Actual source code: dt.c

  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 <petsc/private/petscfeimpl.h>
  8: #include <petscviewer.h>
  9: #include <petscdmplex.h>
 10: #include <petscdmshell.h>

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

 16: const char *const        PetscDTNodeTypes_shifted[] = {"default", "gaussjacobi", "equispaced", "tanhsinh", "PETSCDTNODES_", NULL};
 17: const char *const *const PetscDTNodeTypes           = PetscDTNodeTypes_shifted + 1;

 19: const char *const        PetscDTSimplexQuadratureTypes_shifted[] = {"default", "conic", "minsym", "PETSCDTSIMPLEXQUAD_", NULL};
 20: const char *const *const PetscDTSimplexQuadratureTypes           = PetscDTSimplexQuadratureTypes_shifted + 1;

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

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

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

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

 43: PetscClassId PETSCQUADRATURE_CLASSID = 0;

 45: /*@
 46:   PetscQuadratureCreate - Create a `PetscQuadrature` object

 48:   Collective

 50:   Input Parameter:
 51: . comm - The communicator for the `PetscQuadrature` object

 53:   Output Parameter:
 54: . q - The `PetscQuadrature` object

 56:   Level: beginner

 58: .seealso: `PetscQuadrature`, `Petscquadraturedestroy()`, `PetscQuadratureGetData()`
 59: @*/
 60: PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
 61: {
 62:   PetscFunctionBegin;
 63:   PetscAssertPointer(q, 2);
 64:   PetscCall(DMInitializePackage());
 65:   PetscCall(PetscHeaderCreate(*q, PETSCQUADRATURE_CLASSID, "PetscQuadrature", "Quadrature", "DT", comm, PetscQuadratureDestroy, PetscQuadratureView));
 66:   (*q)->ct        = DM_POLYTOPE_UNKNOWN;
 67:   (*q)->dim       = -1;
 68:   (*q)->Nc        = 1;
 69:   (*q)->order     = -1;
 70:   (*q)->numPoints = 0;
 71:   (*q)->points    = NULL;
 72:   (*q)->weights   = NULL;
 73:   PetscFunctionReturn(PETSC_SUCCESS);
 74: }

 76: /*@
 77:   PetscQuadratureDuplicate - Create a deep copy of the `PetscQuadrature` object

 79:   Collective

 81:   Input Parameter:
 82: . q - The `PetscQuadrature` object

 84:   Output Parameter:
 85: . r - The new `PetscQuadrature` object

 87:   Level: beginner

 89: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureDestroy()`, `PetscQuadratureGetData()`
 90: @*/
 91: PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
 92: {
 93:   DMPolytopeType   ct;
 94:   PetscInt         order, dim, Nc, Nq;
 95:   const PetscReal *points, *weights;
 96:   PetscReal       *p, *w;

 98:   PetscFunctionBegin;
 99:   PetscAssertPointer(q, 1);
100:   PetscCall(PetscQuadratureCreate(PetscObjectComm((PetscObject)q), r));
101:   PetscCall(PetscQuadratureGetCellType(q, &ct));
102:   PetscCall(PetscQuadratureSetCellType(*r, ct));
103:   PetscCall(PetscQuadratureGetOrder(q, &order));
104:   PetscCall(PetscQuadratureSetOrder(*r, order));
105:   PetscCall(PetscQuadratureGetData(q, &dim, &Nc, &Nq, &points, &weights));
106:   PetscCall(PetscMalloc1(Nq * dim, &p));
107:   PetscCall(PetscMalloc1(Nq * Nc, &w));
108:   PetscCall(PetscArraycpy(p, points, Nq * dim));
109:   PetscCall(PetscArraycpy(w, weights, Nc * Nq));
110:   PetscCall(PetscQuadratureSetData(*r, dim, Nc, Nq, p, w));
111:   PetscFunctionReturn(PETSC_SUCCESS);
112: }

114: /*@
115:   PetscQuadratureDestroy - Destroys a `PetscQuadrature` object

117:   Collective

119:   Input Parameter:
120: . q - The `PetscQuadrature` object

122:   Level: beginner

124: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureGetData()`
125: @*/
126: PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
127: {
128:   PetscFunctionBegin;
129:   if (!*q) PetscFunctionReturn(PETSC_SUCCESS);
131:   if (--((PetscObject)(*q))->refct > 0) {
132:     *q = NULL;
133:     PetscFunctionReturn(PETSC_SUCCESS);
134:   }
135:   PetscCall(PetscFree((*q)->points));
136:   PetscCall(PetscFree((*q)->weights));
137:   PetscCall(PetscHeaderDestroy(q));
138:   PetscFunctionReturn(PETSC_SUCCESS);
139: }

141: /*@
142:   PetscQuadratureGetCellType - Return the cell type of the integration domain

144:   Not Collective

146:   Input Parameter:
147: . q - The `PetscQuadrature` object

149:   Output Parameter:
150: . ct - The cell type of the integration domain

152:   Level: intermediate

154: .seealso: `PetscQuadrature`, `PetscQuadratureSetCellType()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
155: @*/
156: PetscErrorCode PetscQuadratureGetCellType(PetscQuadrature q, DMPolytopeType *ct)
157: {
158:   PetscFunctionBegin;
160:   PetscAssertPointer(ct, 2);
161:   *ct = q->ct;
162:   PetscFunctionReturn(PETSC_SUCCESS);
163: }

165: /*@
166:   PetscQuadratureSetCellType - Set the cell type of the integration domain

168:   Not Collective

170:   Input Parameters:
171: + q  - The `PetscQuadrature` object
172: - ct - The cell type of the integration domain

174:   Level: intermediate

176: .seealso: `PetscQuadrature`, `PetscQuadratureGetCellType()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
177: @*/
178: PetscErrorCode PetscQuadratureSetCellType(PetscQuadrature q, DMPolytopeType ct)
179: {
180:   PetscFunctionBegin;
182:   q->ct = ct;
183:   PetscFunctionReturn(PETSC_SUCCESS);
184: }

186: /*@
187:   PetscQuadratureGetOrder - Return the order of the method in the `PetscQuadrature`

189:   Not Collective

191:   Input Parameter:
192: . q - The `PetscQuadrature` object

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

197:   Level: intermediate

199: .seealso: `PetscQuadrature`, `PetscQuadratureSetOrder()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
200: @*/
201: PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature q, PetscInt *order)
202: {
203:   PetscFunctionBegin;
205:   PetscAssertPointer(order, 2);
206:   *order = q->order;
207:   PetscFunctionReturn(PETSC_SUCCESS);
208: }

210: /*@
211:   PetscQuadratureSetOrder - Set the order of the method in the `PetscQuadrature`

213:   Not Collective

215:   Input Parameters:
216: + q     - The `PetscQuadrature` object
217: - order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated

219:   Level: intermediate

221: .seealso: `PetscQuadrature`, `PetscQuadratureGetOrder()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
222: @*/
223: PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature q, PetscInt order)
224: {
225:   PetscFunctionBegin;
227:   q->order = order;
228:   PetscFunctionReturn(PETSC_SUCCESS);
229: }

231: /*@
232:   PetscQuadratureGetNumComponents - Return the number of components for functions to be integrated

234:   Not Collective

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

239:   Output Parameter:
240: . Nc - The number of components

242:   Level: intermediate

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

247: .seealso: `PetscQuadrature`, `PetscQuadratureSetNumComponents()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
248: @*/
249: PetscErrorCode PetscQuadratureGetNumComponents(PetscQuadrature q, PetscInt *Nc)
250: {
251:   PetscFunctionBegin;
253:   PetscAssertPointer(Nc, 2);
254:   *Nc = q->Nc;
255:   PetscFunctionReturn(PETSC_SUCCESS);
256: }

258: /*@
259:   PetscQuadratureSetNumComponents - Sets the number of components for functions to be integrated

261:   Not Collective

263:   Input Parameters:
264: + q  - The `PetscQuadrature` object
265: - Nc - The number of components

267:   Level: intermediate

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

272: .seealso: `PetscQuadrature`, `PetscQuadratureGetNumComponents()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
273: @*/
274: PetscErrorCode PetscQuadratureSetNumComponents(PetscQuadrature q, PetscInt Nc)
275: {
276:   PetscFunctionBegin;
278:   q->Nc = Nc;
279:   PetscFunctionReturn(PETSC_SUCCESS);
280: }

282: /*@C
283:   PetscQuadratureGetData - Returns the data defining the `PetscQuadrature`

285:   Not Collective

287:   Input Parameter:
288: . q - The `PetscQuadrature` object

290:   Output Parameters:
291: + dim     - The spatial dimension
292: . Nc      - The number of components
293: . npoints - The number of quadrature points
294: . points  - The coordinates of each quadrature point
295: - weights - The weight of each quadrature point

297:   Level: intermediate

299:   Fortran Note:
300:   Call `PetscQuadratureRestoreData()` when you are done with the data

302: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureSetData()`
303: @*/
304: PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *Nc, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
305: {
306:   PetscFunctionBegin;
308:   if (dim) {
309:     PetscAssertPointer(dim, 2);
310:     *dim = q->dim;
311:   }
312:   if (Nc) {
313:     PetscAssertPointer(Nc, 3);
314:     *Nc = q->Nc;
315:   }
316:   if (npoints) {
317:     PetscAssertPointer(npoints, 4);
318:     *npoints = q->numPoints;
319:   }
320:   if (points) {
321:     PetscAssertPointer(points, 5);
322:     *points = q->points;
323:   }
324:   if (weights) {
325:     PetscAssertPointer(weights, 6);
326:     *weights = q->weights;
327:   }
328:   PetscFunctionReturn(PETSC_SUCCESS);
329: }

331: /*@
332:   PetscQuadratureEqual - determine whether two quadratures are equivalent

334:   Input Parameters:
335: + A - A `PetscQuadrature` object
336: - B - Another `PetscQuadrature` object

338:   Output Parameter:
339: . equal - `PETSC_TRUE` if the quadratures are the same

341:   Level: intermediate

343: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`
344: @*/
345: PetscErrorCode PetscQuadratureEqual(PetscQuadrature A, PetscQuadrature B, PetscBool *equal)
346: {
347:   PetscFunctionBegin;
350:   PetscAssertPointer(equal, 3);
351:   *equal = PETSC_FALSE;
352:   if (A->ct != B->ct || A->dim != B->dim || A->Nc != B->Nc || A->order != B->order || A->numPoints != B->numPoints) PetscFunctionReturn(PETSC_SUCCESS);
353:   for (PetscInt i = 0; i < A->numPoints * A->dim; i++) {
354:     if (PetscAbsReal(A->points[i] - B->points[i]) > PETSC_SMALL) PetscFunctionReturn(PETSC_SUCCESS);
355:   }
356:   if (!A->weights && !B->weights) {
357:     *equal = PETSC_TRUE;
358:     PetscFunctionReturn(PETSC_SUCCESS);
359:   }
360:   if (A->weights && B->weights) {
361:     for (PetscInt i = 0; i < A->numPoints; i++) {
362:       if (PetscAbsReal(A->weights[i] - B->weights[i]) > PETSC_SMALL) PetscFunctionReturn(PETSC_SUCCESS);
363:     }
364:     *equal = PETSC_TRUE;
365:   }
366:   PetscFunctionReturn(PETSC_SUCCESS);
367: }

369: static PetscErrorCode PetscDTJacobianInverse_Internal(PetscInt m, PetscInt n, const PetscReal J[], PetscReal Jinv[])
370: {
371:   PetscScalar *Js, *Jinvs;
372:   PetscInt     i, j, k;
373:   PetscBLASInt bm, bn, info;

375:   PetscFunctionBegin;
376:   if (!m || !n) PetscFunctionReturn(PETSC_SUCCESS);
377:   PetscCall(PetscBLASIntCast(m, &bm));
378:   PetscCall(PetscBLASIntCast(n, &bn));
379: #if defined(PETSC_USE_COMPLEX)
380:   PetscCall(PetscMalloc2(m * n, &Js, m * n, &Jinvs));
381:   for (i = 0; i < m * n; i++) Js[i] = J[i];
382: #else
383:   Js    = (PetscReal *)J;
384:   Jinvs = Jinv;
385: #endif
386:   if (m == n) {
387:     PetscBLASInt *pivots;
388:     PetscScalar  *W;

390:     PetscCall(PetscMalloc2(m, &pivots, m, &W));

392:     PetscCall(PetscArraycpy(Jinvs, Js, m * m));
393:     PetscCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bm, &bm, Jinvs, &bm, pivots, &info));
394:     PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetrf %" PetscInt_FMT, (PetscInt)info);
395:     PetscCallBLAS("LAPACKgetri", LAPACKgetri_(&bm, Jinvs, &bm, pivots, W, &bm, &info));
396:     PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetri %" PetscInt_FMT, (PetscInt)info);
397:     PetscCall(PetscFree2(pivots, W));
398:   } else if (m < n) {
399:     PetscScalar  *JJT;
400:     PetscBLASInt *pivots;
401:     PetscScalar  *W;

403:     PetscCall(PetscMalloc1(m * m, &JJT));
404:     PetscCall(PetscMalloc2(m, &pivots, m, &W));
405:     for (i = 0; i < m; i++) {
406:       for (j = 0; j < m; j++) {
407:         PetscScalar val = 0.;

409:         for (k = 0; k < n; k++) val += Js[i * n + k] * Js[j * n + k];
410:         JJT[i * m + j] = val;
411:       }
412:     }

414:     PetscCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bm, &bm, JJT, &bm, pivots, &info));
415:     PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetrf %" PetscInt_FMT, (PetscInt)info);
416:     PetscCallBLAS("LAPACKgetri", LAPACKgetri_(&bm, JJT, &bm, pivots, W, &bm, &info));
417:     PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetri %" PetscInt_FMT, (PetscInt)info);
418:     for (i = 0; i < n; i++) {
419:       for (j = 0; j < m; j++) {
420:         PetscScalar val = 0.;

422:         for (k = 0; k < m; k++) val += Js[k * n + i] * JJT[k * m + j];
423:         Jinvs[i * m + j] = val;
424:       }
425:     }
426:     PetscCall(PetscFree2(pivots, W));
427:     PetscCall(PetscFree(JJT));
428:   } else {
429:     PetscScalar  *JTJ;
430:     PetscBLASInt *pivots;
431:     PetscScalar  *W;

433:     PetscCall(PetscMalloc1(n * n, &JTJ));
434:     PetscCall(PetscMalloc2(n, &pivots, n, &W));
435:     for (i = 0; i < n; i++) {
436:       for (j = 0; j < n; j++) {
437:         PetscScalar val = 0.;

439:         for (k = 0; k < m; k++) val += Js[k * n + i] * Js[k * n + j];
440:         JTJ[i * n + j] = val;
441:       }
442:     }

444:     PetscCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bn, &bn, JTJ, &bn, pivots, &info));
445:     PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetrf %" PetscInt_FMT, (PetscInt)info);
446:     PetscCallBLAS("LAPACKgetri", LAPACKgetri_(&bn, JTJ, &bn, pivots, W, &bn, &info));
447:     PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetri %" PetscInt_FMT, (PetscInt)info);
448:     for (i = 0; i < n; i++) {
449:       for (j = 0; j < m; j++) {
450:         PetscScalar val = 0.;

452:         for (k = 0; k < n; k++) val += JTJ[i * n + k] * Js[j * n + k];
453:         Jinvs[i * m + j] = val;
454:       }
455:     }
456:     PetscCall(PetscFree2(pivots, W));
457:     PetscCall(PetscFree(JTJ));
458:   }
459: #if defined(PETSC_USE_COMPLEX)
460:   for (i = 0; i < m * n; i++) Jinv[i] = PetscRealPart(Jinvs[i]);
461:   PetscCall(PetscFree2(Js, Jinvs));
462: #endif
463:   PetscFunctionReturn(PETSC_SUCCESS);
464: }

466: /*@
467:   PetscQuadraturePushForward - Push forward a quadrature functional under an affine transformation.

469:   Collective

471:   Input Parameters:
472: + q           - the quadrature functional
473: . imageDim    - the dimension of the image of the transformation
474: . origin      - a point in the original space
475: . originImage - the image of the origin under the transformation
476: . J           - the Jacobian of the image: an [imageDim x dim] matrix in row major order
477: - formDegree  - transform the quadrature weights as k-forms of this form degree (if the number of components is a multiple of (dim choose `formDegree`),
478:                 it is assumed that they represent multiple k-forms) [see `PetscDTAltVPullback()` for interpretation of `formDegree`]

480:   Output Parameter:
481: . 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
482:   been pulled-back by the pseudoinverse of `J` to the k-form weights in the image space.

484:   Level: intermediate

486:   Note:
487:   The new quadrature rule will have a different number of components if spaces have different dimensions.
488:   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.

490: .seealso: `PetscQuadrature`, `PetscDTAltVPullback()`, `PetscDTAltVPullbackMatrix()`
491: @*/
492: PetscErrorCode PetscQuadraturePushForward(PetscQuadrature q, PetscInt imageDim, const PetscReal origin[], const PetscReal originImage[], const PetscReal J[], PetscInt formDegree, PetscQuadrature *Jinvstarq)
493: {
494:   PetscInt         dim, Nc, imageNc, formSize, Ncopies, imageFormSize, Npoints, pt, i, j, c;
495:   const PetscReal *points;
496:   const PetscReal *weights;
497:   PetscReal       *imagePoints, *imageWeights;
498:   PetscReal       *Jinv;
499:   PetscReal       *Jinvstar;

501:   PetscFunctionBegin;
503:   PetscCheck(imageDim >= PetscAbsInt(formDegree), PetscObjectComm((PetscObject)q), PETSC_ERR_ARG_INCOMP, "Cannot represent a %" PetscInt_FMT "-form in %" PetscInt_FMT " dimensions", PetscAbsInt(formDegree), imageDim);
504:   PetscCall(PetscQuadratureGetData(q, &dim, &Nc, &Npoints, &points, &weights));
505:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &formSize));
506:   PetscCheck(Nc % formSize == 0, PetscObjectComm((PetscObject)q), PETSC_ERR_ARG_INCOMP, "Number of components %" PetscInt_FMT " is not a multiple of formSize %" PetscInt_FMT, Nc, formSize);
507:   Ncopies = Nc / formSize;
508:   PetscCall(PetscDTBinomialInt(imageDim, PetscAbsInt(formDegree), &imageFormSize));
509:   imageNc = Ncopies * imageFormSize;
510:   PetscCall(PetscMalloc1(Npoints * imageDim, &imagePoints));
511:   PetscCall(PetscMalloc1(Npoints * imageNc, &imageWeights));
512:   PetscCall(PetscMalloc2(imageDim * dim, &Jinv, formSize * imageFormSize, &Jinvstar));
513:   PetscCall(PetscDTJacobianInverse_Internal(imageDim, dim, J, Jinv));
514:   PetscCall(PetscDTAltVPullbackMatrix(imageDim, dim, Jinv, formDegree, Jinvstar));
515:   for (pt = 0; pt < Npoints; pt++) {
516:     const PetscReal *point      = &points[pt * dim];
517:     PetscReal       *imagePoint = &imagePoints[pt * imageDim];

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

522:       for (j = 0; j < dim; j++) val += J[i * dim + j] * (point[j] - origin[j]);
523:       imagePoint[i] = val;
524:     }
525:     for (c = 0; c < Ncopies; c++) {
526:       const PetscReal *form      = &weights[pt * Nc + c * formSize];
527:       PetscReal       *imageForm = &imageWeights[pt * imageNc + c * imageFormSize];

529:       for (i = 0; i < imageFormSize; i++) {
530:         PetscReal val = 0.;

532:         for (j = 0; j < formSize; j++) val += Jinvstar[i * formSize + j] * form[j];
533:         imageForm[i] = val;
534:       }
535:     }
536:   }
537:   PetscCall(PetscQuadratureCreate(PetscObjectComm((PetscObject)q), Jinvstarq));
538:   PetscCall(PetscQuadratureSetData(*Jinvstarq, imageDim, imageNc, Npoints, imagePoints, imageWeights));
539:   PetscCall(PetscFree2(Jinv, Jinvstar));
540:   PetscFunctionReturn(PETSC_SUCCESS);
541: }

543: /*@C
544:   PetscQuadratureSetData - Sets the data defining the quadrature

546:   Not Collective

548:   Input Parameters:
549: + q       - The `PetscQuadrature` object
550: . dim     - The spatial dimension
551: . Nc      - The number of components
552: . npoints - The number of quadrature points
553: . points  - The coordinates of each quadrature point
554: - weights - The weight of each quadrature point

556:   Level: intermediate

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

561: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureGetData()`
562: @*/
563: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt Nc, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
564: {
565:   PetscFunctionBegin;
567:   if (dim >= 0) q->dim = dim;
568:   if (Nc >= 0) q->Nc = Nc;
569:   if (npoints >= 0) q->numPoints = npoints;
570:   if (points) {
571:     PetscAssertPointer(points, 5);
572:     q->points = points;
573:   }
574:   if (weights) {
575:     PetscAssertPointer(weights, 6);
576:     q->weights = weights;
577:   }
578:   PetscFunctionReturn(PETSC_SUCCESS);
579: }

581: static PetscErrorCode PetscQuadratureView_Ascii(PetscQuadrature quad, PetscViewer v)
582: {
583:   PetscInt          q, d, c;
584:   PetscViewerFormat format;

586:   PetscFunctionBegin;
587:   if (quad->Nc > 1)
588:     PetscCall(PetscViewerASCIIPrintf(v, "Quadrature on a %s of order %" PetscInt_FMT " on %" PetscInt_FMT " points (dim %" PetscInt_FMT ") with %" PetscInt_FMT " components\n", DMPolytopeTypes[quad->ct], quad->order, quad->numPoints, quad->dim, quad->Nc));
589:   else PetscCall(PetscViewerASCIIPrintf(v, "Quadrature on a %s of order %" PetscInt_FMT " on %" PetscInt_FMT " points (dim %" PetscInt_FMT ")\n", DMPolytopeTypes[quad->ct], quad->order, quad->numPoints, quad->dim));
590:   PetscCall(PetscViewerGetFormat(v, &format));
591:   if (format != PETSC_VIEWER_ASCII_INFO_DETAIL) PetscFunctionReturn(PETSC_SUCCESS);
592:   for (q = 0; q < quad->numPoints; ++q) {
593:     PetscCall(PetscViewerASCIIPrintf(v, "p%" PetscInt_FMT " (", q));
594:     PetscCall(PetscViewerASCIIUseTabs(v, PETSC_FALSE));
595:     for (d = 0; d < quad->dim; ++d) {
596:       if (d) PetscCall(PetscViewerASCIIPrintf(v, ", "));
597:       PetscCall(PetscViewerASCIIPrintf(v, "%+g", (double)quad->points[q * quad->dim + d]));
598:     }
599:     PetscCall(PetscViewerASCIIPrintf(v, ") "));
600:     if (quad->Nc > 1) PetscCall(PetscViewerASCIIPrintf(v, "w%" PetscInt_FMT " (", q));
601:     for (c = 0; c < quad->Nc; ++c) {
602:       if (c) PetscCall(PetscViewerASCIIPrintf(v, ", "));
603:       PetscCall(PetscViewerASCIIPrintf(v, "%+g", (double)quad->weights[q * quad->Nc + c]));
604:     }
605:     if (quad->Nc > 1) PetscCall(PetscViewerASCIIPrintf(v, ")"));
606:     PetscCall(PetscViewerASCIIPrintf(v, "\n"));
607:     PetscCall(PetscViewerASCIIUseTabs(v, PETSC_TRUE));
608:   }
609:   PetscFunctionReturn(PETSC_SUCCESS);
610: }

612: /*@C
613:   PetscQuadratureView - View a `PetscQuadrature` object

615:   Collective

617:   Input Parameters:
618: + quad   - The `PetscQuadrature` object
619: - viewer - The `PetscViewer` object

621:   Level: beginner

623: .seealso: `PetscQuadrature`, `PetscViewer`, `PetscQuadratureCreate()`, `PetscQuadratureGetData()`
624: @*/
625: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
626: {
627:   PetscBool iascii;

629:   PetscFunctionBegin;
632:   if (!viewer) PetscCall(PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject)quad), &viewer));
633:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
634:   PetscCall(PetscViewerASCIIPushTab(viewer));
635:   if (iascii) PetscCall(PetscQuadratureView_Ascii(quad, viewer));
636:   PetscCall(PetscViewerASCIIPopTab(viewer));
637:   PetscFunctionReturn(PETSC_SUCCESS);
638: }

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

643:   Not Collective; No Fortran Support

645:   Input Parameters:
646: + q              - The original `PetscQuadrature`
647: . numSubelements - The number of subelements the original element is divided into
648: . v0             - An array of the initial points for each subelement
649: - jac            - An array of the Jacobian mappings from the reference to each subelement

651:   Output Parameter:
652: . qref - The dimension

654:   Level: intermediate

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

659: .seealso: `PetscQuadrature`, `PetscFECreate()`, `PetscSpaceGetDimension()`, `PetscDualSpaceGetDimension()`
660: @*/
661: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
662: {
663:   DMPolytopeType   ct;
664:   const PetscReal *points, *weights;
665:   PetscReal       *pointsRef, *weightsRef;
666:   PetscInt         dim, Nc, order, npoints, npointsRef, c, p, cp, d, e;

668:   PetscFunctionBegin;
670:   PetscAssertPointer(v0, 3);
671:   PetscAssertPointer(jac, 4);
672:   PetscAssertPointer(qref, 5);
673:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, qref));
674:   PetscCall(PetscQuadratureGetCellType(q, &ct));
675:   PetscCall(PetscQuadratureGetOrder(q, &order));
676:   PetscCall(PetscQuadratureGetData(q, &dim, &Nc, &npoints, &points, &weights));
677:   npointsRef = npoints * numSubelements;
678:   PetscCall(PetscMalloc1(npointsRef * dim, &pointsRef));
679:   PetscCall(PetscMalloc1(npointsRef * Nc, &weightsRef));
680:   for (c = 0; c < numSubelements; ++c) {
681:     for (p = 0; p < npoints; ++p) {
682:       for (d = 0; d < dim; ++d) {
683:         pointsRef[(c * npoints + p) * dim + d] = v0[c * dim + d];
684:         for (e = 0; e < dim; ++e) pointsRef[(c * npoints + p) * dim + d] += jac[(c * dim + d) * dim + e] * (points[p * dim + e] + 1.0);
685:       }
686:       /* Could also use detJ here */
687:       for (cp = 0; cp < Nc; ++cp) weightsRef[(c * npoints + p) * Nc + cp] = weights[p * Nc + cp] / numSubelements;
688:     }
689:   }
690:   PetscCall(PetscQuadratureSetCellType(*qref, ct));
691:   PetscCall(PetscQuadratureSetOrder(*qref, order));
692:   PetscCall(PetscQuadratureSetData(*qref, dim, Nc, npointsRef, pointsRef, weightsRef));
693:   PetscFunctionReturn(PETSC_SUCCESS);
694: }

696: /* Compute the coefficients for the Jacobi polynomial recurrence,

698:    J^{a,b}_n(x) = (cnm1 + cnm1x * x) * J^{a,b}_{n-1}(x) - cnm2 * J^{a,b}_{n-2}(x).
699:  */
700: #define PetscDTJacobiRecurrence_Internal(n, a, b, cnm1, cnm1x, cnm2) \
701:   do { \
702:     PetscReal _a = (a); \
703:     PetscReal _b = (b); \
704:     PetscReal _n = (n); \
705:     if (n == 1) { \
706:       (cnm1)  = (_a - _b) * 0.5; \
707:       (cnm1x) = (_a + _b + 2.) * 0.5; \
708:       (cnm2)  = 0.; \
709:     } else { \
710:       PetscReal _2n  = _n + _n; \
711:       PetscReal _d   = (_2n * (_n + _a + _b) * (_2n + _a + _b - 2)); \
712:       PetscReal _n1  = (_2n + _a + _b - 1.) * (_a * _a - _b * _b); \
713:       PetscReal _n1x = (_2n + _a + _b - 1.) * (_2n + _a + _b) * (_2n + _a + _b - 2); \
714:       PetscReal _n2  = 2. * ((_n + _a - 1.) * (_n + _b - 1.) * (_2n + _a + _b)); \
715:       (cnm1)         = _n1 / _d; \
716:       (cnm1x)        = _n1x / _d; \
717:       (cnm2)         = _n2 / _d; \
718:     } \
719:   } while (0)

721: /*@
722:   PetscDTJacobiNorm - Compute the weighted L2 norm of a Jacobi polynomial.

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

726:   Input Parameters:
727: + alpha - the left exponent > -1
728: . beta  - the right exponent > -1
729: - n     - the polynomial degree

731:   Output Parameter:
732: . norm - the weighted L2 norm

734:   Level: beginner

736: .seealso: `PetscQuadrature`, `PetscDTJacobiEval()`
737: @*/
738: PetscErrorCode PetscDTJacobiNorm(PetscReal alpha, PetscReal beta, PetscInt n, PetscReal *norm)
739: {
740:   PetscReal twoab1;
741:   PetscReal gr;

743:   PetscFunctionBegin;
744:   PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Exponent alpha %g <= -1. invalid", (double)alpha);
745:   PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Exponent beta %g <= -1. invalid", (double)beta);
746:   PetscCheck(n >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "n %" PetscInt_FMT " < 0 invalid", n);
747:   twoab1 = PetscPowReal(2., alpha + beta + 1.);
748: #if defined(PETSC_HAVE_LGAMMA)
749:   if (!n) {
750:     gr = PetscExpReal(PetscLGamma(alpha + 1.) + PetscLGamma(beta + 1.) - PetscLGamma(alpha + beta + 2.));
751:   } else {
752:     gr = PetscExpReal(PetscLGamma(n + alpha + 1.) + PetscLGamma(n + beta + 1.) - (PetscLGamma(n + 1.) + PetscLGamma(n + alpha + beta + 1.))) / (n + n + alpha + beta + 1.);
753:   }
754: #else
755:   {
756:     PetscInt alphai = (PetscInt)alpha;
757:     PetscInt betai  = (PetscInt)beta;
758:     PetscInt i;

760:     gr = n ? (1. / (n + n + alpha + beta + 1.)) : 1.;
761:     if ((PetscReal)alphai == alpha) {
762:       if (!n) {
763:         for (i = 0; i < alphai; i++) gr *= (i + 1.) / (beta + i + 1.);
764:         gr /= (alpha + beta + 1.);
765:       } else {
766:         for (i = 0; i < alphai; i++) gr *= (n + i + 1.) / (n + beta + i + 1.);
767:       }
768:     } else if ((PetscReal)betai == beta) {
769:       if (!n) {
770:         for (i = 0; i < betai; i++) gr *= (i + 1.) / (alpha + i + 2.);
771:         gr /= (alpha + beta + 1.);
772:       } else {
773:         for (i = 0; i < betai; i++) gr *= (n + i + 1.) / (n + alpha + i + 1.);
774:       }
775:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "lgamma() - math routine is unavailable.");
776:   }
777: #endif
778:   *norm = PetscSqrtReal(twoab1 * gr);
779:   PetscFunctionReturn(PETSC_SUCCESS);
780: }

782: static PetscErrorCode PetscDTJacobiEval_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscInt k, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *p)
783: {
784:   PetscReal ak, bk;
785:   PetscReal abk1;
786:   PetscInt  i, l, maxdegree;

788:   PetscFunctionBegin;
789:   maxdegree = degrees[ndegree - 1] - k;
790:   ak        = a + k;
791:   bk        = b + k;
792:   abk1      = a + b + k + 1.;
793:   if (maxdegree < 0) {
794:     for (i = 0; i < npoints; i++)
795:       for (l = 0; l < ndegree; l++) p[i * ndegree + l] = 0.;
796:     PetscFunctionReturn(PETSC_SUCCESS);
797:   }
798:   for (i = 0; i < npoints; i++) {
799:     PetscReal pm1, pm2, x;
800:     PetscReal cnm1, cnm1x, cnm2;
801:     PetscInt  j, m;

803:     x   = points[i];
804:     pm2 = 1.;
805:     PetscDTJacobiRecurrence_Internal(1, ak, bk, cnm1, cnm1x, cnm2);
806:     pm1 = (cnm1 + cnm1x * x);
807:     l   = 0;
808:     while (l < ndegree && degrees[l] - k < 0) p[l++] = 0.;
809:     while (l < ndegree && degrees[l] - k == 0) {
810:       p[l] = pm2;
811:       for (m = 0; m < k; m++) p[l] *= (abk1 + m) * 0.5;
812:       l++;
813:     }
814:     while (l < ndegree && degrees[l] - k == 1) {
815:       p[l] = pm1;
816:       for (m = 0; m < k; m++) p[l] *= (abk1 + 1 + m) * 0.5;
817:       l++;
818:     }
819:     for (j = 2; j <= maxdegree; j++) {
820:       PetscReal pp;

822:       PetscDTJacobiRecurrence_Internal(j, ak, bk, cnm1, cnm1x, cnm2);
823:       pp  = (cnm1 + cnm1x * x) * pm1 - cnm2 * pm2;
824:       pm2 = pm1;
825:       pm1 = pp;
826:       while (l < ndegree && degrees[l] - k == j) {
827:         p[l] = pp;
828:         for (m = 0; m < k; m++) p[l] *= (abk1 + j + m) * 0.5;
829:         l++;
830:       }
831:     }
832:     p += ndegree;
833:   }
834:   PetscFunctionReturn(PETSC_SUCCESS);
835: }

837: /*@
838:   PetscDTJacobiEvalJet - Evaluate the jet (function and derivatives) of the Jacobi polynomials basis up to a given degree.

840:   Input Parameters:
841: + alpha   - the left exponent of the weight
842: . beta    - the right exponetn of the weight
843: . npoints - the number of points to evaluate the polynomials at
844: . points  - [npoints] array of point coordinates
845: . degree  - the maximm degree polynomial space to evaluate, (degree + 1) will be evaluated total.
846: - k       - the maximum derivative to evaluate in the jet, (k + 1) will be evaluated total.

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

854:   Level: advanced

856:   Notes:
857:   The Jacobi polynomials with indices $\alpha$ and $\beta$ are orthogonal with respect to the
858:   weighted inner product $\langle f, g \rangle = \int_{-1}^1 (1+x)^{\alpha} (1-x)^{\beta} f(x)
859:   g(x) dx$.

861: .seealso: `PetscDTJacobiEval()`, `PetscDTPKDEvalJet()`
862: @*/
863: PetscErrorCode PetscDTJacobiEvalJet(PetscReal alpha, PetscReal beta, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
864: {
865:   PetscInt   i, j, l;
866:   PetscInt  *degrees;
867:   PetscReal *psingle;

869:   PetscFunctionBegin;
870:   if (degree == 0) {
871:     PetscInt zero = 0;

873:     for (i = 0; i <= k; i++) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, 1, &zero, &p[i * npoints]));
874:     PetscFunctionReturn(PETSC_SUCCESS);
875:   }
876:   PetscCall(PetscMalloc1(degree + 1, &degrees));
877:   PetscCall(PetscMalloc1((degree + 1) * npoints, &psingle));
878:   for (i = 0; i <= degree; i++) degrees[i] = i;
879:   for (i = 0; i <= k; i++) {
880:     PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, degree + 1, degrees, psingle));
881:     for (j = 0; j <= degree; j++) {
882:       for (l = 0; l < npoints; l++) p[(j * (k + 1) + i) * npoints + l] = psingle[l * (degree + 1) + j];
883:     }
884:   }
885:   PetscCall(PetscFree(psingle));
886:   PetscCall(PetscFree(degrees));
887:   PetscFunctionReturn(PETSC_SUCCESS);
888: }

890: /*@
891:   PetscDTJacobiEval - evaluate Jacobi polynomials for the weight function $(1.+x)^{\alpha} (1.-x)^{\beta}$ at a set of points
892:   at points

894:   Not Collective

896:   Input Parameters:
897: + npoints - number of spatial points to evaluate at
898: . alpha   - the left exponent > -1
899: . beta    - the right exponent > -1
900: . points  - array of locations to evaluate at
901: . ndegree - number of basis degrees to evaluate
902: - degrees - sorted array of degrees to evaluate

904:   Output Parameters:
905: + B  - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension npoints*ndegrees, allocated by caller) (or `NULL`)
906: . D  - row-oriented derivative evaluation matrix (or `NULL`)
907: - D2 - row-oriented second derivative evaluation matrix (or `NULL`)

909:   Level: intermediate

911: .seealso: `PetscDTGaussQuadrature()`, `PetscDTLegendreEval()`
912: @*/
913: PetscErrorCode PetscDTJacobiEval(PetscInt npoints, PetscReal alpha, PetscReal beta, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *B, PetscReal *D, PetscReal *D2)
914: {
915:   PetscFunctionBegin;
916:   PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
917:   PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");
918:   if (!npoints || !ndegree) PetscFunctionReturn(PETSC_SUCCESS);
919:   if (B) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 0, points, ndegree, degrees, B));
920:   if (D) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 1, points, ndegree, degrees, D));
921:   if (D2) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 2, points, ndegree, degrees, D2));
922:   PetscFunctionReturn(PETSC_SUCCESS);
923: }

925: /*@
926:   PetscDTLegendreEval - evaluate Legendre polynomials at points

928:   Not Collective

930:   Input Parameters:
931: + npoints - number of spatial points to evaluate at
932: . points  - array of locations to evaluate at
933: . ndegree - number of basis degrees to evaluate
934: - degrees - sorted array of degrees to evaluate

936:   Output Parameters:
937: + B  - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension `npoints`*`ndegrees`, allocated by caller) (or `NULL`)
938: . D  - row-oriented derivative evaluation matrix (or `NULL`)
939: - D2 - row-oriented second derivative evaluation matrix (or `NULL`)

941:   Level: intermediate

943: .seealso: `PetscDTGaussQuadrature()`
944: @*/
945: PetscErrorCode PetscDTLegendreEval(PetscInt npoints, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *B, PetscReal *D, PetscReal *D2)
946: {
947:   PetscFunctionBegin;
948:   PetscCall(PetscDTJacobiEval(npoints, 0., 0., points, ndegree, degrees, B, D, D2));
949:   PetscFunctionReturn(PETSC_SUCCESS);
950: }

952: /*@
953:   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,
954:   then the index of x is smaller than the index of y)

956:   Input Parameters:
957: + len   - the desired length of the degree tuple
958: - index - the index to convert: should be >= 0

960:   Output Parameter:
961: . degtup - will be filled with a tuple of degrees

963:   Level: beginner

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

970: .seealso: `PetscDTGradedOrderToIndex()`
971: @*/
972: PetscErrorCode PetscDTIndexToGradedOrder(PetscInt len, PetscInt index, PetscInt degtup[])
973: {
974:   PetscInt i, total;
975:   PetscInt sum;

977:   PetscFunctionBeginHot;
978:   PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
979:   PetscCheck(index >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
980:   total = 1;
981:   sum   = 0;
982:   while (index >= total) {
983:     index -= total;
984:     total = (total * (len + sum)) / (sum + 1);
985:     sum++;
986:   }
987:   for (i = 0; i < len; i++) {
988:     PetscInt c;

990:     degtup[i] = sum;
991:     for (c = 0, total = 1; c < sum; c++) {
992:       /* going into the loop, total is the number of way to have a tuple of sum exactly c with length len - 1 - i */
993:       if (index < total) break;
994:       index -= total;
995:       total = (total * (len - 1 - i + c)) / (c + 1);
996:       degtup[i]--;
997:     }
998:     sum -= degtup[i];
999:   }
1000:   PetscFunctionReturn(PETSC_SUCCESS);
1001: }

1003: /*@
1004:   PetscDTGradedOrderToIndex - convert a tuple into an index in a graded order, the inverse of `PetscDTIndexToGradedOrder()`.

1006:   Input Parameters:
1007: + len    - the length of the degree tuple
1008: - degtup - tuple with this length

1010:   Output Parameter:
1011: . index - index in graded order: >= 0

1013:   Level: beginner

1015:   Note:
1016:   For two tuples x and y with the same degree sum, partial degree sums over the final elements of the tuples
1017:   acts as a tiebreaker.  For example, (2, 1, 1) and (1, 2, 1) have the same degree sum, but the degree sum over the
1018:   last two elements is smaller for the former, so (2, 1, 1) < (1, 2, 1).

1020: .seealso: `PetscDTIndexToGradedOrder()`
1021: @*/
1022: PetscErrorCode PetscDTGradedOrderToIndex(PetscInt len, const PetscInt degtup[], PetscInt *index)
1023: {
1024:   PetscInt i, idx, sum, total;

1026:   PetscFunctionBeginHot;
1027:   PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
1028:   for (i = 0, sum = 0; i < len; i++) sum += degtup[i];
1029:   idx   = 0;
1030:   total = 1;
1031:   for (i = 0; i < sum; i++) {
1032:     idx += total;
1033:     total = (total * (len + i)) / (i + 1);
1034:   }
1035:   for (i = 0; i < len - 1; i++) {
1036:     PetscInt c;

1038:     total = 1;
1039:     sum -= degtup[i];
1040:     for (c = 0; c < sum; c++) {
1041:       idx += total;
1042:       total = (total * (len - 1 - i + c)) / (c + 1);
1043:     }
1044:   }
1045:   *index = idx;
1046:   PetscFunctionReturn(PETSC_SUCCESS);
1047: }

1049: static PetscBool PKDCite       = PETSC_FALSE;
1050: const char       PKDCitation[] = "@article{Kirby2010,\n"
1051:                                  "  title={Singularity-free evaluation of collapsed-coordinate orthogonal polynomials},\n"
1052:                                  "  author={Kirby, Robert C},\n"
1053:                                  "  journal={ACM Transactions on Mathematical Software (TOMS)},\n"
1054:                                  "  volume={37},\n"
1055:                                  "  number={1},\n"
1056:                                  "  pages={1--16},\n"
1057:                                  "  year={2010},\n"
1058:                                  "  publisher={ACM New York, NY, USA}\n}\n";

1060: /*@
1061:   PetscDTPKDEvalJet - Evaluate the jet (function and derivatives) of the Proriol-Koornwinder-Dubiner (PKD) basis for
1062:   the space of polynomials up to a given degree.

1064:   Input Parameters:
1065: + dim     - the number of variables in the multivariate polynomials
1066: . npoints - the number of points to evaluate the polynomials at
1067: . points  - [npoints x dim] array of point coordinates
1068: . 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.
1069: - k       - the maximum order partial derivative to evaluate in the jet.  There are (dim + k choose dim) partial derivatives
1070:             in the jet.  Choosing k = 0 means to evaluate just the function and no derivatives

1072:   Output Parameter:
1073: . p - an array containing the evaluations of the PKD polynomials' jets on the points.  The size is ((dim + degree)
1074:       choose dim) x ((dim + k) choose dim) x npoints, which also describes the order of the dimensions of this
1075:       three-dimensional array: the first (slowest varying) dimension is basis function index; the second dimension is jet
1076:       index; the third (fastest varying) dimension is the index of the evaluation point.

1078:   Level: advanced

1080:   Notes:
1081:   The PKD basis is L2-orthonormal on the biunit simplex (which is used as the reference element
1082:   for finite elements in PETSc), which makes it a stable basis to use for evaluating
1083:   polynomials in that domain.

1085:   The ordering of the basis functions, and the ordering of the derivatives in the jet, both follow the graded
1086:   ordering of `PetscDTIndexToGradedOrder()` and `PetscDTGradedOrderToIndex()`.  For example, in 3D, the polynomial with
1087:   leading monomial x^2,y^0,z^1, which has degree tuple (2,0,1), which by `PetscDTGradedOrderToIndex()` has index 12 (it is the 13th basis function in the space);
1088:   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).

1090:   The implementation uses Kirby's singularity-free evaluation algorithm, <https://doi.org/10.1145/1644001.1644006>.

1092: .seealso: `PetscDTGradedOrderToIndex()`, `PetscDTIndexToGradedOrder()`, `PetscDTJacobiEvalJet()`
1093: @*/
1094: PetscErrorCode PetscDTPKDEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
1095: {
1096:   PetscInt   degidx, kidx, d, pt;
1097:   PetscInt   Nk, Ndeg;
1098:   PetscInt  *ktup, *degtup;
1099:   PetscReal *scales, initscale, scaleexp;

1101:   PetscFunctionBegin;
1102:   PetscCall(PetscCitationsRegister(PKDCitation, &PKDCite));
1103:   PetscCall(PetscDTBinomialInt(dim + k, k, &Nk));
1104:   PetscCall(PetscDTBinomialInt(degree + dim, degree, &Ndeg));
1105:   PetscCall(PetscMalloc2(dim, &degtup, dim, &ktup));
1106:   PetscCall(PetscMalloc1(Ndeg, &scales));
1107:   initscale = 1.;
1108:   if (dim > 1) {
1109:     PetscCall(PetscDTBinomial(dim, 2, &scaleexp));
1110:     initscale = PetscPowReal(2., scaleexp * 0.5);
1111:   }
1112:   for (degidx = 0; degidx < Ndeg; degidx++) {
1113:     PetscInt  e, i;
1114:     PetscInt  m1idx = -1, m2idx = -1;
1115:     PetscInt  n;
1116:     PetscInt  degsum;
1117:     PetscReal alpha;
1118:     PetscReal cnm1, cnm1x, cnm2;
1119:     PetscReal norm;

1121:     PetscCall(PetscDTIndexToGradedOrder(dim, degidx, degtup));
1122:     for (d = dim - 1; d >= 0; d--)
1123:       if (degtup[d]) break;
1124:     if (d < 0) { /* constant is 1 everywhere, all derivatives are zero */
1125:       scales[degidx] = initscale;
1126:       for (e = 0; e < dim; e++) {
1127:         PetscCall(PetscDTJacobiNorm(e, 0., 0, &norm));
1128:         scales[degidx] /= norm;
1129:       }
1130:       for (i = 0; i < npoints; i++) p[degidx * Nk * npoints + i] = 1.;
1131:       for (i = 0; i < (Nk - 1) * npoints; i++) p[(degidx * Nk + 1) * npoints + i] = 0.;
1132:       continue;
1133:     }
1134:     n = degtup[d];
1135:     degtup[d]--;
1136:     PetscCall(PetscDTGradedOrderToIndex(dim, degtup, &m1idx));
1137:     if (degtup[d] > 0) {
1138:       degtup[d]--;
1139:       PetscCall(PetscDTGradedOrderToIndex(dim, degtup, &m2idx));
1140:       degtup[d]++;
1141:     }
1142:     degtup[d]++;
1143:     for (e = 0, degsum = 0; e < d; e++) degsum += degtup[e];
1144:     alpha = 2 * degsum + d;
1145:     PetscDTJacobiRecurrence_Internal(n, alpha, 0., cnm1, cnm1x, cnm2);

1147:     scales[degidx] = initscale;
1148:     for (e = 0, degsum = 0; e < dim; e++) {
1149:       PetscInt  f;
1150:       PetscReal ealpha;
1151:       PetscReal enorm;

1153:       ealpha = 2 * degsum + e;
1154:       for (f = 0; f < degsum; f++) scales[degidx] *= 2.;
1155:       PetscCall(PetscDTJacobiNorm(ealpha, 0., degtup[e], &enorm));
1156:       scales[degidx] /= enorm;
1157:       degsum += degtup[e];
1158:     }

1160:     for (pt = 0; pt < npoints; pt++) {
1161:       /* compute the multipliers */
1162:       PetscReal thetanm1, thetanm1x, thetanm2;

1164:       thetanm1x = dim - (d + 1) + 2. * points[pt * dim + d];
1165:       for (e = d + 1; e < dim; e++) thetanm1x += points[pt * dim + e];
1166:       thetanm1x *= 0.5;
1167:       thetanm1 = (2. - (dim - (d + 1)));
1168:       for (e = d + 1; e < dim; e++) thetanm1 -= points[pt * dim + e];
1169:       thetanm1 *= 0.5;
1170:       thetanm2 = thetanm1 * thetanm1;

1172:       for (kidx = 0; kidx < Nk; kidx++) {
1173:         PetscInt f;

1175:         PetscCall(PetscDTIndexToGradedOrder(dim, kidx, ktup));
1176:         /* first sum in the same derivative terms */
1177:         p[(degidx * Nk + kidx) * npoints + pt] = (cnm1 * thetanm1 + cnm1x * thetanm1x) * p[(m1idx * Nk + kidx) * npoints + pt];
1178:         if (m2idx >= 0) p[(degidx * Nk + kidx) * npoints + pt] -= cnm2 * thetanm2 * p[(m2idx * Nk + kidx) * npoints + pt];

1180:         for (f = d; f < dim; f++) {
1181:           PetscInt km1idx, mplty = ktup[f];

1183:           if (!mplty) continue;
1184:           ktup[f]--;
1185:           PetscCall(PetscDTGradedOrderToIndex(dim, ktup, &km1idx));

1187:           /* the derivative of  cnm1x * thetanm1x  wrt x variable f is 0.5 * cnm1x if f > d otherwise it is cnm1x */
1188:           /* the derivative of  cnm1  * thetanm1   wrt x variable f is 0 if f == d, otherwise it is -0.5 * cnm1 */
1189:           /* the derivative of -cnm2  * thetanm2   wrt x variable f is 0 if f == d, otherwise it is cnm2 * thetanm1 */
1190:           if (f > d) {
1191:             PetscInt f2;

1193:             p[(degidx * Nk + kidx) * npoints + pt] += mplty * 0.5 * (cnm1x - cnm1) * p[(m1idx * Nk + km1idx) * npoints + pt];
1194:             if (m2idx >= 0) {
1195:               p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm2 * thetanm1 * p[(m2idx * Nk + km1idx) * npoints + pt];
1196:               /* second derivatives of -cnm2  * thetanm2   wrt x variable f,f2 is like - 0.5 * cnm2 */
1197:               for (f2 = f; f2 < dim; f2++) {
1198:                 PetscInt km2idx, mplty2 = ktup[f2];
1199:                 PetscInt factor;

1201:                 if (!mplty2) continue;
1202:                 ktup[f2]--;
1203:                 PetscCall(PetscDTGradedOrderToIndex(dim, ktup, &km2idx));

1205:                 factor = mplty * mplty2;
1206:                 if (f == f2) factor /= 2;
1207:                 p[(degidx * Nk + kidx) * npoints + pt] -= 0.5 * factor * cnm2 * p[(m2idx * Nk + km2idx) * npoints + pt];
1208:                 ktup[f2]++;
1209:               }
1210:             }
1211:           } else {
1212:             p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm1x * p[(m1idx * Nk + km1idx) * npoints + pt];
1213:           }
1214:           ktup[f]++;
1215:         }
1216:       }
1217:     }
1218:   }
1219:   for (degidx = 0; degidx < Ndeg; degidx++) {
1220:     PetscReal scale = scales[degidx];
1221:     PetscInt  i;

1223:     for (i = 0; i < Nk * npoints; i++) p[degidx * Nk * npoints + i] *= scale;
1224:   }
1225:   PetscCall(PetscFree(scales));
1226:   PetscCall(PetscFree2(degtup, ktup));
1227:   PetscFunctionReturn(PETSC_SUCCESS);
1228: }

1230: /*@
1231:   PetscDTPTrimmedSize - The size of the trimmed polynomial space of k-forms with a given degree and form degree,
1232:   which can be evaluated in `PetscDTPTrimmedEvalJet()`.

1234:   Input Parameters:
1235: + dim        - the number of variables in the multivariate polynomials
1236: . degree     - the degree (sum of degrees on the variables in a monomial) of the trimmed polynomial space.
1237: - formDegree - the degree of the form

1239:   Output Parameter:
1240: . size - The number ((`dim` + `degree`) choose (`dim` + `formDegree`)) x ((`degree` + `formDegree` - 1) choose (`formDegree`))

1242:   Level: advanced

1244: .seealso: `PetscDTPTrimmedEvalJet()`
1245: @*/
1246: PetscErrorCode PetscDTPTrimmedSize(PetscInt dim, PetscInt degree, PetscInt formDegree, PetscInt *size)
1247: {
1248:   PetscInt Nrk, Nbpt; // number of trimmed polynomials

1250:   PetscFunctionBegin;
1251:   formDegree = PetscAbsInt(formDegree);
1252:   PetscCall(PetscDTBinomialInt(degree + dim, degree + formDegree, &Nbpt));
1253:   PetscCall(PetscDTBinomialInt(degree + formDegree - 1, formDegree, &Nrk));
1254:   Nbpt *= Nrk;
1255:   *size = Nbpt;
1256:   PetscFunctionReturn(PETSC_SUCCESS);
1257: }

1259: /* there was a reference implementation based on section 4.4 of Arnold, Falk & Winther (acta numerica, 2006), but it
1260:  * was inferior to this implementation */
1261: static PetscErrorCode PetscDTPTrimmedEvalJet_Internal(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt formDegree, PetscInt jetDegree, PetscReal p[])
1262: {
1263:   PetscInt  formDegreeOrig = formDegree;
1264:   PetscBool formNegative   = (formDegreeOrig < 0) ? PETSC_TRUE : PETSC_FALSE;

1266:   PetscFunctionBegin;
1267:   formDegree = PetscAbsInt(formDegreeOrig);
1268:   if (formDegree == 0) {
1269:     PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree, jetDegree, p));
1270:     PetscFunctionReturn(PETSC_SUCCESS);
1271:   }
1272:   if (formDegree == dim) {
1273:     PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree - 1, jetDegree, p));
1274:     PetscFunctionReturn(PETSC_SUCCESS);
1275:   }
1276:   PetscInt Nbpt;
1277:   PetscCall(PetscDTPTrimmedSize(dim, degree, formDegree, &Nbpt));
1278:   PetscInt Nf;
1279:   PetscCall(PetscDTBinomialInt(dim, formDegree, &Nf));
1280:   PetscInt Nk;
1281:   PetscCall(PetscDTBinomialInt(dim + jetDegree, dim, &Nk));
1282:   PetscCall(PetscArrayzero(p, Nbpt * Nf * Nk * npoints));

1284:   PetscInt Nbpm1; // number of scalar polynomials up to degree - 1;
1285:   PetscCall(PetscDTBinomialInt(dim + degree - 1, dim, &Nbpm1));
1286:   PetscReal *p_scalar;
1287:   PetscCall(PetscMalloc1(Nbpm1 * Nk * npoints, &p_scalar));
1288:   PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree - 1, jetDegree, p_scalar));
1289:   PetscInt total = 0;
1290:   // First add the full polynomials up to degree - 1 into the basis: take the scalar
1291:   // and copy one for each form component
1292:   for (PetscInt i = 0; i < Nbpm1; i++) {
1293:     const PetscReal *src = &p_scalar[i * Nk * npoints];
1294:     for (PetscInt f = 0; f < Nf; f++) {
1295:       PetscReal *dest = &p[(total++ * Nf + f) * Nk * npoints];
1296:       PetscCall(PetscArraycpy(dest, src, Nk * npoints));
1297:     }
1298:   }
1299:   PetscInt *form_atoms;
1300:   PetscCall(PetscMalloc1(formDegree + 1, &form_atoms));
1301:   // construct the interior product pattern
1302:   PetscInt(*pattern)[3];
1303:   PetscInt Nf1; // number of formDegree + 1 forms
1304:   PetscCall(PetscDTBinomialInt(dim, formDegree + 1, &Nf1));
1305:   PetscInt nnz = Nf1 * (formDegree + 1);
1306:   PetscCall(PetscMalloc1(Nf1 * (formDegree + 1), &pattern));
1307:   PetscCall(PetscDTAltVInteriorPattern(dim, formDegree + 1, pattern));
1308:   PetscReal centroid = (1. - dim) / (dim + 1.);
1309:   PetscInt *deriv;
1310:   PetscCall(PetscMalloc1(dim, &deriv));
1311:   for (PetscInt d = dim; d >= formDegree + 1; d--) {
1312:     PetscInt Nfd1; // number of formDegree + 1 forms in dimension d that include dx_0
1313:                    // (equal to the number of formDegree forms in dimension d-1)
1314:     PetscCall(PetscDTBinomialInt(d - 1, formDegree, &Nfd1));
1315:     // The number of homogeneous (degree-1) scalar polynomials in d variables
1316:     PetscInt Nh;
1317:     PetscCall(PetscDTBinomialInt(d - 1 + degree - 1, d - 1, &Nh));
1318:     const PetscReal *h_scalar = &p_scalar[(Nbpm1 - Nh) * Nk * npoints];
1319:     for (PetscInt b = 0; b < Nh; b++) {
1320:       const PetscReal *h_s = &h_scalar[b * Nk * npoints];
1321:       for (PetscInt f = 0; f < Nfd1; f++) {
1322:         // construct all formDegree+1 forms that start with dx_(dim - d) /\ ...
1323:         form_atoms[0] = dim - d;
1324:         PetscCall(PetscDTEnumSubset(d - 1, formDegree, f, &form_atoms[1]));
1325:         for (PetscInt i = 0; i < formDegree; i++) form_atoms[1 + i] += form_atoms[0] + 1;
1326:         PetscInt f_ind; // index of the resulting form
1327:         PetscCall(PetscDTSubsetIndex(dim, formDegree + 1, form_atoms, &f_ind));
1328:         PetscReal *p_f = &p[total++ * Nf * Nk * npoints];
1329:         for (PetscInt nz = 0; nz < nnz; nz++) {
1330:           PetscInt  i     = pattern[nz][0]; // formDegree component
1331:           PetscInt  j     = pattern[nz][1]; // (formDegree + 1) component
1332:           PetscInt  v     = pattern[nz][2]; // coordinate component
1333:           PetscReal scale = v < 0 ? -1. : 1.;

1335:           i     = formNegative ? (Nf - 1 - i) : i;
1336:           scale = (formNegative && (i & 1)) ? -scale : scale;
1337:           v     = v < 0 ? -(v + 1) : v;
1338:           if (j != f_ind) continue;
1339:           PetscReal *p_i = &p_f[i * Nk * npoints];
1340:           for (PetscInt jet = 0; jet < Nk; jet++) {
1341:             const PetscReal *h_jet = &h_s[jet * npoints];
1342:             PetscReal       *p_jet = &p_i[jet * npoints];

1344:             for (PetscInt pt = 0; pt < npoints; pt++) p_jet[pt] += scale * h_jet[pt] * (points[pt * dim + v] - centroid);
1345:             PetscCall(PetscDTIndexToGradedOrder(dim, jet, deriv));
1346:             deriv[v]++;
1347:             PetscReal mult = deriv[v];
1348:             PetscInt  l;
1349:             PetscCall(PetscDTGradedOrderToIndex(dim, deriv, &l));
1350:             if (l >= Nk) continue;
1351:             p_jet = &p_i[l * npoints];
1352:             for (PetscInt pt = 0; pt < npoints; pt++) p_jet[pt] += scale * mult * h_jet[pt];
1353:             deriv[v]--;
1354:           }
1355:         }
1356:       }
1357:     }
1358:   }
1359:   PetscCheck(total == Nbpt, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Incorrectly counted P trimmed polynomials");
1360:   PetscCall(PetscFree(deriv));
1361:   PetscCall(PetscFree(pattern));
1362:   PetscCall(PetscFree(form_atoms));
1363:   PetscCall(PetscFree(p_scalar));
1364:   PetscFunctionReturn(PETSC_SUCCESS);
1365: }

1367: /*@
1368:   PetscDTPTrimmedEvalJet - Evaluate the jet (function and derivatives) of a basis of the trimmed polynomial k-forms up to
1369:   a given degree.

1371:   Input Parameters:
1372: + dim        - the number of variables in the multivariate polynomials
1373: . npoints    - the number of points to evaluate the polynomials at
1374: . points     - [npoints x dim] array of point coordinates
1375: . degree     - the degree (sum of degrees on the variables in a monomial) of the trimmed polynomial space to evaluate.
1376:            There are ((dim + degree) choose (dim + formDegree)) x ((degree + formDegree - 1) choose (formDegree)) polynomials in this space.
1377:            (You can use `PetscDTPTrimmedSize()` to compute this size.)
1378: . formDegree - the degree of the form
1379: - jetDegree  - the maximum order partial derivative to evaluate in the jet.  There are ((dim + jetDegree) choose dim) partial derivatives
1380:               in the jet.  Choosing jetDegree = 0 means to evaluate just the function and no derivatives

1382:   Output Parameter:
1383: . p - an array containing the evaluations of the PKD polynomials' jets on the points.

1385:   Level: advanced

1387:   Notes:
1388:   The size of `p` is `PetscDTPTrimmedSize()` x ((dim + formDegree) choose dim) x ((dim + k)
1389:   choose dim) x npoints,which also describes the order of the dimensions of this
1390:   four-dimensional array\:

1392:   the first (slowest varying) dimension is basis function index;
1393:   the second dimension is component of the form;
1394:   the third dimension is jet index;
1395:   the fourth (fastest varying) dimension is the index of the evaluation point.

1397:   The ordering of the basis functions is not graded, so the basis functions are not nested by degree like `PetscDTPKDEvalJet()`.
1398:   The basis functions are not an L2-orthonormal basis on any particular domain.

1400:   The implementation is based on the description of the trimmed polynomials up to degree r as
1401:   the direct sum of polynomials up to degree (r-1) and the Koszul differential applied to
1402:   homogeneous polynomials of degree (r-1).

1404: .seealso: `PetscDTPKDEvalJet()`, `PetscDTPTrimmedSize()`
1405: @*/
1406: PetscErrorCode PetscDTPTrimmedEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt formDegree, PetscInt jetDegree, PetscReal p[])
1407: {
1408:   PetscFunctionBegin;
1409:   PetscCall(PetscDTPTrimmedEvalJet_Internal(dim, npoints, points, degree, formDegree, jetDegree, p));
1410:   PetscFunctionReturn(PETSC_SUCCESS);
1411: }

1413: /* solve the symmetric tridiagonal eigenvalue system, writing the eigenvalues into eigs and the eigenvectors into V
1414:  * with lds n; diag and subdiag are overwritten */
1415: static PetscErrorCode PetscDTSymmetricTridiagonalEigensolve(PetscInt n, PetscReal diag[], PetscReal subdiag[], PetscReal eigs[], PetscScalar V[])
1416: {
1417:   char          jobz   = 'V'; /* eigenvalues and eigenvectors */
1418:   char          range  = 'A'; /* all eigenvalues will be found */
1419:   PetscReal     VL     = 0.;  /* ignored because range is 'A' */
1420:   PetscReal     VU     = 0.;  /* ignored because range is 'A' */
1421:   PetscBLASInt  IL     = 0;   /* ignored because range is 'A' */
1422:   PetscBLASInt  IU     = 0;   /* ignored because range is 'A' */
1423:   PetscReal     abstol = 0.;  /* unused */
1424:   PetscBLASInt  bn, bm, ldz;  /* bm will equal bn on exit */
1425:   PetscBLASInt *isuppz;
1426:   PetscBLASInt  lwork, liwork;
1427:   PetscReal     workquery;
1428:   PetscBLASInt  iworkquery;
1429:   PetscBLASInt *iwork;
1430:   PetscBLASInt  info;
1431:   PetscReal    *work = NULL;

1433:   PetscFunctionBegin;
1434: #if !defined(PETSCDTGAUSSIANQUADRATURE_EIG)
1435:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
1436: #endif
1437:   PetscCall(PetscBLASIntCast(n, &bn));
1438:   PetscCall(PetscBLASIntCast(n, &ldz));
1439: #if !defined(PETSC_MISSING_LAPACK_STEGR)
1440:   PetscCall(PetscMalloc1(2 * n, &isuppz));
1441:   lwork  = -1;
1442:   liwork = -1;
1443:   PetscCallBLAS("LAPACKstegr", LAPACKstegr_(&jobz, &range, &bn, diag, subdiag, &VL, &VU, &IL, &IU, &abstol, &bm, eigs, V, &ldz, isuppz, &workquery, &lwork, &iworkquery, &liwork, &info));
1444:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEGR error");
1445:   lwork  = (PetscBLASInt)workquery;
1446:   liwork = (PetscBLASInt)iworkquery;
1447:   PetscCall(PetscMalloc2(lwork, &work, liwork, &iwork));
1448:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
1449:   PetscCallBLAS("LAPACKstegr", LAPACKstegr_(&jobz, &range, &bn, diag, subdiag, &VL, &VU, &IL, &IU, &abstol, &bm, eigs, V, &ldz, isuppz, work, &lwork, iwork, &liwork, &info));
1450:   PetscCall(PetscFPTrapPop());
1451:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEGR error");
1452:   PetscCall(PetscFree2(work, iwork));
1453:   PetscCall(PetscFree(isuppz));
1454: #elif !defined(PETSC_MISSING_LAPACK_STEQR)
1455:   jobz = 'I'; /* Compute eigenvalues and eigenvectors of the
1456:                  tridiagonal matrix.  Z is initialized to the identity
1457:                  matrix. */
1458:   PetscCall(PetscMalloc1(PetscMax(1, 2 * n - 2), &work));
1459:   PetscCallBLAS("LAPACKsteqr", LAPACKsteqr_("I", &bn, diag, subdiag, V, &ldz, work, &info));
1460:   PetscCall(PetscFPTrapPop());
1461:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEQR error");
1462:   PetscCall(PetscFree(work));
1463:   PetscCall(PetscArraycpy(eigs, diag, n));
1464: #endif
1465:   PetscFunctionReturn(PETSC_SUCCESS);
1466: }

1468: /* Formula for the weights at the endpoints (-1 and 1) of Gauss-Lobatto-Jacobi
1469:  * quadrature rules on the interval [-1, 1] */
1470: static PetscErrorCode PetscDTGaussLobattoJacobiEndweights_Internal(PetscInt n, PetscReal alpha, PetscReal beta, PetscReal *leftw, PetscReal *rightw)
1471: {
1472:   PetscReal twoab1;
1473:   PetscInt  m = n - 2;
1474:   PetscReal a = alpha + 1.;
1475:   PetscReal b = beta + 1.;
1476:   PetscReal gra, grb;

1478:   PetscFunctionBegin;
1479:   twoab1 = PetscPowReal(2., a + b - 1.);
1480: #if defined(PETSC_HAVE_LGAMMA)
1481:   grb = PetscExpReal(2. * PetscLGamma(b + 1.) + PetscLGamma(m + 1.) + PetscLGamma(m + a + 1.) - (PetscLGamma(m + b + 1) + PetscLGamma(m + a + b + 1.)));
1482:   gra = PetscExpReal(2. * PetscLGamma(a + 1.) + PetscLGamma(m + 1.) + PetscLGamma(m + b + 1.) - (PetscLGamma(m + a + 1) + PetscLGamma(m + a + b + 1.)));
1483: #else
1484:   {
1485:     PetscInt alphai = (PetscInt)alpha;
1486:     PetscInt betai  = (PetscInt)beta;

1488:     if ((PetscReal)alphai == alpha && (PetscReal)betai == beta) {
1489:       PetscReal binom1, binom2;

1491:       PetscCall(PetscDTBinomial(m + b, b, &binom1));
1492:       PetscCall(PetscDTBinomial(m + a + b, b, &binom2));
1493:       grb = 1. / (binom1 * binom2);
1494:       PetscCall(PetscDTBinomial(m + a, a, &binom1));
1495:       PetscCall(PetscDTBinomial(m + a + b, a, &binom2));
1496:       gra = 1. / (binom1 * binom2);
1497:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "lgamma() - math routine is unavailable.");
1498:   }
1499: #endif
1500:   *leftw  = twoab1 * grb / b;
1501:   *rightw = twoab1 * gra / a;
1502:   PetscFunctionReturn(PETSC_SUCCESS);
1503: }

1505: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
1506:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
1507: static inline PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
1508: {
1509:   PetscReal pn1, pn2;
1510:   PetscReal cnm1, cnm1x, cnm2;
1511:   PetscInt  k;

1513:   PetscFunctionBegin;
1514:   if (!n) {
1515:     *P = 1.0;
1516:     PetscFunctionReturn(PETSC_SUCCESS);
1517:   }
1518:   PetscDTJacobiRecurrence_Internal(1, a, b, cnm1, cnm1x, cnm2);
1519:   pn2 = 1.;
1520:   pn1 = cnm1 + cnm1x * x;
1521:   if (n == 1) {
1522:     *P = pn1;
1523:     PetscFunctionReturn(PETSC_SUCCESS);
1524:   }
1525:   *P = 0.0;
1526:   for (k = 2; k < n + 1; ++k) {
1527:     PetscDTJacobiRecurrence_Internal(k, a, b, cnm1, cnm1x, cnm2);

1529:     *P  = (cnm1 + cnm1x * x) * pn1 - cnm2 * pn2;
1530:     pn2 = pn1;
1531:     pn1 = *P;
1532:   }
1533:   PetscFunctionReturn(PETSC_SUCCESS);
1534: }

1536: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
1537: static inline PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscInt k, PetscReal *P)
1538: {
1539:   PetscReal nP;
1540:   PetscInt  i;

1542:   PetscFunctionBegin;
1543:   *P = 0.0;
1544:   if (k > n) PetscFunctionReturn(PETSC_SUCCESS);
1545:   PetscCall(PetscDTComputeJacobi(a + k, b + k, n - k, x, &nP));
1546:   for (i = 0; i < k; i++) nP *= (a + b + n + 1. + i) * 0.5;
1547:   *P = nP;
1548:   PetscFunctionReturn(PETSC_SUCCESS);
1549: }

1551: static PetscErrorCode PetscDTGaussJacobiQuadrature_Newton_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal x[], PetscReal w[])
1552: {
1553:   PetscInt  maxIter = 100;
1554:   PetscReal eps     = PetscExpReal(0.75 * PetscLogReal(PETSC_MACHINE_EPSILON));
1555:   PetscReal a1, a6, gf;
1556:   PetscInt  k;

1558:   PetscFunctionBegin;

1560:   a1 = PetscPowReal(2.0, a + b + 1);
1561: #if defined(PETSC_HAVE_LGAMMA)
1562:   {
1563:     PetscReal a2, a3, a4, a5;
1564:     a2 = PetscLGamma(a + npoints + 1);
1565:     a3 = PetscLGamma(b + npoints + 1);
1566:     a4 = PetscLGamma(a + b + npoints + 1);
1567:     a5 = PetscLGamma(npoints + 1);
1568:     gf = PetscExpReal(a2 + a3 - (a4 + a5));
1569:   }
1570: #else
1571:   {
1572:     PetscInt ia, ib;

1574:     ia = (PetscInt)a;
1575:     ib = (PetscInt)b;
1576:     gf = 1.;
1577:     if (ia == a && ia >= 0) { /* compute ratio of rising factorals wrt a */
1578:       for (k = 0; k < ia; k++) gf *= (npoints + 1. + k) / (npoints + b + 1. + k);
1579:     } else if (b == b && ib >= 0) { /* compute ratio of rising factorials wrt b */
1580:       for (k = 0; k < ib; k++) gf *= (npoints + 1. + k) / (npoints + a + 1. + k);
1581:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "lgamma() - math routine is unavailable.");
1582:   }
1583: #endif

1585:   a6 = a1 * gf;
1586:   /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
1587:    Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
1588:   for (k = 0; k < npoints; ++k) {
1589:     PetscReal r = PetscCosReal(PETSC_PI * (1. - (4. * k + 3. + 2. * b) / (4. * npoints + 2. * (a + b + 1.)))), dP;
1590:     PetscInt  j;

1592:     if (k > 0) r = 0.5 * (r + x[k - 1]);
1593:     for (j = 0; j < maxIter; ++j) {
1594:       PetscReal s = 0.0, delta, f, fp;
1595:       PetscInt  i;

1597:       for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
1598:       PetscCall(PetscDTComputeJacobi(a, b, npoints, r, &f));
1599:       PetscCall(PetscDTComputeJacobiDerivative(a, b, npoints, r, 1, &fp));
1600:       delta = f / (fp - f * s);
1601:       r     = r - delta;
1602:       if (PetscAbsReal(delta) < eps) break;
1603:     }
1604:     x[k] = r;
1605:     PetscCall(PetscDTComputeJacobiDerivative(a, b, npoints, x[k], 1, &dP));
1606:     w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
1607:   }
1608:   PetscFunctionReturn(PETSC_SUCCESS);
1609: }

1611: /* Compute the diagonals of the Jacobi matrix used in Golub & Welsch algorithms for Gauss-Jacobi
1612:  * quadrature weight calculations on [-1,1] for exponents (1. + x)^a (1.-x)^b */
1613: static PetscErrorCode PetscDTJacobiMatrix_Internal(PetscInt nPoints, PetscReal a, PetscReal b, PetscReal *d, PetscReal *s)
1614: {
1615:   PetscInt i;

1617:   PetscFunctionBegin;
1618:   for (i = 0; i < nPoints; i++) {
1619:     PetscReal A, B, C;

1621:     PetscDTJacobiRecurrence_Internal(i + 1, a, b, A, B, C);
1622:     d[i] = -A / B;
1623:     if (i) s[i - 1] *= C / B;
1624:     if (i < nPoints - 1) s[i] = 1. / B;
1625:   }
1626:   PetscFunctionReturn(PETSC_SUCCESS);
1627: }

1629: static PetscErrorCode PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
1630: {
1631:   PetscReal mu0;
1632:   PetscReal ga, gb, gab;
1633:   PetscInt  i;

1635:   PetscFunctionBegin;
1636:   PetscCall(PetscCitationsRegister(GolubWelschCitation, &GolubWelschCite));

1638: #if defined(PETSC_HAVE_TGAMMA)
1639:   ga  = PetscTGamma(a + 1);
1640:   gb  = PetscTGamma(b + 1);
1641:   gab = PetscTGamma(a + b + 2);
1642: #else
1643:   {
1644:     PetscInt ia, ib;

1646:     ia = (PetscInt)a;
1647:     ib = (PetscInt)b;
1648:     if (ia == a && ib == b && ia + 1 > 0 && ib + 1 > 0 && ia + ib + 2 > 0) { /* All gamma(x) terms are (x-1)! terms */
1649:       PetscCall(PetscDTFactorial(ia, &ga));
1650:       PetscCall(PetscDTFactorial(ib, &gb));
1651:       PetscCall(PetscDTFactorial(ia + ib + 1, &gab));
1652:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "tgamma() - math routine is unavailable.");
1653:   }
1654: #endif
1655:   mu0 = PetscPowReal(2., a + b + 1.) * ga * gb / gab;

1657: #if defined(PETSCDTGAUSSIANQUADRATURE_EIG)
1658:   {
1659:     PetscReal   *diag, *subdiag;
1660:     PetscScalar *V;

1662:     PetscCall(PetscMalloc2(npoints, &diag, npoints, &subdiag));
1663:     PetscCall(PetscMalloc1(npoints * npoints, &V));
1664:     PetscCall(PetscDTJacobiMatrix_Internal(npoints, a, b, diag, subdiag));
1665:     for (i = 0; i < npoints - 1; i++) subdiag[i] = PetscSqrtReal(subdiag[i]);
1666:     PetscCall(PetscDTSymmetricTridiagonalEigensolve(npoints, diag, subdiag, x, V));
1667:     for (i = 0; i < npoints; i++) w[i] = PetscSqr(PetscRealPart(V[i * npoints])) * mu0;
1668:     PetscCall(PetscFree(V));
1669:     PetscCall(PetscFree2(diag, subdiag));
1670:   }
1671: #else
1672:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
1673: #endif
1674:   { /* As of March 2, 2020, The Sun Performance Library breaks the LAPACK contract for xstegr and xsteqr: the
1675:        eigenvalues are not guaranteed to be in ascending order.  So we heave a passive aggressive sigh and check that
1676:        the eigenvalues are sorted */
1677:     PetscBool sorted;

1679:     PetscCall(PetscSortedReal(npoints, x, &sorted));
1680:     if (!sorted) {
1681:       PetscInt  *order, i;
1682:       PetscReal *tmp;

1684:       PetscCall(PetscMalloc2(npoints, &order, npoints, &tmp));
1685:       for (i = 0; i < npoints; i++) order[i] = i;
1686:       PetscCall(PetscSortRealWithPermutation(npoints, x, order));
1687:       PetscCall(PetscArraycpy(tmp, x, npoints));
1688:       for (i = 0; i < npoints; i++) x[i] = tmp[order[i]];
1689:       PetscCall(PetscArraycpy(tmp, w, npoints));
1690:       for (i = 0; i < npoints; i++) w[i] = tmp[order[i]];
1691:       PetscCall(PetscFree2(order, tmp));
1692:     }
1693:   }
1694:   PetscFunctionReturn(PETSC_SUCCESS);
1695: }

1697: static PetscErrorCode PetscDTGaussJacobiQuadrature_Internal(PetscInt npoints, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
1698: {
1699:   PetscFunctionBegin;
1700:   PetscCheck(npoints >= 1, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of points must be positive");
1701:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
1702:   PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
1703:   PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");

1705:   if (newton) PetscCall(PetscDTGaussJacobiQuadrature_Newton_Internal(npoints, alpha, beta, x, w));
1706:   else PetscCall(PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(npoints, alpha, beta, x, w));
1707:   if (alpha == beta) { /* symmetrize */
1708:     PetscInt i;
1709:     for (i = 0; i < (npoints + 1) / 2; i++) {
1710:       PetscInt  j  = npoints - 1 - i;
1711:       PetscReal xi = x[i];
1712:       PetscReal xj = x[j];
1713:       PetscReal wi = w[i];
1714:       PetscReal wj = w[j];

1716:       x[i] = (xi - xj) / 2.;
1717:       x[j] = (xj - xi) / 2.;
1718:       w[i] = w[j] = (wi + wj) / 2.;
1719:     }
1720:   }
1721:   PetscFunctionReturn(PETSC_SUCCESS);
1722: }

1724: /*@
1725:   PetscDTGaussJacobiQuadrature - quadrature for the interval $[a, b]$ with the weight function
1726:   $(x-a)^\alpha (x-b)^\beta$.

1728:   Not Collective

1730:   Input Parameters:
1731: + npoints - the number of points in the quadrature rule
1732: . a       - the left endpoint of the interval
1733: . b       - the right endpoint of the interval
1734: . alpha   - the left exponent
1735: - beta    - the right exponent

1737:   Output Parameters:
1738: + x - array of length `npoints`, the locations of the quadrature points
1739: - w - array of length `npoints`, the weights of the quadrature points

1741:   Level: intermediate

1743:   Note:
1744:   This quadrature rule is exact for polynomials up to degree 2*`npoints` - 1.

1746: .seealso: `PetscDTGaussQuadrature()`
1747: @*/
1748: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
1749: {
1750:   PetscInt i;

1752:   PetscFunctionBegin;
1753:   PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal));
1754:   if (a != -1. || b != 1.) { /* shift */
1755:     for (i = 0; i < npoints; i++) {
1756:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1757:       w[i] *= (b - a) / 2.;
1758:     }
1759:   }
1760:   PetscFunctionReturn(PETSC_SUCCESS);
1761: }

1763: static PetscErrorCode PetscDTGaussLobattoJacobiQuadrature_Internal(PetscInt npoints, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
1764: {
1765:   PetscInt i;

1767:   PetscFunctionBegin;
1768:   PetscCheck(npoints >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of points must be positive");
1769:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
1770:   PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
1771:   PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");

1773:   x[0]           = -1.;
1774:   x[npoints - 1] = 1.;
1775:   if (npoints > 2) PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints - 2, alpha + 1., beta + 1., &x[1], &w[1], newton));
1776:   for (i = 1; i < npoints - 1; i++) w[i] /= (1. - x[i] * x[i]);
1777:   PetscCall(PetscDTGaussLobattoJacobiEndweights_Internal(npoints, alpha, beta, &w[0], &w[npoints - 1]));
1778:   PetscFunctionReturn(PETSC_SUCCESS);
1779: }

1781: /*@
1782:   PetscDTGaussLobattoJacobiQuadrature - quadrature for the interval $[a, b]$ with the weight function
1783:   $(x-a)^\alpha (x-b)^\beta$, with endpoints `a` and `b` included as quadrature points.

1785:   Not Collective

1787:   Input Parameters:
1788: + npoints - the number of points in the quadrature rule
1789: . a       - the left endpoint of the interval
1790: . b       - the right endpoint of the interval
1791: . alpha   - the left exponent
1792: - beta    - the right exponent

1794:   Output Parameters:
1795: + x - array of length `npoints`, the locations of the quadrature points
1796: - w - array of length `npoints`, the weights of the quadrature points

1798:   Level: intermediate

1800:   Note:
1801:   This quadrature rule is exact for polynomials up to degree 2*`npoints` - 3.

1803: .seealso: `PetscDTGaussJacobiQuadrature()`
1804: @*/
1805: PetscErrorCode PetscDTGaussLobattoJacobiQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
1806: {
1807:   PetscInt i;

1809:   PetscFunctionBegin;
1810:   PetscCall(PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal));
1811:   if (a != -1. || b != 1.) { /* shift */
1812:     for (i = 0; i < npoints; i++) {
1813:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1814:       w[i] *= (b - a) / 2.;
1815:     }
1816:   }
1817:   PetscFunctionReturn(PETSC_SUCCESS);
1818: }

1820: /*@
1821:   PetscDTGaussQuadrature - create Gauss-Legendre quadrature

1823:   Not Collective

1825:   Input Parameters:
1826: + npoints - number of points
1827: . a       - left end of interval (often-1)
1828: - b       - right end of interval (often +1)

1830:   Output Parameters:
1831: + x - quadrature points
1832: - w - quadrature weights

1834:   Level: intermediate

1836:   Note:
1837:   See {cite}`golub1969calculation`

1839: .seealso: `PetscDTLegendreEval()`, `PetscDTGaussJacobiQuadrature()`
1840: @*/
1841: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
1842: {
1843:   PetscInt i;

1845:   PetscFunctionBegin;
1846:   PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints, 0., 0., x, w, PetscDTGaussQuadratureNewton_Internal));
1847:   if (a != -1. || b != 1.) { /* shift */
1848:     for (i = 0; i < npoints; i++) {
1849:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1850:       w[i] *= (b - a) / 2.;
1851:     }
1852:   }
1853:   PetscFunctionReturn(PETSC_SUCCESS);
1854: }

1856: /*@C
1857:   PetscDTGaussLobattoLegendreQuadrature - creates a set of the locations and weights of the Gauss-Lobatto-Legendre
1858:   nodes of a given size on the domain $[-1,1]$

1860:   Not Collective

1862:   Input Parameters:
1863: + npoints - number of grid nodes
1864: - type    - `PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA` or `PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON`

1866:   Output Parameters:
1867: + x - quadrature points
1868: - w - quadrature weights

1870:   Level: intermediate

1872:   Notes:
1873:   For n > 30  the Newton approach computes duplicate (incorrect) values for some nodes because the initial guess is apparently not
1874:   close enough to the desired solution

1876:   These are useful for implementing spectral methods based on Gauss-Lobatto-Legendre (GLL) nodes

1878:   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

1880: .seealso: `PetscDTGaussQuadrature()`, `PetscGaussLobattoLegendreCreateType`

1882: @*/
1883: PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt npoints, PetscGaussLobattoLegendreCreateType type, PetscReal *x, PetscReal *w)
1884: {
1885:   PetscBool newton;

1887:   PetscFunctionBegin;
1888:   PetscCheck(npoints >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must provide at least 2 grid points per element");
1889:   newton = (PetscBool)(type == PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON);
1890:   PetscCall(PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, 0., 0., x, w, newton));
1891:   PetscFunctionReturn(PETSC_SUCCESS);
1892: }

1894: /*@
1895:   PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

1897:   Not Collective

1899:   Input Parameters:
1900: + dim     - The spatial dimension
1901: . Nc      - The number of components
1902: . npoints - number of points in one dimension
1903: . a       - left end of interval (often-1)
1904: - b       - right end of interval (often +1)

1906:   Output Parameter:
1907: . q - A `PetscQuadrature` object

1909:   Level: intermediate

1911: .seealso: `PetscDTGaussQuadrature()`, `PetscDTLegendreEval()`
1912: @*/
1913: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
1914: {
1915:   DMPolytopeType ct;
1916:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints * PetscSqr(npoints) : PetscSqr(npoints) : npoints;
1917:   PetscReal     *x, *w, *xw, *ww;

1919:   PetscFunctionBegin;
1920:   PetscCall(PetscMalloc1(totpoints * dim, &x));
1921:   PetscCall(PetscMalloc1(totpoints * Nc, &w));
1922:   /* Set up the Golub-Welsch system */
1923:   switch (dim) {
1924:   case 0:
1925:     ct = DM_POLYTOPE_POINT;
1926:     PetscCall(PetscFree(x));
1927:     PetscCall(PetscFree(w));
1928:     PetscCall(PetscMalloc1(1, &x));
1929:     PetscCall(PetscMalloc1(Nc, &w));
1930:     totpoints = 1;
1931:     x[0]      = 0.0;
1932:     for (PetscInt c = 0; c < Nc; ++c) w[c] = 1.0;
1933:     break;
1934:   case 1:
1935:     ct = DM_POLYTOPE_SEGMENT;
1936:     PetscCall(PetscMalloc1(npoints, &ww));
1937:     PetscCall(PetscDTGaussQuadrature(npoints, a, b, x, ww));
1938:     for (PetscInt i = 0; i < npoints; ++i)
1939:       for (PetscInt c = 0; c < Nc; ++c) w[i * Nc + c] = ww[i];
1940:     PetscCall(PetscFree(ww));
1941:     break;
1942:   case 2:
1943:     ct = DM_POLYTOPE_QUADRILATERAL;
1944:     PetscCall(PetscMalloc2(npoints, &xw, npoints, &ww));
1945:     PetscCall(PetscDTGaussQuadrature(npoints, a, b, xw, ww));
1946:     for (PetscInt i = 0; i < npoints; ++i) {
1947:       for (PetscInt j = 0; j < npoints; ++j) {
1948:         x[(i * npoints + j) * dim + 0] = xw[i];
1949:         x[(i * npoints + j) * dim + 1] = xw[j];
1950:         for (PetscInt c = 0; c < Nc; ++c) w[(i * npoints + j) * Nc + c] = ww[i] * ww[j];
1951:       }
1952:     }
1953:     PetscCall(PetscFree2(xw, ww));
1954:     break;
1955:   case 3:
1956:     ct = DM_POLYTOPE_HEXAHEDRON;
1957:     PetscCall(PetscMalloc2(npoints, &xw, npoints, &ww));
1958:     PetscCall(PetscDTGaussQuadrature(npoints, a, b, xw, ww));
1959:     for (PetscInt i = 0; i < npoints; ++i) {
1960:       for (PetscInt j = 0; j < npoints; ++j) {
1961:         for (PetscInt k = 0; k < npoints; ++k) {
1962:           x[((i * npoints + j) * npoints + k) * dim + 0] = xw[i];
1963:           x[((i * npoints + j) * npoints + k) * dim + 1] = xw[j];
1964:           x[((i * npoints + j) * npoints + k) * dim + 2] = xw[k];
1965:           for (PetscInt c = 0; c < Nc; ++c) w[((i * npoints + j) * npoints + k) * Nc + c] = ww[i] * ww[j] * ww[k];
1966:         }
1967:       }
1968:     }
1969:     PetscCall(PetscFree2(xw, ww));
1970:     break;
1971:   default:
1972:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %" PetscInt_FMT, dim);
1973:   }
1974:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
1975:   PetscCall(PetscQuadratureSetCellType(*q, ct));
1976:   PetscCall(PetscQuadratureSetOrder(*q, 2 * npoints - 1));
1977:   PetscCall(PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w));
1978:   PetscCall(PetscObjectChangeTypeName((PetscObject)*q, "GaussTensor"));
1979:   PetscFunctionReturn(PETSC_SUCCESS);
1980: }

1982: /*@
1983:   PetscDTStroudConicalQuadrature - create Stroud conical quadrature for a simplex {cite}`karniadakis2005spectral`

1985:   Not Collective

1987:   Input Parameters:
1988: + dim     - The simplex dimension
1989: . Nc      - The number of components
1990: . npoints - The number of points in one dimension
1991: . a       - left end of interval (often-1)
1992: - b       - right end of interval (often +1)

1994:   Output Parameter:
1995: . q - A `PetscQuadrature` object

1997:   Level: intermediate

1999:   Note:
2000:   For `dim` == 1, this is Gauss-Legendre quadrature

2002: .seealso: `PetscDTGaussTensorQuadrature()`, `PetscDTGaussQuadrature()`
2003: @*/
2004: PetscErrorCode PetscDTStroudConicalQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
2005: {
2006:   DMPolytopeType ct;
2007:   PetscInt       totpoints;
2008:   PetscReal     *p1, *w1;
2009:   PetscReal     *x, *w;

2011:   PetscFunctionBegin;
2012:   PetscCheck(!(a != -1.0) && !(b != 1.0), PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
2013:   switch (dim) {
2014:   case 0:
2015:     ct = DM_POLYTOPE_POINT;
2016:     break;
2017:   case 1:
2018:     ct = DM_POLYTOPE_SEGMENT;
2019:     break;
2020:   case 2:
2021:     ct = DM_POLYTOPE_TRIANGLE;
2022:     break;
2023:   case 3:
2024:     ct = DM_POLYTOPE_TETRAHEDRON;
2025:     break;
2026:   default:
2027:     ct = DM_POLYTOPE_UNKNOWN;
2028:   }
2029:   totpoints = 1;
2030:   for (PetscInt i = 0; i < dim; ++i) totpoints *= npoints;
2031:   PetscCall(PetscMalloc1(totpoints * dim, &x));
2032:   PetscCall(PetscMalloc1(totpoints * Nc, &w));
2033:   PetscCall(PetscMalloc2(npoints, &p1, npoints, &w1));
2034:   for (PetscInt i = 0; i < totpoints * Nc; ++i) w[i] = 1.;
2035:   for (PetscInt i = 0, totprev = 1, totrem = totpoints / npoints; i < dim; ++i) {
2036:     PetscReal mul;

2038:     mul = PetscPowReal(2., -i);
2039:     PetscCall(PetscDTGaussJacobiQuadrature(npoints, -1., 1., i, 0.0, p1, w1));
2040:     for (PetscInt pt = 0, l = 0; l < totprev; l++) {
2041:       for (PetscInt j = 0; j < npoints; j++) {
2042:         for (PetscInt m = 0; m < totrem; m++, pt++) {
2043:           for (PetscInt k = 0; k < i; k++) x[pt * dim + k] = (x[pt * dim + k] + 1.) * (1. - p1[j]) * 0.5 - 1.;
2044:           x[pt * dim + i] = p1[j];
2045:           for (PetscInt c = 0; c < Nc; c++) w[pt * Nc + c] *= mul * w1[j];
2046:         }
2047:       }
2048:     }
2049:     totprev *= npoints;
2050:     totrem /= npoints;
2051:   }
2052:   PetscCall(PetscFree2(p1, w1));
2053:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
2054:   PetscCall(PetscQuadratureSetCellType(*q, ct));
2055:   PetscCall(PetscQuadratureSetOrder(*q, 2 * npoints - 1));
2056:   PetscCall(PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w));
2057:   PetscCall(PetscObjectChangeTypeName((PetscObject)*q, "StroudConical"));
2058:   PetscFunctionReturn(PETSC_SUCCESS);
2059: }

2061: static PetscBool MinSymTriQuadCite       = PETSC_FALSE;
2062: const char       MinSymTriQuadCitation[] = "@article{WitherdenVincent2015,\n"
2063:                                            "  title = {On the identification of symmetric quadrature rules for finite element methods},\n"
2064:                                            "  journal = {Computers & Mathematics with Applications},\n"
2065:                                            "  volume = {69},\n"
2066:                                            "  number = {10},\n"
2067:                                            "  pages = {1232-1241},\n"
2068:                                            "  year = {2015},\n"
2069:                                            "  issn = {0898-1221},\n"
2070:                                            "  doi = {10.1016/j.camwa.2015.03.017},\n"
2071:                                            "  url = {https://www.sciencedirect.com/science/article/pii/S0898122115001224},\n"
2072:                                            "  author = {F.D. Witherden and P.E. Vincent},\n"
2073:                                            "}\n";

2075: #include "petscdttriquadrules.h"

2077: static PetscBool MinSymTetQuadCite       = PETSC_FALSE;
2078: const char       MinSymTetQuadCitation[] = "@article{JaskowiecSukumar2021\n"
2079:                                            "  author = {Jaskowiec, Jan and Sukumar, N.},\n"
2080:                                            "  title = {High-order symmetric cubature rules for tetrahedra and pyramids},\n"
2081:                                            "  journal = {International Journal for Numerical Methods in Engineering},\n"
2082:                                            "  volume = {122},\n"
2083:                                            "  number = {1},\n"
2084:                                            "  pages = {148-171},\n"
2085:                                            "  doi = {10.1002/nme.6528},\n"
2086:                                            "  url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.6528},\n"
2087:                                            "  eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.6528},\n"
2088:                                            "  year = {2021}\n"
2089:                                            "}\n";

2091: #include "petscdttetquadrules.h"

2093: // https://en.wikipedia.org/wiki/Partition_(number_theory)
2094: static PetscErrorCode PetscDTPartitionNumber(PetscInt n, PetscInt *p)
2095: {
2096:   // sequence A000041 in the OEIS
2097:   const PetscInt partition[]   = {1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604};
2098:   PetscInt       tabulated_max = PETSC_STATIC_ARRAY_LENGTH(partition) - 1;

2100:   PetscFunctionBegin;
2101:   PetscCheck(n >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Partition number not defined for negative number %" PetscInt_FMT, n);
2102:   // not implementing the pentagonal number recurrence, we don't need partition numbers for n that high
2103:   PetscCheck(n <= tabulated_max, PETSC_COMM_SELF, PETSC_ERR_SUP, "Partition numbers only tabulated up to %" PetscInt_FMT ", not computed for %" PetscInt_FMT, tabulated_max, n);
2104:   *p = partition[n];
2105:   PetscFunctionReturn(PETSC_SUCCESS);
2106: }

2108: /*@
2109:   PetscDTSimplexQuadrature - Create a quadrature rule for a simplex that exactly integrates polynomials up to a given degree.

2111:   Not Collective

2113:   Input Parameters:
2114: + dim    - The spatial dimension of the simplex (1 = segment, 2 = triangle, 3 = tetrahedron)
2115: . degree - The largest polynomial degree that is required to be integrated exactly
2116: - type   - left end of interval (often-1)

2118:   Output Parameter:
2119: . quad - A `PetscQuadrature` object for integration over the biunit simplex
2120:             (defined by the bounds $x_i >= -1$ and $\sum_i x_i <= 2 - d$) that is exact for
2121:             polynomials up to the given degree

2123:   Level: intermediate

2125: .seealso: `PetscDTSimplexQuadratureType`, `PetscDTGaussQuadrature()`, `PetscDTStroudCononicalQuadrature()`, `PetscQuadrature`
2126: @*/
2127: PetscErrorCode PetscDTSimplexQuadrature(PetscInt dim, PetscInt degree, PetscDTSimplexQuadratureType type, PetscQuadrature *quad)
2128: {
2129:   PetscDTSimplexQuadratureType orig_type = type;

2131:   PetscFunctionBegin;
2132:   PetscCheck(dim >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Negative dimension %" PetscInt_FMT, dim);
2133:   PetscCheck(degree >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Negative degree %" PetscInt_FMT, degree);
2134:   if (type == PETSCDTSIMPLEXQUAD_DEFAULT) type = PETSCDTSIMPLEXQUAD_MINSYM;
2135:   if (type == PETSCDTSIMPLEXQUAD_CONIC || dim < 2) {
2136:     PetscInt points_per_dim = (degree + 2) / 2; // ceil((degree + 1) / 2);
2137:     PetscCall(PetscDTStroudConicalQuadrature(dim, 1, points_per_dim, -1, 1, quad));
2138:   } else {
2139:     DMPolytopeType    ct;
2140:     PetscInt          n    = dim + 1;
2141:     PetscInt          fact = 1;
2142:     PetscInt         *part, *perm;
2143:     PetscInt          p = 0;
2144:     PetscInt          max_degree;
2145:     const PetscInt   *nodes_per_type     = NULL;
2146:     const PetscInt   *all_num_full_nodes = NULL;
2147:     const PetscReal **weights_list       = NULL;
2148:     const PetscReal **compact_nodes_list = NULL;
2149:     const char       *citation           = NULL;
2150:     PetscBool        *cited              = NULL;

2152:     switch (dim) {
2153:     case 0:
2154:       ct = DM_POLYTOPE_POINT;
2155:       break;
2156:     case 1:
2157:       ct = DM_POLYTOPE_SEGMENT;
2158:       break;
2159:     case 2:
2160:       ct = DM_POLYTOPE_TRIANGLE;
2161:       break;
2162:     case 3:
2163:       ct = DM_POLYTOPE_TETRAHEDRON;
2164:       break;
2165:     default:
2166:       ct = DM_POLYTOPE_UNKNOWN;
2167:     }
2168:     switch (dim) {
2169:     case 2:
2170:       cited              = &MinSymTriQuadCite;
2171:       citation           = MinSymTriQuadCitation;
2172:       max_degree         = PetscDTWVTriQuad_max_degree;
2173:       nodes_per_type     = PetscDTWVTriQuad_num_orbits;
2174:       all_num_full_nodes = PetscDTWVTriQuad_num_nodes;
2175:       weights_list       = PetscDTWVTriQuad_weights;
2176:       compact_nodes_list = PetscDTWVTriQuad_orbits;
2177:       break;
2178:     case 3:
2179:       cited              = &MinSymTetQuadCite;
2180:       citation           = MinSymTetQuadCitation;
2181:       max_degree         = PetscDTJSTetQuad_max_degree;
2182:       nodes_per_type     = PetscDTJSTetQuad_num_orbits;
2183:       all_num_full_nodes = PetscDTJSTetQuad_num_nodes;
2184:       weights_list       = PetscDTJSTetQuad_weights;
2185:       compact_nodes_list = PetscDTJSTetQuad_orbits;
2186:       break;
2187:     default:
2188:       max_degree = -1;
2189:       break;
2190:     }

2192:     if (degree > max_degree) {
2193:       if (orig_type == PETSCDTSIMPLEXQUAD_DEFAULT) {
2194:         // fall back to conic
2195:         PetscCall(PetscDTSimplexQuadrature(dim, degree, PETSCDTSIMPLEXQUAD_CONIC, quad));
2196:         PetscFunctionReturn(PETSC_SUCCESS);
2197:       } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Minimal symmetric quadrature for dim %" PetscInt_FMT ", degree %" PetscInt_FMT " unsupported", dim, degree);
2198:     }

2200:     PetscCall(PetscCitationsRegister(citation, cited));

2202:     PetscCall(PetscDTPartitionNumber(n, &p));
2203:     for (PetscInt d = 2; d <= n; d++) fact *= d;

2205:     PetscInt         num_full_nodes      = all_num_full_nodes[degree];
2206:     const PetscReal *all_compact_nodes   = compact_nodes_list[degree];
2207:     const PetscReal *all_compact_weights = weights_list[degree];
2208:     nodes_per_type                       = &nodes_per_type[p * degree];

2210:     PetscReal      *points;
2211:     PetscReal      *counts;
2212:     PetscReal      *weights;
2213:     PetscReal      *bary_to_biunit; // row-major transformation of barycentric coordinate to biunit
2214:     PetscQuadrature q;

2216:     // compute the transformation
2217:     PetscCall(PetscMalloc1(n * dim, &bary_to_biunit));
2218:     for (PetscInt d = 0; d < dim; d++) {
2219:       for (PetscInt b = 0; b < n; b++) bary_to_biunit[d * n + b] = (d == b) ? 1.0 : -1.0;
2220:     }

2222:     PetscCall(PetscMalloc3(n, &part, n, &perm, n, &counts));
2223:     PetscCall(PetscCalloc1(num_full_nodes * dim, &points));
2224:     PetscCall(PetscMalloc1(num_full_nodes, &weights));

2226:     // (0, 0, ...) is the first partition lexicographically
2227:     PetscCall(PetscArrayzero(part, n));
2228:     PetscCall(PetscArrayzero(counts, n));
2229:     counts[0] = n;

2231:     // for each partition
2232:     for (PetscInt s = 0, node_offset = 0; s < p; s++) {
2233:       PetscInt num_compact_coords = part[n - 1] + 1;

2235:       const PetscReal *compact_nodes   = all_compact_nodes;
2236:       const PetscReal *compact_weights = all_compact_weights;
2237:       all_compact_nodes += num_compact_coords * nodes_per_type[s];
2238:       all_compact_weights += nodes_per_type[s];

2240:       // for every permutation of the vertices
2241:       for (PetscInt f = 0; f < fact; f++) {
2242:         PetscCall(PetscDTEnumPerm(n, f, perm, NULL));

2244:         // check if it is a valid permutation
2245:         PetscInt digit;
2246:         for (digit = 1; digit < n; digit++) {
2247:           // skip permutations that would duplicate a node because it has a smaller symmetry group
2248:           if (part[digit - 1] == part[digit] && perm[digit - 1] > perm[digit]) break;
2249:         }
2250:         if (digit < n) continue;

2252:         // create full nodes from this permutation of the compact nodes
2253:         PetscReal *full_nodes   = &points[node_offset * dim];
2254:         PetscReal *full_weights = &weights[node_offset];

2256:         PetscCall(PetscArraycpy(full_weights, compact_weights, nodes_per_type[s]));
2257:         for (PetscInt b = 0; b < n; b++) {
2258:           for (PetscInt d = 0; d < dim; d++) {
2259:             for (PetscInt node = 0; node < nodes_per_type[s]; node++) full_nodes[node * dim + d] += bary_to_biunit[d * n + perm[b]] * compact_nodes[node * num_compact_coords + part[b]];
2260:           }
2261:         }
2262:         node_offset += nodes_per_type[s];
2263:       }

2265:       if (s < p - 1) { // Generate the next partition
2266:         /* A partition is described by the number of coordinates that are in
2267:          * each set of duplicates (counts) and redundantly by mapping each
2268:          * index to its set of duplicates (part)
2269:          *
2270:          * Counts should always be in nonincreasing order
2271:          *
2272:          * We want to generate the partitions lexically by part, which means
2273:          * finding the last index where count > 1 and reducing by 1.
2274:          *
2275:          * For the new counts beyond that index, we eagerly assign the remaining
2276:          * capacity of the partition to smaller indices (ensures lexical ordering),
2277:          * while respecting the nonincreasing invariant of the counts
2278:          */
2279:         PetscInt last_digit            = part[n - 1];
2280:         PetscInt last_digit_with_extra = last_digit;
2281:         while (counts[last_digit_with_extra] == 1) last_digit_with_extra--;
2282:         PetscInt limit               = --counts[last_digit_with_extra];
2283:         PetscInt total_to_distribute = last_digit - last_digit_with_extra + 1;
2284:         for (PetscInt digit = last_digit_with_extra + 1; digit < n; digit++) {
2285:           counts[digit] = PetscMin(limit, total_to_distribute);
2286:           total_to_distribute -= PetscMin(limit, total_to_distribute);
2287:         }
2288:         for (PetscInt digit = 0, offset = 0; digit < n; digit++) {
2289:           PetscInt count = counts[digit];
2290:           for (PetscInt c = 0; c < count; c++) part[offset++] = digit;
2291:         }
2292:       }
2293:     }
2294:     PetscCall(PetscFree3(part, perm, counts));
2295:     PetscCall(PetscFree(bary_to_biunit));
2296:     PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, &q));
2297:     PetscCall(PetscQuadratureSetCellType(q, ct));
2298:     PetscCall(PetscQuadratureSetOrder(q, degree));
2299:     PetscCall(PetscQuadratureSetData(q, dim, 1, num_full_nodes, points, weights));
2300:     *quad = q;
2301:   }
2302:   PetscFunctionReturn(PETSC_SUCCESS);
2303: }

2305: /*@
2306:   PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell

2308:   Not Collective

2310:   Input Parameters:
2311: + dim   - The cell dimension
2312: . level - The number of points in one dimension, $2^l$
2313: . a     - left end of interval (often-1)
2314: - b     - right end of interval (often +1)

2316:   Output Parameter:
2317: . q - A `PetscQuadrature` object

2319:   Level: intermediate

2321: .seealso: `PetscDTGaussTensorQuadrature()`, `PetscQuadrature`
2322: @*/
2323: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
2324: {
2325:   DMPolytopeType  ct;
2326:   const PetscInt  p     = 16;                        /* Digits of precision in the evaluation */
2327:   const PetscReal alpha = (b - a) / 2.;              /* Half-width of the integration interval */
2328:   const PetscReal beta  = (b + a) / 2.;              /* Center of the integration interval */
2329:   const PetscReal h     = PetscPowReal(2.0, -level); /* Step size, length between x_k */
2330:   PetscReal       xk;                                /* Quadrature point x_k on reference domain [-1, 1] */
2331:   PetscReal       wk = 0.5 * PETSC_PI;               /* Quadrature weight at x_k */
2332:   PetscReal      *x, *w;
2333:   PetscInt        K, k, npoints;

2335:   PetscFunctionBegin;
2336:   PetscCheck(dim <= 1, PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %" PetscInt_FMT " not yet implemented", dim);
2337:   PetscCheck(level, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
2338:   switch (dim) {
2339:   case 0:
2340:     ct = DM_POLYTOPE_POINT;
2341:     break;
2342:   case 1:
2343:     ct = DM_POLYTOPE_SEGMENT;
2344:     break;
2345:   case 2:
2346:     ct = DM_POLYTOPE_QUADRILATERAL;
2347:     break;
2348:   case 3:
2349:     ct = DM_POLYTOPE_HEXAHEDRON;
2350:     break;
2351:   default:
2352:     ct = DM_POLYTOPE_UNKNOWN;
2353:   }
2354:   /* Find K such that the weights are < 32 digits of precision */
2355:   for (K = 1; PetscAbsReal(PetscLog10Real(wk)) < 2 * p; ++K) wk = 0.5 * h * PETSC_PI * PetscCoshReal(K * h) / PetscSqr(PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(K * h)));
2356:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
2357:   PetscCall(PetscQuadratureSetCellType(*q, ct));
2358:   PetscCall(PetscQuadratureSetOrder(*q, 2 * K + 1));
2359:   npoints = 2 * K - 1;
2360:   PetscCall(PetscMalloc1(npoints * dim, &x));
2361:   PetscCall(PetscMalloc1(npoints, &w));
2362:   /* Center term */
2363:   x[0] = beta;
2364:   w[0] = 0.5 * alpha * PETSC_PI;
2365:   for (k = 1; k < K; ++k) {
2366:     wk           = 0.5 * alpha * h * PETSC_PI * PetscCoshReal(k * h) / PetscSqr(PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2367:     xk           = PetscTanhReal(0.5 * PETSC_PI * PetscSinhReal(k * h));
2368:     x[2 * k - 1] = -alpha * xk + beta;
2369:     w[2 * k - 1] = wk;
2370:     x[2 * k + 0] = alpha * xk + beta;
2371:     w[2 * k + 0] = wk;
2372:   }
2373:   PetscCall(PetscQuadratureSetData(*q, dim, 1, npoints, x, w));
2374:   PetscFunctionReturn(PETSC_SUCCESS);
2375: }

2377: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2378: {
2379:   const PetscInt  p     = 16;           /* Digits of precision in the evaluation */
2380:   const PetscReal alpha = (b - a) / 2.; /* Half-width of the integration interval */
2381:   const PetscReal beta  = (b + a) / 2.; /* Center of the integration interval */
2382:   PetscReal       h     = 1.0;          /* Step size, length between x_k */
2383:   PetscInt        l     = 0;            /* Level of refinement, h = 2^{-l} */
2384:   PetscReal       osum  = 0.0;          /* Integral on last level */
2385:   PetscReal       psum  = 0.0;          /* Integral on the level before the last level */
2386:   PetscReal       sum;                  /* Integral on current level */
2387:   PetscReal       yk;                   /* Quadrature point 1 - x_k on reference domain [-1, 1] */
2388:   PetscReal       lx, rx;               /* Quadrature points to the left and right of 0 on the real domain [a, b] */
2389:   PetscReal       wk;                   /* Quadrature weight at x_k */
2390:   PetscReal       lval, rval;           /* Terms in the quadature sum to the left and right of 0 */
2391:   PetscInt        d;                    /* Digits of precision in the integral */

2393:   PetscFunctionBegin;
2394:   PetscCheck(digits > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
2395:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
2396:   /* Center term */
2397:   func(&beta, ctx, &lval);
2398:   sum = 0.5 * alpha * PETSC_PI * lval;
2399:   /* */
2400:   do {
2401:     PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
2402:     PetscInt  k = 1;

2404:     ++l;
2405:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %" PetscInt_FMT " sum: %15.15f\n", l, sum); */
2406:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
2407:     psum = osum;
2408:     osum = sum;
2409:     h *= 0.5;
2410:     sum *= 0.5;
2411:     do {
2412:       wk = 0.5 * h * PETSC_PI * PetscCoshReal(k * h) / PetscSqr(PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2413:       yk = 1.0 / (PetscExpReal(0.5 * PETSC_PI * PetscSinhReal(k * h)) * PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2414:       lx = -alpha * (1.0 - yk) + beta;
2415:       rx = alpha * (1.0 - yk) + beta;
2416:       func(&lx, ctx, &lval);
2417:       func(&rx, ctx, &rval);
2418:       lterm   = alpha * wk * lval;
2419:       maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
2420:       sum += lterm;
2421:       rterm   = alpha * wk * rval;
2422:       maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
2423:       sum += rterm;
2424:       ++k;
2425:       /* Only need to evaluate every other point on refined levels */
2426:       if (l != 1) ++k;
2427:     } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */

2429:     d1 = PetscLog10Real(PetscAbsReal(sum - osum));
2430:     d2 = PetscLog10Real(PetscAbsReal(sum - psum));
2431:     d3 = PetscLog10Real(maxTerm) - p;
2432:     if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
2433:     else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
2434:     d = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1) / d2, 2 * d1), d3), d4)));
2435:   } while (d < digits && l < 12);
2436:   *sol = sum;
2437:   PetscCall(PetscFPTrapPop());
2438:   PetscFunctionReturn(PETSC_SUCCESS);
2439: }

2441: #if defined(PETSC_HAVE_MPFR)
2442: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2443: {
2444:   const PetscInt safetyFactor = 2; /* Calculate abcissa until 2*p digits */
2445:   PetscInt       l            = 0; /* Level of refinement, h = 2^{-l} */
2446:   mpfr_t         alpha;            /* Half-width of the integration interval */
2447:   mpfr_t         beta;             /* Center of the integration interval */
2448:   mpfr_t         h;                /* Step size, length between x_k */
2449:   mpfr_t         osum;             /* Integral on last level */
2450:   mpfr_t         psum;             /* Integral on the level before the last level */
2451:   mpfr_t         sum;              /* Integral on current level */
2452:   mpfr_t         yk;               /* Quadrature point 1 - x_k on reference domain [-1, 1] */
2453:   mpfr_t         lx, rx;           /* Quadrature points to the left and right of 0 on the real domain [a, b] */
2454:   mpfr_t         wk;               /* Quadrature weight at x_k */
2455:   PetscReal      lval, rval, rtmp; /* Terms in the quadature sum to the left and right of 0 */
2456:   PetscInt       d;                /* Digits of precision in the integral */
2457:   mpfr_t         pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;

2459:   PetscFunctionBegin;
2460:   PetscCheck(digits > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
2461:   /* Create high precision storage */
2462:   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);
2463:   /* Initialization */
2464:   mpfr_set_d(alpha, 0.5 * (b - a), MPFR_RNDN);
2465:   mpfr_set_d(beta, 0.5 * (b + a), MPFR_RNDN);
2466:   mpfr_set_d(osum, 0.0, MPFR_RNDN);
2467:   mpfr_set_d(psum, 0.0, MPFR_RNDN);
2468:   mpfr_set_d(h, 1.0, MPFR_RNDN);
2469:   mpfr_const_pi(pi2, MPFR_RNDN);
2470:   mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
2471:   /* Center term */
2472:   rtmp = 0.5 * (b + a);
2473:   func(&rtmp, ctx, &lval);
2474:   mpfr_set(sum, pi2, MPFR_RNDN);
2475:   mpfr_mul(sum, sum, alpha, MPFR_RNDN);
2476:   mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
2477:   /* */
2478:   do {
2479:     PetscReal d1, d2, d3, d4;
2480:     PetscInt  k = 1;

2482:     ++l;
2483:     mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
2484:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %" PetscInt_FMT " sum: %15.15f\n", l, sum); */
2485:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
2486:     mpfr_set(psum, osum, MPFR_RNDN);
2487:     mpfr_set(osum, sum, MPFR_RNDN);
2488:     mpfr_mul_d(h, h, 0.5, MPFR_RNDN);
2489:     mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
2490:     do {
2491:       mpfr_set_si(kh, k, MPFR_RNDN);
2492:       mpfr_mul(kh, kh, h, MPFR_RNDN);
2493:       /* Weight */
2494:       mpfr_set(wk, h, MPFR_RNDN);
2495:       mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
2496:       mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
2497:       mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
2498:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
2499:       mpfr_sqr(tmp, tmp, MPFR_RNDN);
2500:       mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
2501:       mpfr_div(wk, wk, tmp, MPFR_RNDN);
2502:       /* Abscissa */
2503:       mpfr_set_d(yk, 1.0, MPFR_RNDZ);
2504:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
2505:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
2506:       mpfr_exp(tmp, msinh, MPFR_RNDN);
2507:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
2508:       /* Quadrature points */
2509:       mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
2510:       mpfr_mul(lx, lx, alpha, MPFR_RNDU);
2511:       mpfr_add(lx, lx, beta, MPFR_RNDU);
2512:       mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
2513:       mpfr_mul(rx, rx, alpha, MPFR_RNDD);
2514:       mpfr_add(rx, rx, beta, MPFR_RNDD);
2515:       /* Evaluation */
2516:       rtmp = mpfr_get_d(lx, MPFR_RNDU);
2517:       func(&rtmp, ctx, &lval);
2518:       rtmp = mpfr_get_d(rx, MPFR_RNDD);
2519:       func(&rtmp, ctx, &rval);
2520:       /* Update */
2521:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
2522:       mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
2523:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
2524:       mpfr_abs(tmp, tmp, MPFR_RNDN);
2525:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
2526:       mpfr_set(curTerm, tmp, MPFR_RNDN);
2527:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
2528:       mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
2529:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
2530:       mpfr_abs(tmp, tmp, MPFR_RNDN);
2531:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
2532:       mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
2533:       ++k;
2534:       /* Only need to evaluate every other point on refined levels */
2535:       if (l != 1) ++k;
2536:       mpfr_log10(tmp, wk, MPFR_RNDN);
2537:       mpfr_abs(tmp, tmp, MPFR_RNDN);
2538:     } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor * digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
2539:     mpfr_sub(tmp, sum, osum, MPFR_RNDN);
2540:     mpfr_abs(tmp, tmp, MPFR_RNDN);
2541:     mpfr_log10(tmp, tmp, MPFR_RNDN);
2542:     d1 = mpfr_get_d(tmp, MPFR_RNDN);
2543:     mpfr_sub(tmp, sum, psum, MPFR_RNDN);
2544:     mpfr_abs(tmp, tmp, MPFR_RNDN);
2545:     mpfr_log10(tmp, tmp, MPFR_RNDN);
2546:     d2 = mpfr_get_d(tmp, MPFR_RNDN);
2547:     mpfr_log10(tmp, maxTerm, MPFR_RNDN);
2548:     d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
2549:     mpfr_log10(tmp, curTerm, MPFR_RNDN);
2550:     d4 = mpfr_get_d(tmp, MPFR_RNDN);
2551:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1) / d2, 2 * d1), d3), d4)));
2552:   } while (d < digits && l < 8);
2553:   *sol = mpfr_get_d(sum, MPFR_RNDN);
2554:   /* Cleanup */
2555:   mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
2556:   PetscFunctionReturn(PETSC_SUCCESS);
2557: }
2558: #else

2560: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2561: {
2562:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
2563: }
2564: #endif

2566: /*@
2567:   PetscDTTensorQuadratureCreate - create the tensor product quadrature from two lower-dimensional quadratures

2569:   Not Collective

2571:   Input Parameters:
2572: + q1 - The first quadrature
2573: - q2 - The second quadrature

2575:   Output Parameter:
2576: . q - A `PetscQuadrature` object

2578:   Level: intermediate

2580: .seealso: `PetscQuadrature`, `PetscDTGaussTensorQuadrature()`
2581: @*/
2582: PetscErrorCode PetscDTTensorQuadratureCreate(PetscQuadrature q1, PetscQuadrature q2, PetscQuadrature *q)
2583: {
2584:   DMPolytopeType   ct1, ct2, ct;
2585:   const PetscReal *x1, *w1, *x2, *w2;
2586:   PetscReal       *x, *w;
2587:   PetscInt         dim1, Nc1, Np1, order1, qa, d1;
2588:   PetscInt         dim2, Nc2, Np2, order2, qb, d2;
2589:   PetscInt         dim, Nc, Np, order, qc, d;

2591:   PetscFunctionBegin;
2594:   PetscAssertPointer(q, 3);
2595:   PetscCall(PetscQuadratureGetOrder(q1, &order1));
2596:   PetscCall(PetscQuadratureGetOrder(q2, &order2));
2597:   PetscCheck(order1 == order2, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Order1 %" PetscInt_FMT " != %" PetscInt_FMT " Order2", order1, order2);
2598:   PetscCall(PetscQuadratureGetData(q1, &dim1, &Nc1, &Np1, &x1, &w1));
2599:   PetscCall(PetscQuadratureGetCellType(q1, &ct1));
2600:   PetscCall(PetscQuadratureGetData(q2, &dim2, &Nc2, &Np2, &x2, &w2));
2601:   PetscCall(PetscQuadratureGetCellType(q2, &ct2));
2602:   PetscCheck(Nc1 == Nc2, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "NumComp1 %" PetscInt_FMT " != %" PetscInt_FMT " NumComp2", Nc1, Nc2);

2604:   switch (ct1) {
2605:   case DM_POLYTOPE_POINT:
2606:     ct = ct2;
2607:     break;
2608:   case DM_POLYTOPE_SEGMENT:
2609:     switch (ct2) {
2610:     case DM_POLYTOPE_POINT:
2611:       ct = ct1;
2612:       break;
2613:     case DM_POLYTOPE_SEGMENT:
2614:       ct = DM_POLYTOPE_QUADRILATERAL;
2615:       break;
2616:     case DM_POLYTOPE_TRIANGLE:
2617:       ct = DM_POLYTOPE_TRI_PRISM;
2618:       break;
2619:     case DM_POLYTOPE_QUADRILATERAL:
2620:       ct = DM_POLYTOPE_HEXAHEDRON;
2621:       break;
2622:     case DM_POLYTOPE_TETRAHEDRON:
2623:       ct = DM_POLYTOPE_UNKNOWN;
2624:       break;
2625:     case DM_POLYTOPE_HEXAHEDRON:
2626:       ct = DM_POLYTOPE_UNKNOWN;
2627:       break;
2628:     default:
2629:       ct = DM_POLYTOPE_UNKNOWN;
2630:     }
2631:     break;
2632:   case DM_POLYTOPE_TRIANGLE:
2633:     switch (ct2) {
2634:     case DM_POLYTOPE_POINT:
2635:       ct = ct1;
2636:       break;
2637:     case DM_POLYTOPE_SEGMENT:
2638:       ct = DM_POLYTOPE_TRI_PRISM;
2639:       break;
2640:     case DM_POLYTOPE_TRIANGLE:
2641:       ct = DM_POLYTOPE_UNKNOWN;
2642:       break;
2643:     case DM_POLYTOPE_QUADRILATERAL:
2644:       ct = DM_POLYTOPE_UNKNOWN;
2645:       break;
2646:     case DM_POLYTOPE_TETRAHEDRON:
2647:       ct = DM_POLYTOPE_UNKNOWN;
2648:       break;
2649:     case DM_POLYTOPE_HEXAHEDRON:
2650:       ct = DM_POLYTOPE_UNKNOWN;
2651:       break;
2652:     default:
2653:       ct = DM_POLYTOPE_UNKNOWN;
2654:     }
2655:     break;
2656:   case DM_POLYTOPE_QUADRILATERAL:
2657:     switch (ct2) {
2658:     case DM_POLYTOPE_POINT:
2659:       ct = ct1;
2660:       break;
2661:     case DM_POLYTOPE_SEGMENT:
2662:       ct = DM_POLYTOPE_HEXAHEDRON;
2663:       break;
2664:     case DM_POLYTOPE_TRIANGLE:
2665:       ct = DM_POLYTOPE_UNKNOWN;
2666:       break;
2667:     case DM_POLYTOPE_QUADRILATERAL:
2668:       ct = DM_POLYTOPE_UNKNOWN;
2669:       break;
2670:     case DM_POLYTOPE_TETRAHEDRON:
2671:       ct = DM_POLYTOPE_UNKNOWN;
2672:       break;
2673:     case DM_POLYTOPE_HEXAHEDRON:
2674:       ct = DM_POLYTOPE_UNKNOWN;
2675:       break;
2676:     default:
2677:       ct = DM_POLYTOPE_UNKNOWN;
2678:     }
2679:     break;
2680:   case DM_POLYTOPE_TETRAHEDRON:
2681:     switch (ct2) {
2682:     case DM_POLYTOPE_POINT:
2683:       ct = ct1;
2684:       break;
2685:     case DM_POLYTOPE_SEGMENT:
2686:       ct = DM_POLYTOPE_UNKNOWN;
2687:       break;
2688:     case DM_POLYTOPE_TRIANGLE:
2689:       ct = DM_POLYTOPE_UNKNOWN;
2690:       break;
2691:     case DM_POLYTOPE_QUADRILATERAL:
2692:       ct = DM_POLYTOPE_UNKNOWN;
2693:       break;
2694:     case DM_POLYTOPE_TETRAHEDRON:
2695:       ct = DM_POLYTOPE_UNKNOWN;
2696:       break;
2697:     case DM_POLYTOPE_HEXAHEDRON:
2698:       ct = DM_POLYTOPE_UNKNOWN;
2699:       break;
2700:     default:
2701:       ct = DM_POLYTOPE_UNKNOWN;
2702:     }
2703:     break;
2704:   case DM_POLYTOPE_HEXAHEDRON:
2705:     switch (ct2) {
2706:     case DM_POLYTOPE_POINT:
2707:       ct = ct1;
2708:       break;
2709:     case DM_POLYTOPE_SEGMENT:
2710:       ct = DM_POLYTOPE_UNKNOWN;
2711:       break;
2712:     case DM_POLYTOPE_TRIANGLE:
2713:       ct = DM_POLYTOPE_UNKNOWN;
2714:       break;
2715:     case DM_POLYTOPE_QUADRILATERAL:
2716:       ct = DM_POLYTOPE_UNKNOWN;
2717:       break;
2718:     case DM_POLYTOPE_TETRAHEDRON:
2719:       ct = DM_POLYTOPE_UNKNOWN;
2720:       break;
2721:     case DM_POLYTOPE_HEXAHEDRON:
2722:       ct = DM_POLYTOPE_UNKNOWN;
2723:       break;
2724:     default:
2725:       ct = DM_POLYTOPE_UNKNOWN;
2726:     }
2727:     break;
2728:   default:
2729:     ct = DM_POLYTOPE_UNKNOWN;
2730:   }
2731:   dim   = dim1 + dim2;
2732:   Nc    = Nc1;
2733:   Np    = Np1 * Np2;
2734:   order = order1;
2735:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
2736:   PetscCall(PetscQuadratureSetCellType(*q, ct));
2737:   PetscCall(PetscQuadratureSetOrder(*q, order));
2738:   PetscCall(PetscMalloc1(Np * dim, &x));
2739:   PetscCall(PetscMalloc1(Np, &w));
2740:   for (qa = 0, qc = 0; qa < Np1; ++qa) {
2741:     for (qb = 0; qb < Np2; ++qb, ++qc) {
2742:       for (d1 = 0, d = 0; d1 < dim1; ++d1, ++d) x[qc * dim + d] = x1[qa * dim1 + d1];
2743:       for (d2 = 0; d2 < dim2; ++d2, ++d) x[qc * dim + d] = x2[qb * dim2 + d2];
2744:       w[qc] = w1[qa] * w2[qb];
2745:     }
2746:   }
2747:   PetscCall(PetscQuadratureSetData(*q, dim, Nc, Np, x, w));
2748:   PetscFunctionReturn(PETSC_SUCCESS);
2749: }

2751: /* Overwrites A. Can only handle full-rank problems with m>=n
2752:    A in column-major format
2753:    Ainv in row-major format
2754:    tau has length m
2755:    worksize must be >= max(1,n)
2756:  */
2757: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m, PetscInt mstride, PetscInt n, PetscReal *A_in, PetscReal *Ainv_out, PetscScalar *tau, PetscInt worksize, PetscScalar *work)
2758: {
2759:   PetscBLASInt M, N, K, lda, ldb, ldwork, info;
2760:   PetscScalar *A, *Ainv, *R, *Q, Alpha;

2762:   PetscFunctionBegin;
2763: #if defined(PETSC_USE_COMPLEX)
2764:   {
2765:     PetscInt i, j;
2766:     PetscCall(PetscMalloc2(m * n, &A, m * n, &Ainv));
2767:     for (j = 0; j < n; j++) {
2768:       for (i = 0; i < m; i++) A[i + m * j] = A_in[i + mstride * j];
2769:     }
2770:     mstride = m;
2771:   }
2772: #else
2773:   A    = A_in;
2774:   Ainv = Ainv_out;
2775: #endif

2777:   PetscCall(PetscBLASIntCast(m, &M));
2778:   PetscCall(PetscBLASIntCast(n, &N));
2779:   PetscCall(PetscBLASIntCast(mstride, &lda));
2780:   PetscCall(PetscBLASIntCast(worksize, &ldwork));
2781:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
2782:   PetscCallBLAS("LAPACKgeqrf", LAPACKgeqrf_(&M, &N, A, &lda, tau, work, &ldwork, &info));
2783:   PetscCall(PetscFPTrapPop());
2784:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "xGEQRF error");
2785:   R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

2787:   /* Extract an explicit representation of Q */
2788:   Q = Ainv;
2789:   PetscCall(PetscArraycpy(Q, A, mstride * n));
2790:   K = N; /* full rank */
2791:   PetscCallBLAS("LAPACKorgqr", LAPACKorgqr_(&M, &N, &K, Q, &lda, tau, work, &ldwork, &info));
2792:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "xORGQR/xUNGQR error");

2794:   /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
2795:   Alpha = 1.0;
2796:   ldb   = lda;
2797:   PetscCallBLAS("BLAStrsm", BLAStrsm_("Right", "Upper", "ConjugateTranspose", "NotUnitTriangular", &M, &N, &Alpha, R, &lda, Q, &ldb));
2798:   /* Ainv is Q, overwritten with inverse */

2800: #if defined(PETSC_USE_COMPLEX)
2801:   {
2802:     PetscInt i;
2803:     for (i = 0; i < m * n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
2804:     PetscCall(PetscFree2(A, Ainv));
2805:   }
2806: #endif
2807:   PetscFunctionReturn(PETSC_SUCCESS);
2808: }

2810: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
2811: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval, const PetscReal *x, PetscInt ndegree, const PetscInt *degrees, PetscBool Transpose, PetscReal *B)
2812: {
2813:   PetscReal *Bv;
2814:   PetscInt   i, j;

2816:   PetscFunctionBegin;
2817:   PetscCall(PetscMalloc1((ninterval + 1) * ndegree, &Bv));
2818:   /* Point evaluation of L_p on all the source vertices */
2819:   PetscCall(PetscDTLegendreEval(ninterval + 1, x, ndegree, degrees, Bv, NULL, NULL));
2820:   /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
2821:   for (i = 0; i < ninterval; i++) {
2822:     for (j = 0; j < ndegree; j++) {
2823:       if (Transpose) B[i + ninterval * j] = Bv[(i + 1) * ndegree + j] - Bv[i * ndegree + j];
2824:       else B[i * ndegree + j] = Bv[(i + 1) * ndegree + j] - Bv[i * ndegree + j];
2825:     }
2826:   }
2827:   PetscCall(PetscFree(Bv));
2828:   PetscFunctionReturn(PETSC_SUCCESS);
2829: }

2831: /*@
2832:   PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

2834:   Not Collective

2836:   Input Parameters:
2837: + degree  - degree of reconstruction polynomial
2838: . nsource - number of source intervals
2839: . sourcex - sorted coordinates of source cell boundaries (length `nsource`+1)
2840: . ntarget - number of target intervals
2841: - targetx - sorted coordinates of target cell boundaries (length `ntarget`+1)

2843:   Output Parameter:
2844: . R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]

2846:   Level: advanced

2848: .seealso: `PetscDTLegendreEval()`
2849: @*/
2850: PetscErrorCode PetscDTReconstructPoly(PetscInt degree, PetscInt nsource, const PetscReal *sourcex, PetscInt ntarget, const PetscReal *targetx, PetscReal *R)
2851: {
2852:   PetscInt     i, j, k, *bdegrees, worksize;
2853:   PetscReal    xmin, xmax, center, hscale, *sourcey, *targety, *Bsource, *Bsinv, *Btarget;
2854:   PetscScalar *tau, *work;

2856:   PetscFunctionBegin;
2857:   PetscAssertPointer(sourcex, 3);
2858:   PetscAssertPointer(targetx, 5);
2859:   PetscAssertPointer(R, 6);
2860:   PetscCheck(degree < nsource, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Reconstruction degree %" PetscInt_FMT " must be less than number of source intervals %" PetscInt_FMT, degree, nsource);
2861:   if (PetscDefined(USE_DEBUG)) {
2862:     for (i = 0; i < nsource; i++) PetscCheck(sourcex[i] < sourcex[i + 1], PETSC_COMM_SELF, PETSC_ERR_ARG_CORRUPT, "Source interval %" PetscInt_FMT " has negative orientation (%g,%g)", i, (double)sourcex[i], (double)sourcex[i + 1]);
2863:     for (i = 0; i < ntarget; i++) PetscCheck(targetx[i] < targetx[i + 1], PETSC_COMM_SELF, PETSC_ERR_ARG_CORRUPT, "Target interval %" PetscInt_FMT " has negative orientation (%g,%g)", i, (double)targetx[i], (double)targetx[i + 1]);
2864:   }
2865:   xmin     = PetscMin(sourcex[0], targetx[0]);
2866:   xmax     = PetscMax(sourcex[nsource], targetx[ntarget]);
2867:   center   = (xmin + xmax) / 2;
2868:   hscale   = (xmax - xmin) / 2;
2869:   worksize = nsource;
2870:   PetscCall(PetscMalloc4(degree + 1, &bdegrees, nsource + 1, &sourcey, nsource * (degree + 1), &Bsource, worksize, &work));
2871:   PetscCall(PetscMalloc4(nsource, &tau, nsource * (degree + 1), &Bsinv, ntarget + 1, &targety, ntarget * (degree + 1), &Btarget));
2872:   for (i = 0; i <= nsource; i++) sourcey[i] = (sourcex[i] - center) / hscale;
2873:   for (i = 0; i <= degree; i++) bdegrees[i] = i + 1;
2874:   PetscCall(PetscDTLegendreIntegrate(nsource, sourcey, degree + 1, bdegrees, PETSC_TRUE, Bsource));
2875:   PetscCall(PetscDTPseudoInverseQR(nsource, nsource, degree + 1, Bsource, Bsinv, tau, nsource, work));
2876:   for (i = 0; i <= ntarget; i++) targety[i] = (targetx[i] - center) / hscale;
2877:   PetscCall(PetscDTLegendreIntegrate(ntarget, targety, degree + 1, bdegrees, PETSC_FALSE, Btarget));
2878:   for (i = 0; i < ntarget; i++) {
2879:     PetscReal rowsum = 0;
2880:     for (j = 0; j < nsource; j++) {
2881:       PetscReal sum = 0;
2882:       for (k = 0; k < degree + 1; k++) sum += Btarget[i * (degree + 1) + k] * Bsinv[k * nsource + j];
2883:       R[i * nsource + j] = sum;
2884:       rowsum += sum;
2885:     }
2886:     for (j = 0; j < nsource; j++) R[i * nsource + j] /= rowsum; /* normalize each row */
2887:   }
2888:   PetscCall(PetscFree4(bdegrees, sourcey, Bsource, work));
2889:   PetscCall(PetscFree4(tau, Bsinv, targety, Btarget));
2890:   PetscFunctionReturn(PETSC_SUCCESS);
2891: }

2893: /*@C
2894:   PetscGaussLobattoLegendreIntegrate - Compute the L2 integral of a function on the GLL points

2896:   Not Collective

2898:   Input Parameters:
2899: + n       - the number of GLL nodes
2900: . nodes   - the GLL nodes
2901: . weights - the GLL weights
2902: - f       - the function values at the nodes

2904:   Output Parameter:
2905: . in - the value of the integral

2907:   Level: beginner

2909: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`
2910: @*/
2911: PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt n, PetscReal *nodes, PetscReal *weights, const PetscReal *f, PetscReal *in)
2912: {
2913:   PetscInt i;

2915:   PetscFunctionBegin;
2916:   *in = 0.;
2917:   for (i = 0; i < n; i++) *in += f[i] * f[i] * weights[i];
2918:   PetscFunctionReturn(PETSC_SUCCESS);
2919: }

2921: /*@C
2922:   PetscGaussLobattoLegendreElementLaplacianCreate - computes the Laplacian for a single 1d GLL element

2924:   Not Collective

2926:   Input Parameters:
2927: + n       - the number of GLL nodes
2928: . nodes   - the GLL nodes
2929: - weights - the GLL weights

2931:   Output Parameter:
2932: . AA - the stiffness element

2934:   Level: beginner

2936:   Notes:
2937:   Destroy this with `PetscGaussLobattoLegendreElementLaplacianDestroy()`

2939:   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)

2941: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianDestroy()`
2942: @*/
2943: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
2944: {
2945:   PetscReal      **A;
2946:   const PetscReal *gllnodes = nodes;
2947:   const PetscInt   p        = n - 1;
2948:   PetscReal        z0, z1, z2 = -1, x, Lpj, Lpr;
2949:   PetscInt         i, j, nn, r;

2951:   PetscFunctionBegin;
2952:   PetscCall(PetscMalloc1(n, &A));
2953:   PetscCall(PetscMalloc1(n * n, &A[0]));
2954:   for (i = 1; i < n; i++) A[i] = A[i - 1] + n;

2956:   for (j = 1; j < p; j++) {
2957:     x  = gllnodes[j];
2958:     z0 = 1.;
2959:     z1 = x;
2960:     for (nn = 1; nn < p; nn++) {
2961:       z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2962:       z0 = z1;
2963:       z1 = z2;
2964:     }
2965:     Lpj = z2;
2966:     for (r = 1; r < p; r++) {
2967:       if (r == j) {
2968:         A[j][j] = 2. / (3. * (1. - gllnodes[j] * gllnodes[j]) * Lpj * Lpj);
2969:       } else {
2970:         x  = gllnodes[r];
2971:         z0 = 1.;
2972:         z1 = x;
2973:         for (nn = 1; nn < p; nn++) {
2974:           z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2975:           z0 = z1;
2976:           z1 = z2;
2977:         }
2978:         Lpr     = z2;
2979:         A[r][j] = 4. / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * Lpr * (gllnodes[j] - gllnodes[r]) * (gllnodes[j] - gllnodes[r]));
2980:       }
2981:     }
2982:   }
2983:   for (j = 1; j < p + 1; j++) {
2984:     x  = gllnodes[j];
2985:     z0 = 1.;
2986:     z1 = x;
2987:     for (nn = 1; nn < p; nn++) {
2988:       z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2989:       z0 = z1;
2990:       z1 = z2;
2991:     }
2992:     Lpj     = z2;
2993:     A[j][0] = 4. * PetscPowRealInt(-1., p) / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * (1. + gllnodes[j]) * (1. + gllnodes[j]));
2994:     A[0][j] = A[j][0];
2995:   }
2996:   for (j = 0; j < p; j++) {
2997:     x  = gllnodes[j];
2998:     z0 = 1.;
2999:     z1 = x;
3000:     for (nn = 1; nn < p; nn++) {
3001:       z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
3002:       z0 = z1;
3003:       z1 = z2;
3004:     }
3005:     Lpj = z2;

3007:     A[p][j] = 4. / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * (1. - gllnodes[j]) * (1. - gllnodes[j]));
3008:     A[j][p] = A[p][j];
3009:   }
3010:   A[0][0] = 0.5 + (((PetscReal)p) * (((PetscReal)p) + 1.) - 2.) / 6.;
3011:   A[p][p] = A[0][0];
3012:   *AA     = A;
3013:   PetscFunctionReturn(PETSC_SUCCESS);
3014: }

3016: /*@C
3017:   PetscGaussLobattoLegendreElementLaplacianDestroy - frees the Laplacian for a single 1d GLL element created with `PetscGaussLobattoLegendreElementLaplacianCreate()`

3019:   Not Collective

3021:   Input Parameters:
3022: + n       - the number of GLL nodes
3023: . nodes   - the GLL nodes
3024: . weights - the GLL weightss
3025: - AA      - the stiffness element

3027:   Level: beginner

3029: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`
3030: @*/
3031: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3032: {
3033:   PetscFunctionBegin;
3034:   PetscCall(PetscFree((*AA)[0]));
3035:   PetscCall(PetscFree(*AA));
3036:   *AA = NULL;
3037:   PetscFunctionReturn(PETSC_SUCCESS);
3038: }

3040: /*@C
3041:   PetscGaussLobattoLegendreElementGradientCreate - computes the gradient for a single 1d GLL element

3043:   Not Collective

3045:   Input Parameters:
3046: + n       - the number of GLL nodes
3047: . nodes   - the GLL nodes
3048: - weights - the GLL weights

3050:   Output Parameters:
3051: + AA  - the stiffness element
3052: - AAT - the transpose of AA (pass in `NULL` if you do not need this array)

3054:   Level: beginner

3056:   Notes:
3057:   Destroy this with `PetscGaussLobattoLegendreElementGradientDestroy()`

3059:   You can access entries in these arrays with AA[i][j] but in memory it is stored in contiguous memory, row oriented

3061: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianDestroy()`, `PetscGaussLobattoLegendreElementGradientDestroy()`
3062: @*/
3063: PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA, PetscReal ***AAT)
3064: {
3065:   PetscReal      **A, **AT = NULL;
3066:   const PetscReal *gllnodes = nodes;
3067:   const PetscInt   p        = n - 1;
3068:   PetscReal        Li, Lj, d0;
3069:   PetscInt         i, j;

3071:   PetscFunctionBegin;
3072:   PetscCall(PetscMalloc1(n, &A));
3073:   PetscCall(PetscMalloc1(n * n, &A[0]));
3074:   for (i = 1; i < n; i++) A[i] = A[i - 1] + n;

3076:   if (AAT) {
3077:     PetscCall(PetscMalloc1(n, &AT));
3078:     PetscCall(PetscMalloc1(n * n, &AT[0]));
3079:     for (i = 1; i < n; i++) AT[i] = AT[i - 1] + n;
3080:   }

3082:   if (n == 1) A[0][0] = 0.;
3083:   d0 = (PetscReal)p * ((PetscReal)p + 1.) / 4.;
3084:   for (i = 0; i < n; i++) {
3085:     for (j = 0; j < n; j++) {
3086:       A[i][j] = 0.;
3087:       PetscCall(PetscDTComputeJacobi(0., 0., p, gllnodes[i], &Li));
3088:       PetscCall(PetscDTComputeJacobi(0., 0., p, gllnodes[j], &Lj));
3089:       if (i != j) A[i][j] = Li / (Lj * (gllnodes[i] - gllnodes[j]));
3090:       if ((j == i) && (i == 0)) A[i][j] = -d0;
3091:       if (j == i && i == p) A[i][j] = d0;
3092:       if (AT) AT[j][i] = A[i][j];
3093:     }
3094:   }
3095:   if (AAT) *AAT = AT;
3096:   *AA = A;
3097:   PetscFunctionReturn(PETSC_SUCCESS);
3098: }

3100: /*@C
3101:   PetscGaussLobattoLegendreElementGradientDestroy - frees the gradient for a single 1d GLL element obtained with `PetscGaussLobattoLegendreElementGradientCreate()`

3103:   Not Collective

3105:   Input Parameters:
3106: + n       - the number of GLL nodes
3107: . nodes   - the GLL nodes
3108: . weights - the GLL weights
3109: . AA      - the stiffness element
3110: - AAT     - the transpose of the element

3112:   Level: beginner

3114: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`, `PetscGaussLobattoLegendreElementAdvectionCreate()`
3115: @*/
3116: PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA, PetscReal ***AAT)
3117: {
3118:   PetscFunctionBegin;
3119:   PetscCall(PetscFree((*AA)[0]));
3120:   PetscCall(PetscFree(*AA));
3121:   *AA = NULL;
3122:   if (AAT) {
3123:     PetscCall(PetscFree((*AAT)[0]));
3124:     PetscCall(PetscFree(*AAT));
3125:     *AAT = NULL;
3126:   }
3127:   PetscFunctionReturn(PETSC_SUCCESS);
3128: }

3130: /*@C
3131:   PetscGaussLobattoLegendreElementAdvectionCreate - computes the advection operator for a single 1d GLL element

3133:   Not Collective

3135:   Input Parameters:
3136: + n       - the number of GLL nodes
3137: . nodes   - the GLL nodes
3138: - weights - the GLL weightss

3140:   Output Parameter:
3141: . AA - the stiffness element

3143:   Level: beginner

3145:   Notes:
3146:   Destroy this with `PetscGaussLobattoLegendreElementAdvectionDestroy()`

3148:   This is the same as the Gradient operator multiplied by the diagonal mass matrix

3150:   You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented

3152: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`, `PetscGaussLobattoLegendreElementAdvectionDestroy()`
3153: @*/
3154: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3155: {
3156:   PetscReal      **D;
3157:   const PetscReal *gllweights = weights;
3158:   const PetscInt   glln       = n;
3159:   PetscInt         i, j;

3161:   PetscFunctionBegin;
3162:   PetscCall(PetscGaussLobattoLegendreElementGradientCreate(n, nodes, weights, &D, NULL));
3163:   for (i = 0; i < glln; i++) {
3164:     for (j = 0; j < glln; j++) D[i][j] = gllweights[i] * D[i][j];
3165:   }
3166:   *AA = D;
3167:   PetscFunctionReturn(PETSC_SUCCESS);
3168: }

3170: /*@C
3171:   PetscGaussLobattoLegendreElementAdvectionDestroy - frees the advection stiffness for a single 1d GLL element created with `PetscGaussLobattoLegendreElementAdvectionCreate()`

3173:   Not Collective

3175:   Input Parameters:
3176: + n       - the number of GLL nodes
3177: . nodes   - the GLL nodes
3178: . weights - the GLL weights
3179: - AA      - advection

3181:   Level: beginner

3183: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementAdvectionCreate()`
3184: @*/
3185: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3186: {
3187:   PetscFunctionBegin;
3188:   PetscCall(PetscFree((*AA)[0]));
3189:   PetscCall(PetscFree(*AA));
3190:   *AA = NULL;
3191:   PetscFunctionReturn(PETSC_SUCCESS);
3192: }

3194: PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3195: {
3196:   PetscReal      **A;
3197:   const PetscReal *gllweights = weights;
3198:   const PetscInt   glln       = n;
3199:   PetscInt         i, j;

3201:   PetscFunctionBegin;
3202:   PetscCall(PetscMalloc1(glln, &A));
3203:   PetscCall(PetscMalloc1(glln * glln, &A[0]));
3204:   for (i = 1; i < glln; i++) A[i] = A[i - 1] + glln;
3205:   if (glln == 1) A[0][0] = 0.;
3206:   for (i = 0; i < glln; i++) {
3207:     for (j = 0; j < glln; j++) {
3208:       A[i][j] = 0.;
3209:       if (j == i) A[i][j] = gllweights[i];
3210:     }
3211:   }
3212:   *AA = A;
3213:   PetscFunctionReturn(PETSC_SUCCESS);
3214: }

3216: PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3217: {
3218:   PetscFunctionBegin;
3219:   PetscCall(PetscFree((*AA)[0]));
3220:   PetscCall(PetscFree(*AA));
3221:   *AA = NULL;
3222:   PetscFunctionReturn(PETSC_SUCCESS);
3223: }

3225: /*@
3226:   PetscDTIndexToBary - convert an index into a barycentric coordinate.

3228:   Input Parameters:
3229: + 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)
3230: . sum   - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
3231: - index - the index to convert: should be >= 0 and < Binomial(len - 1 + sum, sum)

3233:   Output Parameter:
3234: . coord - will be filled with the barycentric coordinate

3236:   Level: beginner

3238:   Note:
3239:   The indices map to barycentric coordinates in lexicographic order, where the first index is the
3240:   least significant and the last index is the most significant.

3242: .seealso: `PetscDTBaryToIndex()`
3243: @*/
3244: PetscErrorCode PetscDTIndexToBary(PetscInt len, PetscInt sum, PetscInt index, PetscInt coord[])
3245: {
3246:   PetscInt c, d, s, total, subtotal, nexttotal;

3248:   PetscFunctionBeginHot;
3249:   PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
3250:   PetscCheck(index >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
3251:   if (!len) {
3252:     if (!sum && !index) PetscFunctionReturn(PETSC_SUCCESS);
3253:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
3254:   }
3255:   for (c = 1, total = 1; c <= len; c++) {
3256:     /* total is the number of ways to have a tuple of length c with sum */
3257:     if (index < total) break;
3258:     total = (total * (sum + c)) / c;
3259:   }
3260:   PetscCheck(c <= len, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index out of range");
3261:   for (d = c; d < len; d++) coord[d] = 0;
3262:   for (s = 0, subtotal = 1, nexttotal = 1; c > 0;) {
3263:     /* subtotal is the number of ways to have a tuple of length c with sum s */
3264:     /* nexttotal is the number of ways to have a tuple of length c-1 with sum s */
3265:     if ((index + subtotal) >= total) {
3266:       coord[--c] = sum - s;
3267:       index -= (total - subtotal);
3268:       sum       = s;
3269:       total     = nexttotal;
3270:       subtotal  = 1;
3271:       nexttotal = 1;
3272:       s         = 0;
3273:     } else {
3274:       subtotal  = (subtotal * (c + s)) / (s + 1);
3275:       nexttotal = (nexttotal * (c - 1 + s)) / (s + 1);
3276:       s++;
3277:     }
3278:   }
3279:   PetscFunctionReturn(PETSC_SUCCESS);
3280: }

3282: /*@
3283:   PetscDTBaryToIndex - convert a barycentric coordinate to an index

3285:   Input Parameters:
3286: + 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)
3287: . sum   - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
3288: - coord - a barycentric coordinate with the given length and sum

3290:   Output Parameter:
3291: . index - the unique index for the coordinate, >= 0 and < Binomial(len - 1 + sum, sum)

3293:   Level: beginner

3295:   Note:
3296:   The indices map to barycentric coordinates in lexicographic order, where the first index is the
3297:   least significant and the last index is the most significant.

3299: .seealso: `PetscDTIndexToBary`
3300: @*/
3301: PetscErrorCode PetscDTBaryToIndex(PetscInt len, PetscInt sum, const PetscInt coord[], PetscInt *index)
3302: {
3303:   PetscInt c;
3304:   PetscInt i;
3305:   PetscInt total;

3307:   PetscFunctionBeginHot;
3308:   PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
3309:   if (!len) {
3310:     if (!sum) {
3311:       *index = 0;
3312:       PetscFunctionReturn(PETSC_SUCCESS);
3313:     }
3314:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
3315:   }
3316:   for (c = 1, total = 1; c < len; c++) total = (total * (sum + c)) / c;
3317:   i = total - 1;
3318:   c = len - 1;
3319:   sum -= coord[c];
3320:   while (sum > 0) {
3321:     PetscInt subtotal;
3322:     PetscInt s;

3324:     for (s = 1, subtotal = 1; s < sum; s++) subtotal = (subtotal * (c + s)) / s;
3325:     i -= subtotal;
3326:     sum -= coord[--c];
3327:   }
3328:   *index = i;
3329:   PetscFunctionReturn(PETSC_SUCCESS);
3330: }

3332: /*@
3333:   PetscQuadratureComputePermutations - Compute permutations of quadrature points corresponding to domain orientations

3335:   Input Parameter:
3336: . quad - The `PetscQuadrature`

3338:   Output Parameters:
3339: + Np   - The number of domain orientations
3340: - perm - An array of `IS` permutations, one for ech orientation,

3342:   Level: developer

3344: .seealso: `PetscQuadratureSetCellType()`, `PetscQuadrature`
3345: @*/
3346: PetscErrorCode PetscQuadratureComputePermutations(PetscQuadrature quad, PetscInt *Np, IS *perm[])
3347: {
3348:   DMPolytopeType   ct;
3349:   const PetscReal *xq, *wq;
3350:   PetscInt         dim, qdim, d, Na, o, Nq, q, qp;

3352:   PetscFunctionBegin;
3353:   PetscCall(PetscQuadratureGetData(quad, &qdim, NULL, &Nq, &xq, &wq));
3354:   PetscCall(PetscQuadratureGetCellType(quad, &ct));
3355:   dim = DMPolytopeTypeGetDim(ct);
3356:   Na  = DMPolytopeTypeGetNumArrangments(ct);
3357:   PetscCall(PetscMalloc1(Na, perm));
3358:   if (Np) *Np = Na;
3359:   Na /= 2;
3360:   for (o = -Na; o < Na; ++o) {
3361:     DM        refdm;
3362:     PetscInt *idx;
3363:     PetscReal xi0[3] = {-1., -1., -1.}, v0[3], J[9], detJ, txq[3];
3364:     PetscBool flg;

3366:     PetscCall(DMPlexCreateReferenceCell(PETSC_COMM_SELF, ct, &refdm));
3367:     PetscCall(DMPlexOrientPoint(refdm, 0, o));
3368:     PetscCall(DMPlexComputeCellGeometryFEM(refdm, 0, NULL, v0, J, NULL, &detJ));
3369:     PetscCall(PetscMalloc1(Nq, &idx));
3370:     for (q = 0; q < Nq; ++q) {
3371:       CoordinatesRefToReal(dim, dim, xi0, v0, J, &xq[q * dim], txq);
3372:       for (qp = 0; qp < Nq; ++qp) {
3373:         PetscReal diff = 0.;

3375:         for (d = 0; d < dim; ++d) diff += PetscAbsReal(txq[d] - xq[qp * dim + d]);
3376:         if (diff < PETSC_SMALL) break;
3377:       }
3378:       PetscCheck(qp < Nq, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Transformed quad point %" PetscInt_FMT " does not match another quad point", q);
3379:       idx[q] = qp;
3380:     }
3381:     PetscCall(DMDestroy(&refdm));
3382:     PetscCall(ISCreateGeneral(PETSC_COMM_SELF, Nq, idx, PETSC_OWN_POINTER, &(*perm)[o + Na]));
3383:     PetscCall(ISGetInfo((*perm)[o + Na], IS_PERMUTATION, IS_LOCAL, PETSC_TRUE, &flg));
3384:     PetscCheck(flg, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Ordering for orientation %" PetscInt_FMT " was not a permutation", o);
3385:     PetscCall(ISSetPermutation((*perm)[o + Na]));
3386:   }
3387:   if (!Na) (*perm)[0] = NULL;
3388:   PetscFunctionReturn(PETSC_SUCCESS);
3389: }

3391: /*@
3392:   PetscDTCreateDefaultQuadrature - Create default quadrature for a given cell

3394:   Not collective

3396:   Input Parameters:
3397: + ct     - The integration domain
3398: - qorder - The desired quadrature order

3400:   Output Parameters:
3401: + q  - The cell quadrature
3402: - fq - The face quadrature

3404:   Level: developer

3406: .seealso: `PetscFECreateDefault()`, `PetscDTGaussTensorQuadrature()`, `PetscDTSimplexQuadrature()`, `PetscDTTensorQuadratureCreate()`
3407: @*/
3408: PetscErrorCode PetscDTCreateDefaultQuadrature(DMPolytopeType ct, PetscInt qorder, PetscQuadrature *q, PetscQuadrature *fq)
3409: {
3410:   const PetscInt quadPointsPerEdge = PetscMax(qorder + 1, 1);
3411:   const PetscInt dim               = DMPolytopeTypeGetDim(ct);

3413:   PetscFunctionBegin;
3414:   switch (ct) {
3415:   case DM_POLYTOPE_SEGMENT:
3416:   case DM_POLYTOPE_POINT_PRISM_TENSOR:
3417:   case DM_POLYTOPE_QUADRILATERAL:
3418:   case DM_POLYTOPE_SEG_PRISM_TENSOR:
3419:   case DM_POLYTOPE_HEXAHEDRON:
3420:   case DM_POLYTOPE_QUAD_PRISM_TENSOR:
3421:     PetscCall(PetscDTGaussTensorQuadrature(dim, 1, quadPointsPerEdge, -1.0, 1.0, q));
3422:     PetscCall(PetscDTGaussTensorQuadrature(dim - 1, 1, quadPointsPerEdge, -1.0, 1.0, fq));
3423:     break;
3424:   case DM_POLYTOPE_TRIANGLE:
3425:   case DM_POLYTOPE_TETRAHEDRON:
3426:     PetscCall(PetscDTSimplexQuadrature(dim, 2 * qorder, PETSCDTSIMPLEXQUAD_DEFAULT, q));
3427:     PetscCall(PetscDTSimplexQuadrature(dim - 1, 2 * qorder, PETSCDTSIMPLEXQUAD_DEFAULT, fq));
3428:     break;
3429:   case DM_POLYTOPE_TRI_PRISM:
3430:   case DM_POLYTOPE_TRI_PRISM_TENSOR: {
3431:     PetscQuadrature q1, q2;

3433:     // TODO: this should be able to use symmetric rules, but doing so causes tests to fail
3434:     PetscCall(PetscDTSimplexQuadrature(2, 2 * qorder, PETSCDTSIMPLEXQUAD_CONIC, &q1));
3435:     PetscCall(PetscDTGaussTensorQuadrature(1, 1, quadPointsPerEdge, -1.0, 1.0, &q2));
3436:     PetscCall(PetscDTTensorQuadratureCreate(q1, q2, q));
3437:     PetscCall(PetscQuadratureDestroy(&q2));
3438:     *fq = q1;
3439:     /* TODO Need separate quadratures for each face */
3440:   } break;
3441:   default:
3442:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "No quadrature for celltype %s", DMPolytopeTypes[PetscMin(ct, DM_POLYTOPE_UNKNOWN)]);
3443:   }
3444:   PetscFunctionReturn(PETSC_SUCCESS);
3445: }