBLASIntCast(n,&N);
2051:   PetscBLASIntCast(mstride,&lda);
2052:   PetscBLASIntCast(worksize,&ldwork);
2053:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
2054:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info));
2055:   PetscFPTrapPop();
2056:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
2057:   R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

2059:   /* Extract an explicit representation of Q */
2060:   Q = Ainv;
2061:   PetscArraycpy(Q,A,mstride*n);
2062:   K = N;                        /* full rank */
2063:   PetscStackCallBLAS("LAPACKorgqr",LAPACKorgqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info));
2064:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");

2066:   /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
2067:   Alpha = 1.0;
2068:   ldb = lda;
2069:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb));
2070:   /* Ainv is Q, overwritten with inverse */

2072: #if defined(PETSC_USE_COMPLEX)
2073:   {
2074:     PetscInt i;
2075:     for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
2076:     PetscFree2(A,Ainv);
2077:   }
2078: #endif
2079:   return(0);
2080: }

2082: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
2083: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
2084: {
2086:   PetscReal      *Bv;
2087:   PetscInt       i,j;

2090:   PetscMalloc1((ninterval+1)*ndegree,&Bv);
2091:   /* Point evaluation of L_p on all the source vertices */
2092:   PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
2093:   /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
2094:   for (i=0; i<ninterval; i++) {
2095:     for (j=0; j<ndegree; j++) {
2096:       if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
2097:       else           B[i*ndegree+j]   = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
2098:     }
2099:   }
2100:   PetscFree(Bv);
2101:   return(0);
2102: }

2104: /*@
2105:    PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

2107:    Not Collective

2109:    Input Arguments:
2110: +  degree - degree of reconstruction polynomial
2111: .  nsource - number of source intervals
2112: .  sourcex - sorted coordinates of source cell boundaries (length nsource+1)
2113: .  ntarget - number of target intervals
2114: -  targetx - sorted coordinates of target cell boundaries (length ntarget+1)

2116:    Output Arguments:
2117: .  R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]

2119:    Level: advanced

2121: .seealso: PetscDTLegendreEval()
2122: @*/
2123: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
2124: {
2126:   PetscInt       i,j,k,*bdegrees,worksize;
2127:   PetscReal      xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
2128:   PetscScalar    *tau,*work;

2134:   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);
2135:   if (PetscDefined(USE_DEBUG)) {
2136:     for (i=0; i<nsource; i++) {
2137:       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]);
2138:     }
2139:     for (i=0; i<ntarget; i++) {
2140:       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]);
2141:     }
2142:   }
2143:   xmin = PetscMin(sourcex[0],targetx[0]);
2144:   xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
2145:   center = (xmin + xmax)/2;
2146:   hscale = (xmax - xmin)/2;
2147:   worksize = nsource;
2148:   PetscMalloc4(degree+1,&bdegrees,nsource+1,&sourcey,nsource*(degree+1),&Bsource,worksize,&work);
2149:   PetscMalloc4(nsource,&tau,nsource*(degree+1),&Bsinv,ntarget+1,&targety,ntarget*(degree+1),&Btarget);
2150:   for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
2151:   for (i=0; i<=degree; i++) bdegrees[i] = i+1;
2152:   PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
2153:   PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
2154:   for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
2155:   PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
2156:   for (i=0; i<ntarget; i++) {
2157:     PetscReal rowsum = 0;
2158:     for (j=0; j<nsource; j++) {
2159:       PetscReal sum = 0;
2160:       for (k=0; k<degree+1; k++) {
2161:         sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
2162:       }
2163:       R[i*nsource+j] = sum;
2164:       rowsum += sum;
2165:     }
2166:     for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
2167:   }
2168:   PetscFree4(bdegrees,sourcey,Bsource,work);
2169:   PetscFree4(tau,Bsinv,targety,Btarget);
2170:   return(0);
2171: }

2173: /*@C
2174:    PetscGaussLobattoLegendreIntegrate - Compute the L2 integral of a function on the GLL points

2176:    Not Collective

2178:    Input Parameter:
2179: +  n - the number of GLL nodes
2180: .  nodes - the GLL nodes
2181: .  weights - the GLL weights
2182: -  f - the function values at the nodes

2184:    Output Parameter:
2185: .  in - the value of the integral

2187:    Level: beginner

2189: .seealso: PetscDTGaussLobattoLegendreQuadrature()

2191: @*/
2192: PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt n,PetscReal *nodes,PetscReal *weights,const PetscReal *f,PetscReal *in)
2193: {
2194:   PetscInt          i;

2197:   *in = 0.;
2198:   for (i=0; i<n; i++) {
2199:     *in += f[i]*f[i]*weights[i];
2200:   }
2201:   return(0);
2202: }

2204: /*@C
2205:    PetscGaussLobattoLegendreElementLaplacianCreate - computes the Laplacian for a single 1d GLL element

2207:    Not Collective

2209:    Input Parameter:
2210: +  n - the number of GLL nodes
2211: .  nodes - the GLL nodes
2212: -  weights - the GLL weights

2214:    Output Parameter:
2215: .  A - the stiffness element

2217:    Level: beginner

2219:    Notes:
2220:     Destroy this with PetscGaussLobattoLegendreElementLaplacianDestroy()

2222:    You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented (the array is symmetric)

2224: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianDestroy()

2226: @*/
2227: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
2228: {
2229:   PetscReal        **A;
2230:   PetscErrorCode  ierr;
2231:   const PetscReal  *gllnodes = nodes;
2232:   const PetscInt   p = n-1;
2233:   PetscReal        z0,z1,z2 = -1,x,Lpj,Lpr;
2234:   PetscInt         i,j,nn,r;

2237:   PetscMalloc1(n,&A);
2238:   PetscMalloc1(n*n,&A[0]);
2239:   for (i=1; i<n; i++) A[i] = A[i-1]+n;

2241:   for (j=1; j<p; j++) {
2242:     x  = gllnodes[j];
2243:     z0 = 1.;
2244:     z1 = x;
2245:     for (nn=1; nn<p; nn++) {
2246:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
2247:       z0 = z1;
2248:       z1 = z2;
2249:     }
2250:     Lpj=z2;
2251:     for (r=1; r<p; r++) {
2252:       if (r == j) {
2253:         A[j][j]=2./(3.*(1.-gllnodes[j]*gllnodes[j])*Lpj*Lpj);
2254:       } else {
2255:         x  = gllnodes[r];
2256:         z0 = 1.;
2257:         z1 = x;
2258:         for (nn=1; nn<p; nn++) {
2259:           z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
2260:           z0 = z1;
2261:           z1 = z2;
2262:         }
2263:         Lpr     = z2;
2264:         A[r][j] = 4./(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*Lpr*(gllnodes[j]-gllnodes[r])*(gllnodes[j]-gllnodes[r]));
2265:       }
2266:     }
2267:   }
2268:   for (j=1; j<p+1; j++) {
2269:     x  = gllnodes[j];
2270:     z0 = 1.;
2271:     z1 = x;
2272:     for (nn=1; nn<p; nn++) {
2273:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
2274:       z0 = z1;
2275:       z1 = z2;
2276:     }
2277:     Lpj     = z2;
2278:     A[j][0] = 4.*PetscPowRealInt(-1.,p)/(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*(1.+gllnodes[j])*(1.+gllnodes[j]));
2279:     A[0][j] = A[j][0];
2280:   }
2281:   for (j=0; j<p; j++) {
2282:     x  = gllnodes[j];
2283:     z0 = 1.;
2284:     z1 = x;
2285:     for (nn=1; nn<p; nn++) {
2286:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
2287:       z0 = z1;
2288:       z1 = z2;
2289:     }
2290:     Lpj=z2;

2292:     A[p][j] = 4./(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*(1.-gllnodes[j])*(1.-gllnodes[j]));
2293:     A[j][p] = A[p][j];
2294:   }
2295:   A[0][0]=0.5+(((PetscReal)p)*(((PetscReal)p)+1.)-2.)/6.;
2296:   A[p][p]=A[0][0];
2297:   *AA = A;
2298:   return(0);
2299: }

2301: /*@C
2302:    PetscGaussLobattoLegendreElementLaplacianDestroy - frees the Laplacian for a single 1d GLL element

2304:    Not Collective

2306:    Input Parameter:
2307: +  n - the number of GLL nodes
2308: .  nodes - the GLL nodes
2309: .  weights - the GLL weightss
2310: -  A - the stiffness element

2312:    Level: beginner

2314: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate()

2316: @*/
2317: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
2318: {

2322:   PetscFree((*AA)[0]);
2323:   PetscFree(*AA);
2324:   *AA  = NULL;
2325:   return(0);
2326: }

2328: /*@C
2329:    PetscGaussLobattoLegendreElementGradientCreate - computes the gradient for a single 1d GLL element

2331:    Not Collective

2333:    Input Parameter:
2334: +  n - the number of GLL nodes
2335: .  nodes - the GLL nodes
2336: .  weights - the GLL weights

2338:    Output Parameter:
2339: .  AA - the stiffness element
2340: -  AAT - the transpose of AA (pass in NULL if you do not need this array)

2342:    Level: beginner

2344:    Notes:
2345:     Destroy this with PetscGaussLobattoLegendreElementGradientDestroy()

2347:    You can access entries in these arrays with AA[i][j] but in memory it is stored in contiguous memory, row oriented

2349: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianDestroy()

2351: @*/
2352: PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA,PetscReal ***AAT)
2353: {
2354:   PetscReal        **A, **AT = NULL;
2355:   PetscErrorCode  ierr;
2356:   const PetscReal  *gllnodes = nodes;
2357:   const PetscInt   p = n-1;
2358:   PetscReal        Li, Lj,d0;
2359:   PetscInt         i,j;

2362:   PetscMalloc1(n,&A);
2363:   PetscMalloc1(n*n,&A[0]);
2364:   for (i=1; i<n; i++) A[i] = A[i-1]+n;

2366:   if (AAT) {
2367:     PetscMalloc1(n,&AT);
2368:     PetscMalloc1(n*n,&AT[0]);
2369:     for (i=1; i<n; i++) AT[i] = AT[i-1]+n;
2370:   }

2372:   if (n==1) {A[0][0] = 0.;}
2373:   d0 = (PetscReal)p*((PetscReal)p+1.)/4.;
2374:   for  (i=0; i<n; i++) {
2375:     for  (j=0; j<n; j++) {
2376:       A[i][j] = 0.;
2377:       PetscDTComputeJacobi(0., 0., p, gllnodes[i], &Li);
2378:       PetscDTComputeJacobi(0., 0., p, gllnodes[j], &Lj);
2379:       if (i!=j)             A[i][j] = Li/(Lj*(gllnodes[i]-gllnodes[j]));
2380:       if ((j==i) && (i==0)) A[i][j] = -d0;
2381:       if (j==i && i==p)     A[i][j] = d0;
2382:       if (AT) AT[j][i] = A[i][j];
2383:     }
2384:   }
2385:   if (AAT) *AAT = AT;
2386:   *AA  = A;
2387:   return(0);
2388: }

2390: /*@C
2391:    PetscGaussLobattoLegendreElementGradientDestroy - frees the gradient for a single 1d GLL element obtained with PetscGaussLobattoLegendreElementGradientCreate()

2393:    Not Collective

2395:    Input Parameter:
2396: +  n - the number of GLL nodes
2397: .  nodes - the GLL nodes
2398: .  weights - the GLL weights
2399: .  AA - the stiffness element
2400: -  AAT - the transpose of the element

2402:    Level: beginner

2404: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate(), PetscGaussLobattoLegendreElementAdvectionCreate()

2406: @*/
2407: PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA,PetscReal ***AAT)
2408: {

2412:   PetscFree((*AA)[0]);
2413:   PetscFree(*AA);
2414:   *AA  = NULL;
2415:   if (*AAT) {
2416:     PetscFree((*AAT)[0]);
2417:     PetscFree(*AAT);
2418:     *AAT  = NULL;
2419:   }
2420:   return(0);
2421: }

2423: /*@C
2424:    PetscGaussLobattoLegendreElementAdvectionCreate - computes the advection operator for a single 1d GLL element

2426:    Not Collective

2428:    Input Parameter:
2429: +  n - the number of GLL nodes
2430: .  nodes - the GLL nodes
2431: -  weights - the GLL weightss

2433:    Output Parameter:
2434: .  AA - the stiffness element

2436:    Level: beginner

2438:    Notes:
2439:     Destroy this with PetscGaussLobattoLegendreElementAdvectionDestroy()

2441:    This is the same as the Gradient operator multiplied by the diagonal mass matrix

2443:    You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented

2445: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate(), PetscGaussLobattoLegendreElementAdvectionDestroy()

2447: @*/
2448: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
2449: {
2450:   PetscReal       **D;
2451:   PetscErrorCode  ierr;
2452:   const PetscReal  *gllweights = weights;
2453:   const PetscInt   glln = n;
2454:   PetscInt         i,j;

2457:   PetscGaussLobattoLegendreElementGradientCreate(n,nodes,weights,&D,NULL);
2458:   for (i=0; i<glln; i++){
2459:     for (j=0; j<glln; j++) {
2460:       D[i][j] = gllweights[i]*D[i][j];
2461:     }
2462:   }
2463:   *AA = D;
2464:   return(0);
2465: }

2467: /*@C
2468:    PetscGaussLobattoLegendreElementAdvectionDestroy - frees the advection stiffness for a single 1d GLL element

2470:    Not Collective

2472:    Input Parameter:
2473: +  n - the number of GLL nodes
2474: .  nodes - the GLL nodes
2475: .  weights - the GLL weights
2476: -  A - advection

2478:    Level: beginner

2480: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementAdvectionCreate()

2482: @*/
2483: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
2484: {

2488:   PetscFree((*AA)[0]);
2489:   PetscFree(*AA);
2490:   *AA  = NULL;
2491:   return(0);
2492: }

2494: PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
2495: {
2496:   PetscReal        **A;
2497:   PetscErrorCode  ierr;
2498:   const PetscReal  *gllweights = weights;
2499:   const PetscInt   glln = n;
2500:   PetscInt         i,j;

2503:   PetscMalloc1(glln,&A);
2504:   PetscMalloc1(glln*glln,&A[0]);
2505:   for (i=1; i<glln; i++) A[i] = A[i-1]+glln;
2506:   if (glln==1) {A[0][0] = 0.;}
2507:   for  (i=0; i<glln; i++) {
2508:     for  (j=0; j<glln; j++) {
2509:       A[i][j] = 0.;
2510:       if (j==i)     A[i][j] = gllweights[i];
2511:     }
2512:   }
2513:   *AA  = A;
2514:   return(0);
2515: }

2517: PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
2518: {

2522:   PetscFree((*AA)[0]);
2523:   PetscFree(*AA);
2524:   *AA  = NULL;
2525:   return(0);
2526: }

2528: /*@
2529:   PetscDTIndexToBary - convert an index into a barycentric coordinate.

2531:   Input Parameters:
2532: + len - the desired length of the barycentric tuple (usually 1 more than the dimension it represents, so a barycentric coordinate in a triangle has length 3)
2533: . sum - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
2534: - index - the index to convert: should be >= 0 and < Binomial(len - 1 + sum, sum)

2536:   Output Parameter:
2537: . coord - will be filled with the barycentric coordinate

2539:   Level: beginner

2541:   Note: the indices map to barycentric coordinates in lexicographic order, where the first index is the
2542:   least significant and the last index is the most significant.

2544: .seealso: PetscDTBaryToIndex()
2545: @*/
2546: PetscErrorCode PetscDTIndexToBary(PetscInt len, PetscInt sum, PetscInt index, PetscInt coord[])
2547: {
2548:   PetscInt c, d, s, total, subtotal, nexttotal;

2551:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
2552:   if (index < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
2553:   if (!len) {
2554:     if (!sum && !index) return(0);
2555:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
2556:   }
2557:   for (c = 1, total = 1; c <= len; c++) {
2558:     /* total is the number of ways to have a tuple of length c with sum */
2559:     if (index < total) break;
2560:     total = (total * (sum + c)) / c;
2561:   }
2562:   if (c > len) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index out of range");
2563:   for (d = c; d < len; d++) coord[d] = 0;
2564:   for (s = 0, subtotal = 1, nexttotal = 1; c > 0;) {
2565:     /* subtotal is the number of ways to have a tuple of length c with sum s */
2566:     /* nexttotal is the number of ways to have a tuple of length c-1 with sum s */
2567:     if ((index + subtotal) >= total) {
2568:       coord[--c] = sum - s;
2569:       index -= (total - subtotal);
2570:       sum = s;
2571:       total = nexttotal;
2572:       subtotal = 1;
2573:       nexttotal = 1;
2574:       s = 0;
2575:     } else {
2576:       subtotal = (subtotal * (c + s)) / (s + 1);
2577:       nexttotal = (nexttotal * (c - 1 + s)) / (s + 1);
2578:       s++;
2579:     }
2580:   }
2581:   return(0);
2582: }

2584: /*@
2585:   PetscDTBaryToIndex - convert a barycentric coordinate to an index

2587:   Input Parameters:
2588: + len - the desired length of the barycentric tuple (usually 1 more than the dimension it represents, so a barycentric coordinate in a triangle has length 3)
2589: . sum - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
2590: - coord - a barycentric coordinate with the given length and sum

2592:   Output Parameter:
2593: . index - the unique index for the coordinate, >= 0 and < Binomial(len - 1 + sum, sum)

2595:   Level: beginner

2597:   Note: the indices map to barycentric coordinates in lexicographic order, where the first index is the
2598:   least significant and the last index is the most significant.

2600: .seealso: PetscDTIndexToBary
2601: @*/
2602: PetscErrorCode PetscDTBaryToIndex(PetscInt len, PetscInt sum, const PetscInt coord[], PetscInt *index)
2603: {
2604:   PetscInt c;
2605:   PetscInt i;
2606:   PetscInt total;

2609:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
2610:   if (!len) {
2611:     if (!sum) {
2612:       *index = 0;
2613:       return(0);
2614:     }
2615:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
2616:   }
2617:   for (c = 1, total = 1; c < len; c++) total = (total * (sum + c)) / c;
2618:   i = total - 1;
2619:   c = len - 1;
2620:   sum -= coord[c];
2621:   while (sum > 0) {
2622:     PetscInt subtotal;
2623:     PetscInt s;

2625:     for (s = 1, subtotal = 1; s < sum; s++) subtotal = (subtotal * (c + s)) / s;
2626:     i   -= subtotal;
2627:     sum -= coord[--c];
2628:   }
2629:   *index = i;
2630:   return(0);
2631: }