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 - Return 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 Notes:
300:   From Fortran you must 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), it is assumed that they represent multiple k-forms) [see `PetscDTAltVPullback()` for interpretation of formDegree]

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

482:   Level: intermediate

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

487: .seealso: `PetscQuadrature`, `PetscDTAltVPullback()`, `PetscDTAltVPullbackMatrix()`
488: @*/
489: PetscErrorCode PetscQuadraturePushForward(PetscQuadrature q, PetscInt imageDim, const PetscReal origin[], const PetscReal originImage[], const PetscReal J[], PetscInt formDegree, PetscQuadrature *Jinvstarq)
490: {
491:   PetscInt         dim, Nc, imageNc, formSize, Ncopies, imageFormSize, Npoints, pt, i, j, c;
492:   const PetscReal *points;
493:   const PetscReal *weights;
494:   PetscReal       *imagePoints, *imageWeights;
495:   PetscReal       *Jinv;
496:   PetscReal       *Jinvstar;

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

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

519:       for (j = 0; j < dim; j++) val += J[i * dim + j] * (point[j] - origin[j]);
520:       imagePoint[i] = val;
521:     }
522:     for (c = 0; c < Ncopies; c++) {
523:       const PetscReal *form      = &weights[pt * Nc + c * formSize];
524:       PetscReal       *imageForm = &imageWeights[pt * imageNc + c * imageFormSize];

526:       for (i = 0; i < imageFormSize; i++) {
527:         PetscReal val = 0.;

529:         for (j = 0; j < formSize; j++) val += Jinvstar[i * formSize + j] * form[j];
530:         imageForm[i] = val;
531:       }
532:     }
533:   }
534:   PetscCall(PetscQuadratureCreate(PetscObjectComm((PetscObject)q), Jinvstarq));
535:   PetscCall(PetscQuadratureSetData(*Jinvstarq, imageDim, imageNc, Npoints, imagePoints, imageWeights));
536:   PetscCall(PetscFree2(Jinv, Jinvstar));
537:   PetscFunctionReturn(PETSC_SUCCESS);
538: }

540: /*@C
541:   PetscQuadratureSetData - Sets the data defining the quadrature

543:   Not Collective

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

553:   Level: intermediate

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

558: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureGetData()`
559: @*/
560: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt Nc, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
561: {
562:   PetscFunctionBegin;
564:   if (dim >= 0) q->dim = dim;
565:   if (Nc >= 0) q->Nc = Nc;
566:   if (npoints >= 0) q->numPoints = npoints;
567:   if (points) {
568:     PetscAssertPointer(points, 5);
569:     q->points = points;
570:   }
571:   if (weights) {
572:     PetscAssertPointer(weights, 6);
573:     q->weights = weights;
574:   }
575:   PetscFunctionReturn(PETSC_SUCCESS);
576: }

578: static PetscErrorCode PetscQuadratureView_Ascii(PetscQuadrature quad, PetscViewer v)
579: {
580:   PetscInt          q, d, c;
581:   PetscViewerFormat format;

583:   PetscFunctionBegin;
584:   if (quad->Nc > 1)
585:     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));
586:   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));
587:   PetscCall(PetscViewerGetFormat(v, &format));
588:   if (format != PETSC_VIEWER_ASCII_INFO_DETAIL) PetscFunctionReturn(PETSC_SUCCESS);
589:   for (q = 0; q < quad->numPoints; ++q) {
590:     PetscCall(PetscViewerASCIIPrintf(v, "p%" PetscInt_FMT " (", q));
591:     PetscCall(PetscViewerASCIIUseTabs(v, PETSC_FALSE));
592:     for (d = 0; d < quad->dim; ++d) {
593:       if (d) PetscCall(PetscViewerASCIIPrintf(v, ", "));
594:       PetscCall(PetscViewerASCIIPrintf(v, "%+g", (double)quad->points[q * quad->dim + d]));
595:     }
596:     PetscCall(PetscViewerASCIIPrintf(v, ") "));
597:     if (quad->Nc > 1) PetscCall(PetscViewerASCIIPrintf(v, "w%" PetscInt_FMT " (", q));
598:     for (c = 0; c < quad->Nc; ++c) {
599:       if (c) PetscCall(PetscViewerASCIIPrintf(v, ", "));
600:       PetscCall(PetscViewerASCIIPrintf(v, "%+g", (double)quad->weights[q * quad->Nc + c]));
601:     }
602:     if (quad->Nc > 1) PetscCall(PetscViewerASCIIPrintf(v, ")"));
603:     PetscCall(PetscViewerASCIIPrintf(v, "\n"));
604:     PetscCall(PetscViewerASCIIUseTabs(v, PETSC_TRUE));
605:   }
606:   PetscFunctionReturn(PETSC_SUCCESS);
607: }

609: /*@C
610:   PetscQuadratureView - View a `PetscQuadrature` object

612:   Collective

614:   Input Parameters:
615: + quad   - The `PetscQuadrature` object
616: - viewer - The `PetscViewer` object

618:   Level: beginner

620: .seealso: `PetscQuadrature`, `PetscViewer`, `PetscQuadratureCreate()`, `PetscQuadratureGetData()`
621: @*/
622: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
623: {
624:   PetscBool iascii;

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

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

640:   Not Collective; No Fortran Support

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

648:   Output Parameter:
649: . qref - The dimension

651:   Level: intermediate

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

656: .seealso: `PetscQuadrature`, `PetscFECreate()`, `PetscSpaceGetDimension()`, `PetscDualSpaceGetDimension()`
657: @*/
658: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
659: {
660:   DMPolytopeType   ct;
661:   const PetscReal *points, *weights;
662:   PetscReal       *pointsRef, *weightsRef;
663:   PetscInt         dim, Nc, order, npoints, npointsRef, c, p, cp, d, e;

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

693: /* Compute the coefficients for the Jacobi polynomial recurrence,
694:  *
695:  * J^{a,b}_n(x) = (cnm1 + cnm1x * x) * J^{a,b}_{n-1}(x) - cnm2 * J^{a,b}_{n-2}(x).
696:  */
697: #define PetscDTJacobiRecurrence_Internal(n, a, b, cnm1, cnm1x, cnm2) \
698:   do { \
699:     PetscReal _a = (a); \
700:     PetscReal _b = (b); \
701:     PetscReal _n = (n); \
702:     if (n == 1) { \
703:       (cnm1)  = (_a - _b) * 0.5; \
704:       (cnm1x) = (_a + _b + 2.) * 0.5; \
705:       (cnm2)  = 0.; \
706:     } else { \
707:       PetscReal _2n  = _n + _n; \
708:       PetscReal _d   = (_2n * (_n + _a + _b) * (_2n + _a + _b - 2)); \
709:       PetscReal _n1  = (_2n + _a + _b - 1.) * (_a * _a - _b * _b); \
710:       PetscReal _n1x = (_2n + _a + _b - 1.) * (_2n + _a + _b) * (_2n + _a + _b - 2); \
711:       PetscReal _n2  = 2. * ((_n + _a - 1.) * (_n + _b - 1.) * (_2n + _a + _b)); \
712:       (cnm1)         = _n1 / _d; \
713:       (cnm1x)        = _n1x / _d; \
714:       (cnm2)         = _n2 / _d; \
715:     } \
716:   } while (0)

718: /*@
719:   PetscDTJacobiNorm - Compute the weighted L2 norm of a Jacobi polynomial.

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

723:   Input Parameters:
724: + alpha - the left exponent > -1
725: . beta  - the right exponent > -1
726: - n     - the polynomial degree

728:   Output Parameter:
729: . norm - the weighted L2 norm

731:   Level: beginner

733: .seealso: `PetscQuadrature`, `PetscDTJacobiEval()`
734: @*/
735: PetscErrorCode PetscDTJacobiNorm(PetscReal alpha, PetscReal beta, PetscInt n, PetscReal *norm)
736: {
737:   PetscReal twoab1;
738:   PetscReal gr;

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

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

779: static PetscErrorCode PetscDTJacobiEval_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscInt k, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *p)
780: {
781:   PetscReal ak, bk;
782:   PetscReal abk1;
783:   PetscInt  i, l, maxdegree;

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

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

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

834: /*@
835:   PetscDTJacobiEvalJet - Evaluate the jet (function and derivatives) of the Jacobi polynomials basis up to a given degree.

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

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

851:   Level: advanced

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

858: .seealso: `PetscDTJacobiEval()`, `PetscDTPKDEvalJet()`
859: @*/
860: PetscErrorCode PetscDTJacobiEvalJet(PetscReal alpha, PetscReal beta, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
861: {
862:   PetscInt   i, j, l;
863:   PetscInt  *degrees;
864:   PetscReal *psingle;

866:   PetscFunctionBegin;
867:   if (degree == 0) {
868:     PetscInt zero = 0;

870:     for (i = 0; i <= k; i++) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, 1, &zero, &p[i * npoints]));
871:     PetscFunctionReturn(PETSC_SUCCESS);
872:   }
873:   PetscCall(PetscMalloc1(degree + 1, &degrees));
874:   PetscCall(PetscMalloc1((degree + 1) * npoints, &psingle));
875:   for (i = 0; i <= degree; i++) degrees[i] = i;
876:   for (i = 0; i <= k; i++) {
877:     PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, degree + 1, degrees, psingle));
878:     for (j = 0; j <= degree; j++) {
879:       for (l = 0; l < npoints; l++) p[(j * (k + 1) + i) * npoints + l] = psingle[l * (degree + 1) + j];
880:     }
881:   }
882:   PetscCall(PetscFree(psingle));
883:   PetscCall(PetscFree(degrees));
884:   PetscFunctionReturn(PETSC_SUCCESS);
885: }

887: /*@
888:   PetscDTJacobiEval - evaluate Jacobi polynomials for the weight function $(1.+x)^{\alpha} (1.-x)^{\beta}$ at a set of points
889:   at points

891:   Not Collective

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

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

906:   Level: intermediate

908: .seealso: `PetscDTGaussQuadrature()`, `PetscDTLegendreEval()`
909: @*/
910: PetscErrorCode PetscDTJacobiEval(PetscInt npoints, PetscReal alpha, PetscReal beta, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *B, PetscReal *D, PetscReal *D2)
911: {
912:   PetscFunctionBegin;
913:   PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
914:   PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");
915:   if (!npoints || !ndegree) PetscFunctionReturn(PETSC_SUCCESS);
916:   if (B) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 0, points, ndegree, degrees, B));
917:   if (D) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 1, points, ndegree, degrees, D));
918:   if (D2) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 2, points, ndegree, degrees, D2));
919:   PetscFunctionReturn(PETSC_SUCCESS);
920: }

922: /*@
923:   PetscDTLegendreEval - evaluate Legendre polynomials at points

925:   Not Collective

927:   Input Parameters:
928: + npoints - number of spatial points to evaluate at
929: . points  - array of locations to evaluate at
930: . ndegree - number of basis degrees to evaluate
931: - degrees - sorted array of degrees to evaluate

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

938:   Level: intermediate

940: .seealso: `PetscDTGaussQuadrature()`
941: @*/
942: PetscErrorCode PetscDTLegendreEval(PetscInt npoints, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *B, PetscReal *D, PetscReal *D2)
943: {
944:   PetscFunctionBegin;
945:   PetscCall(PetscDTJacobiEval(npoints, 0., 0., points, ndegree, degrees, B, D, D2));
946:   PetscFunctionReturn(PETSC_SUCCESS);
947: }

949: /*@
950:   PetscDTIndexToGradedOrder - convert an index into a tuple of monomial degrees in a graded order (that is, if the degree sum of tuple x is less than the degree sum of tuple y, then the index of x is smaller than the index of y)

952:   Input Parameters:
953: + len   - the desired length of the degree tuple
954: - index - the index to convert: should be >= 0

956:   Output Parameter:
957: . degtup - will be filled with a tuple of degrees

959:   Level: beginner

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

966: .seealso: `PetscDTGradedOrderToIndex()`
967: @*/
968: PetscErrorCode PetscDTIndexToGradedOrder(PetscInt len, PetscInt index, PetscInt degtup[])
969: {
970:   PetscInt i, total;
971:   PetscInt sum;

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

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

999: /*@
1000:   PetscDTGradedOrderToIndex - convert a tuple into an index in a graded order, the inverse of `PetscDTIndexToGradedOrder()`.

1002:   Input Parameters:
1003: + len    - the length of the degree tuple
1004: - degtup - tuple with this length

1006:   Output Parameter:
1007: . index - index in graded order: >= 0

1009:   Level: beginner

1011:   Note:
1012:   For two tuples x and y with the same degree sum, partial degree sums over the final elements of the tuples
1013:   acts as a tiebreaker.  For example, (2, 1, 1) and (1, 2, 1) have the same degree sum, but the degree sum over the
1014:   last two elements is smaller for the former, so (2, 1, 1) < (1, 2, 1).

1016: .seealso: `PetscDTIndexToGradedOrder()`
1017: @*/
1018: PetscErrorCode PetscDTGradedOrderToIndex(PetscInt len, const PetscInt degtup[], PetscInt *index)
1019: {
1020:   PetscInt i, idx, sum, total;

1022:   PetscFunctionBeginHot;
1023:   PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
1024:   for (i = 0, sum = 0; i < len; i++) sum += degtup[i];
1025:   idx   = 0;
1026:   total = 1;
1027:   for (i = 0; i < sum; i++) {
1028:     idx += total;
1029:     total = (total * (len + i)) / (i + 1);
1030:   }
1031:   for (i = 0; i < len - 1; i++) {
1032:     PetscInt c;

1034:     total = 1;
1035:     sum -= degtup[i];
1036:     for (c = 0; c < sum; c++) {
1037:       idx += total;
1038:       total = (total * (len - 1 - i + c)) / (c + 1);
1039:     }
1040:   }
1041:   *index = idx;
1042:   PetscFunctionReturn(PETSC_SUCCESS);
1043: }

1045: static PetscBool PKDCite       = PETSC_FALSE;
1046: const char       PKDCitation[] = "@article{Kirby2010,\n"
1047:                                  "  title={Singularity-free evaluation of collapsed-coordinate orthogonal polynomials},\n"
1048:                                  "  author={Kirby, Robert C},\n"
1049:                                  "  journal={ACM Transactions on Mathematical Software (TOMS)},\n"
1050:                                  "  volume={37},\n"
1051:                                  "  number={1},\n"
1052:                                  "  pages={1--16},\n"
1053:                                  "  year={2010},\n"
1054:                                  "  publisher={ACM New York, NY, USA}\n}\n";

1056: /*@
1057:   PetscDTPKDEvalJet - Evaluate the jet (function and derivatives) of the Proriol-Koornwinder-Dubiner (PKD) basis for
1058:   the space of polynomials up to a given degree.

1060:   Input Parameters:
1061: + dim     - the number of variables in the multivariate polynomials
1062: . npoints - the number of points to evaluate the polynomials at
1063: . points  - [npoints x dim] array of point coordinates
1064: . 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.
1065: - k       - the maximum order partial derivative to evaluate in the jet.  There are (dim + k choose dim) partial derivatives
1066:   in the jet.  Choosing k = 0 means to evaluate just the function and no derivatives

1068:   Output Parameter:
1069: . p - an array containing the evaluations of the PKD polynomials' jets on the points.  The size is ((dim + degree)
1070:   choose dim) x ((dim + k) choose dim) x npoints, which also describes the order of the dimensions of this
1071:   three-dimensional array: the first (slowest varying) dimension is basis function index; the second dimension is jet
1072:   index; the third (fastest varying) dimension is the index of the evaluation point.

1074:   Level: advanced

1076:   Notes:
1077:   The PKD basis is L2-orthonormal on the biunit simplex (which is used as the reference element
1078:   for finite elements in PETSc), which makes it a stable basis to use for evaluating
1079:   polynomials in that domain.

1081:   The ordering of the basis functions, and the ordering of the derivatives in the jet, both follow the graded
1082:   ordering of `PetscDTIndexToGradedOrder()` and `PetscDTGradedOrderToIndex()`.  For example, in 3D, the polynomial with
1083:   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);
1084:   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).

1086:   The implementation uses Kirby's singularity-free evaluation algorithm, https://doi.org/10.1145/1644001.1644006.

1088: .seealso: `PetscDTGradedOrderToIndex()`, `PetscDTIndexToGradedOrder()`, `PetscDTJacobiEvalJet()`
1089: @*/
1090: PetscErrorCode PetscDTPKDEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
1091: {
1092:   PetscInt   degidx, kidx, d, pt;
1093:   PetscInt   Nk, Ndeg;
1094:   PetscInt  *ktup, *degtup;
1095:   PetscReal *scales, initscale, scaleexp;

1097:   PetscFunctionBegin;
1098:   PetscCall(PetscCitationsRegister(PKDCitation, &PKDCite));
1099:   PetscCall(PetscDTBinomialInt(dim + k, k, &Nk));
1100:   PetscCall(PetscDTBinomialInt(degree + dim, degree, &Ndeg));
1101:   PetscCall(PetscMalloc2(dim, &degtup, dim, &ktup));
1102:   PetscCall(PetscMalloc1(Ndeg, &scales));
1103:   initscale = 1.;
1104:   if (dim > 1) {
1105:     PetscCall(PetscDTBinomial(dim, 2, &scaleexp));
1106:     initscale = PetscPowReal(2., scaleexp * 0.5);
1107:   }
1108:   for (degidx = 0; degidx < Ndeg; degidx++) {
1109:     PetscInt  e, i;
1110:     PetscInt  m1idx = -1, m2idx = -1;
1111:     PetscInt  n;
1112:     PetscInt  degsum;
1113:     PetscReal alpha;
1114:     PetscReal cnm1, cnm1x, cnm2;
1115:     PetscReal norm;

1117:     PetscCall(PetscDTIndexToGradedOrder(dim, degidx, degtup));
1118:     for (d = dim - 1; d >= 0; d--)
1119:       if (degtup[d]) break;
1120:     if (d < 0) { /* constant is 1 everywhere, all derivatives are zero */
1121:       scales[degidx] = initscale;
1122:       for (e = 0; e < dim; e++) {
1123:         PetscCall(PetscDTJacobiNorm(e, 0., 0, &norm));
1124:         scales[degidx] /= norm;
1125:       }
1126:       for (i = 0; i < npoints; i++) p[degidx * Nk * npoints + i] = 1.;
1127:       for (i = 0; i < (Nk - 1) * npoints; i++) p[(degidx * Nk + 1) * npoints + i] = 0.;
1128:       continue;
1129:     }
1130:     n = degtup[d];
1131:     degtup[d]--;
1132:     PetscCall(PetscDTGradedOrderToIndex(dim, degtup, &m1idx));
1133:     if (degtup[d] > 0) {
1134:       degtup[d]--;
1135:       PetscCall(PetscDTGradedOrderToIndex(dim, degtup, &m2idx));
1136:       degtup[d]++;
1137:     }
1138:     degtup[d]++;
1139:     for (e = 0, degsum = 0; e < d; e++) degsum += degtup[e];
1140:     alpha = 2 * degsum + d;
1141:     PetscDTJacobiRecurrence_Internal(n, alpha, 0., cnm1, cnm1x, cnm2);

1143:     scales[degidx] = initscale;
1144:     for (e = 0, degsum = 0; e < dim; e++) {
1145:       PetscInt  f;
1146:       PetscReal ealpha;
1147:       PetscReal enorm;

1149:       ealpha = 2 * degsum + e;
1150:       for (f = 0; f < degsum; f++) scales[degidx] *= 2.;
1151:       PetscCall(PetscDTJacobiNorm(ealpha, 0., degtup[e], &enorm));
1152:       scales[degidx] /= enorm;
1153:       degsum += degtup[e];
1154:     }

1156:     for (pt = 0; pt < npoints; pt++) {
1157:       /* compute the multipliers */
1158:       PetscReal thetanm1, thetanm1x, thetanm2;

1160:       thetanm1x = dim - (d + 1) + 2. * points[pt * dim + d];
1161:       for (e = d + 1; e < dim; e++) thetanm1x += points[pt * dim + e];
1162:       thetanm1x *= 0.5;
1163:       thetanm1 = (2. - (dim - (d + 1)));
1164:       for (e = d + 1; e < dim; e++) thetanm1 -= points[pt * dim + e];
1165:       thetanm1 *= 0.5;
1166:       thetanm2 = thetanm1 * thetanm1;

1168:       for (kidx = 0; kidx < Nk; kidx++) {
1169:         PetscInt f;

1171:         PetscCall(PetscDTIndexToGradedOrder(dim, kidx, ktup));
1172:         /* first sum in the same derivative terms */
1173:         p[(degidx * Nk + kidx) * npoints + pt] = (cnm1 * thetanm1 + cnm1x * thetanm1x) * p[(m1idx * Nk + kidx) * npoints + pt];
1174:         if (m2idx >= 0) p[(degidx * Nk + kidx) * npoints + pt] -= cnm2 * thetanm2 * p[(m2idx * Nk + kidx) * npoints + pt];

1176:         for (f = d; f < dim; f++) {
1177:           PetscInt km1idx, mplty = ktup[f];

1179:           if (!mplty) continue;
1180:           ktup[f]--;
1181:           PetscCall(PetscDTGradedOrderToIndex(dim, ktup, &km1idx));

1183:           /* the derivative of  cnm1x * thetanm1x  wrt x variable f is 0.5 * cnm1x if f > d otherwise it is cnm1x */
1184:           /* the derivative of  cnm1  * thetanm1   wrt x variable f is 0 if f == d, otherwise it is -0.5 * cnm1 */
1185:           /* the derivative of -cnm2  * thetanm2   wrt x variable f is 0 if f == d, otherwise it is cnm2 * thetanm1 */
1186:           if (f > d) {
1187:             PetscInt f2;

1189:             p[(degidx * Nk + kidx) * npoints + pt] += mplty * 0.5 * (cnm1x - cnm1) * p[(m1idx * Nk + km1idx) * npoints + pt];
1190:             if (m2idx >= 0) {
1191:               p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm2 * thetanm1 * p[(m2idx * Nk + km1idx) * npoints + pt];
1192:               /* second derivatives of -cnm2  * thetanm2   wrt x variable f,f2 is like - 0.5 * cnm2 */
1193:               for (f2 = f; f2 < dim; f2++) {
1194:                 PetscInt km2idx, mplty2 = ktup[f2];
1195:                 PetscInt factor;

1197:                 if (!mplty2) continue;
1198:                 ktup[f2]--;
1199:                 PetscCall(PetscDTGradedOrderToIndex(dim, ktup, &km2idx));

1201:                 factor = mplty * mplty2;
1202:                 if (f == f2) factor /= 2;
1203:                 p[(degidx * Nk + kidx) * npoints + pt] -= 0.5 * factor * cnm2 * p[(m2idx * Nk + km2idx) * npoints + pt];
1204:                 ktup[f2]++;
1205:               }
1206:             }
1207:           } else {
1208:             p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm1x * p[(m1idx * Nk + km1idx) * npoints + pt];
1209:           }
1210:           ktup[f]++;
1211:         }
1212:       }
1213:     }
1214:   }
1215:   for (degidx = 0; degidx < Ndeg; degidx++) {
1216:     PetscReal scale = scales[degidx];
1217:     PetscInt  i;

1219:     for (i = 0; i < Nk * npoints; i++) p[degidx * Nk * npoints + i] *= scale;
1220:   }
1221:   PetscCall(PetscFree(scales));
1222:   PetscCall(PetscFree2(degtup, ktup));
1223:   PetscFunctionReturn(PETSC_SUCCESS);
1224: }

1226: /*@
1227:   PetscDTPTrimmedSize - The size of the trimmed polynomial space of k-forms with a given degree and form degree,
1228:   which can be evaluated in `PetscDTPTrimmedEvalJet()`.

1230:   Input Parameters:
1231: + dim        - the number of variables in the multivariate polynomials
1232: . degree     - the degree (sum of degrees on the variables in a monomial) of the trimmed polynomial space.
1233: - formDegree - the degree of the form

1235:   Output Parameter:
1236: . size - The number ((`dim` + `degree`) choose (`dim` + `formDegree`)) x ((`degree` + `formDegree` - 1) choose (`formDegree`))

1238:   Level: advanced

1240: .seealso: `PetscDTPTrimmedEvalJet()`
1241: @*/
1242: PetscErrorCode PetscDTPTrimmedSize(PetscInt dim, PetscInt degree, PetscInt formDegree, PetscInt *size)
1243: {
1244:   PetscInt Nrk, Nbpt; // number of trimmed polynomials

1246:   PetscFunctionBegin;
1247:   formDegree = PetscAbsInt(formDegree);
1248:   PetscCall(PetscDTBinomialInt(degree + dim, degree + formDegree, &Nbpt));
1249:   PetscCall(PetscDTBinomialInt(degree + formDegree - 1, formDegree, &Nrk));
1250:   Nbpt *= Nrk;
1251:   *size = Nbpt;
1252:   PetscFunctionReturn(PETSC_SUCCESS);
1253: }

1255: /* there was a reference implementation based on section 4.4 of Arnold, Falk & Winther (acta numerica, 2006), but it
1256:  * was inferior to this implementation */
1257: static PetscErrorCode PetscDTPTrimmedEvalJet_Internal(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt formDegree, PetscInt jetDegree, PetscReal p[])
1258: {
1259:   PetscInt  formDegreeOrig = formDegree;
1260:   PetscBool formNegative   = (formDegreeOrig < 0) ? PETSC_TRUE : PETSC_FALSE;

1262:   PetscFunctionBegin;
1263:   formDegree = PetscAbsInt(formDegreeOrig);
1264:   if (formDegree == 0) {
1265:     PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree, jetDegree, p));
1266:     PetscFunctionReturn(PETSC_SUCCESS);
1267:   }
1268:   if (formDegree == dim) {
1269:     PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree - 1, jetDegree, p));
1270:     PetscFunctionReturn(PETSC_SUCCESS);
1271:   }
1272:   PetscInt Nbpt;
1273:   PetscCall(PetscDTPTrimmedSize(dim, degree, formDegree, &Nbpt));
1274:   PetscInt Nf;
1275:   PetscCall(PetscDTBinomialInt(dim, formDegree, &Nf));
1276:   PetscInt Nk;
1277:   PetscCall(PetscDTBinomialInt(dim + jetDegree, dim, &Nk));
1278:   PetscCall(PetscArrayzero(p, Nbpt * Nf * Nk * npoints));

1280:   PetscInt Nbpm1; // number of scalar polynomials up to degree - 1;
1281:   PetscCall(PetscDTBinomialInt(dim + degree - 1, dim, &Nbpm1));
1282:   PetscReal *p_scalar;
1283:   PetscCall(PetscMalloc1(Nbpm1 * Nk * npoints, &p_scalar));
1284:   PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree - 1, jetDegree, p_scalar));
1285:   PetscInt total = 0;
1286:   // First add the full polynomials up to degree - 1 into the basis: take the scalar
1287:   // and copy one for each form component
1288:   for (PetscInt i = 0; i < Nbpm1; i++) {
1289:     const PetscReal *src = &p_scalar[i * Nk * npoints];
1290:     for (PetscInt f = 0; f < Nf; f++) {
1291:       PetscReal *dest = &p[(total++ * Nf + f) * Nk * npoints];
1292:       PetscCall(PetscArraycpy(dest, src, Nk * npoints));
1293:     }
1294:   }
1295:   PetscInt *form_atoms;
1296:   PetscCall(PetscMalloc1(formDegree + 1, &form_atoms));
1297:   // construct the interior product pattern
1298:   PetscInt(*pattern)[3];
1299:   PetscInt Nf1; // number of formDegree + 1 forms
1300:   PetscCall(PetscDTBinomialInt(dim, formDegree + 1, &Nf1));
1301:   PetscInt nnz = Nf1 * (formDegree + 1);
1302:   PetscCall(PetscMalloc1(Nf1 * (formDegree + 1), &pattern));
1303:   PetscCall(PetscDTAltVInteriorPattern(dim, formDegree + 1, pattern));
1304:   PetscReal centroid = (1. - dim) / (dim + 1.);
1305:   PetscInt *deriv;
1306:   PetscCall(PetscMalloc1(dim, &deriv));
1307:   for (PetscInt d = dim; d >= formDegree + 1; d--) {
1308:     PetscInt Nfd1; // number of formDegree + 1 forms in dimension d that include dx_0
1309:                    // (equal to the number of formDegree forms in dimension d-1)
1310:     PetscCall(PetscDTBinomialInt(d - 1, formDegree, &Nfd1));
1311:     // The number of homogeneous (degree-1) scalar polynomials in d variables
1312:     PetscInt Nh;
1313:     PetscCall(PetscDTBinomialInt(d - 1 + degree - 1, d - 1, &Nh));
1314:     const PetscReal *h_scalar = &p_scalar[(Nbpm1 - Nh) * Nk * npoints];
1315:     for (PetscInt b = 0; b < Nh; b++) {
1316:       const PetscReal *h_s = &h_scalar[b * Nk * npoints];
1317:       for (PetscInt f = 0; f < Nfd1; f++) {
1318:         // construct all formDegree+1 forms that start with dx_(dim - d) /\ ...
1319:         form_atoms[0] = dim - d;
1320:         PetscCall(PetscDTEnumSubset(d - 1, formDegree, f, &form_atoms[1]));
1321:         for (PetscInt i = 0; i < formDegree; i++) form_atoms[1 + i] += form_atoms[0] + 1;
1322:         PetscInt f_ind; // index of the resulting form
1323:         PetscCall(PetscDTSubsetIndex(dim, formDegree + 1, form_atoms, &f_ind));
1324:         PetscReal *p_f = &p[total++ * Nf * Nk * npoints];
1325:         for (PetscInt nz = 0; nz < nnz; nz++) {
1326:           PetscInt  i     = pattern[nz][0]; // formDegree component
1327:           PetscInt  j     = pattern[nz][1]; // (formDegree + 1) component
1328:           PetscInt  v     = pattern[nz][2]; // coordinate component
1329:           PetscReal scale = v < 0 ? -1. : 1.;

1331:           i     = formNegative ? (Nf - 1 - i) : i;
1332:           scale = (formNegative && (i & 1)) ? -scale : scale;
1333:           v     = v < 0 ? -(v + 1) : v;
1334:           if (j != f_ind) continue;
1335:           PetscReal *p_i = &p_f[i * Nk * npoints];
1336:           for (PetscInt jet = 0; jet < Nk; jet++) {
1337:             const PetscReal *h_jet = &h_s[jet * npoints];
1338:             PetscReal       *p_jet = &p_i[jet * npoints];

1340:             for (PetscInt pt = 0; pt < npoints; pt++) p_jet[pt] += scale * h_jet[pt] * (points[pt * dim + v] - centroid);
1341:             PetscCall(PetscDTIndexToGradedOrder(dim, jet, deriv));
1342:             deriv[v]++;
1343:             PetscReal mult = deriv[v];
1344:             PetscInt  l;
1345:             PetscCall(PetscDTGradedOrderToIndex(dim, deriv, &l));
1346:             if (l >= Nk) continue;
1347:             p_jet = &p_i[l * npoints];
1348:             for (PetscInt pt = 0; pt < npoints; pt++) p_jet[pt] += scale * mult * h_jet[pt];
1349:             deriv[v]--;
1350:           }
1351:         }
1352:       }
1353:     }
1354:   }
1355:   PetscCheck(total == Nbpt, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Incorrectly counted P trimmed polynomials");
1356:   PetscCall(PetscFree(deriv));
1357:   PetscCall(PetscFree(pattern));
1358:   PetscCall(PetscFree(form_atoms));
1359:   PetscCall(PetscFree(p_scalar));
1360:   PetscFunctionReturn(PETSC_SUCCESS);
1361: }

1363: /*@
1364:   PetscDTPTrimmedEvalJet - Evaluate the jet (function and derivatives) of a basis of the trimmed polynomial k-forms up to
1365:   a given degree.

1367:   Input Parameters:
1368: + dim        - the number of variables in the multivariate polynomials
1369: . npoints    - the number of points to evaluate the polynomials at
1370: . points     - [npoints x dim] array of point coordinates
1371: . degree     - the degree (sum of degrees on the variables in a monomial) of the trimmed polynomial space to evaluate.
1372:            There are ((dim + degree) choose (dim + formDegree)) x ((degree + formDegree - 1) choose (formDegree)) polynomials in this space.
1373:            (You can use `PetscDTPTrimmedSize()` to compute this size.)
1374: . formDegree - the degree of the form
1375: - jetDegree  - the maximum order partial derivative to evaluate in the jet.  There are ((dim + jetDegree) choose dim) partial derivatives
1376:               in the jet.  Choosing jetDegree = 0 means to evaluate just the function and no derivatives

1378:   Output Parameter:
1379: . p - an array containing the evaluations of the PKD polynomials' jets on the points.

1381:   Level: advanced

1383:   Notes:
1384:   The size of `p` is `PetscDTPTrimmedSize()` x ((dim + formDegree) choose dim) x ((dim + k)
1385:   choose dim) x npoints,which also describes the order of the dimensions of this
1386:   four-dimensional array\:

1388:   the first (slowest varying) dimension is basis function index;
1389:   the second dimension is component of the form;
1390:   the third dimension is jet index;
1391:   the fourth (fastest varying) dimension is the index of the evaluation point.

1393:   The ordering of the basis functions is not graded, so the basis functions are not nested by degree like `PetscDTPKDEvalJet()`.
1394:   The basis functions are not an L2-orthonormal basis on any particular domain.

1396:   The implementation is based on the description of the trimmed polynomials up to degree r as
1397:   the direct sum of polynomials up to degree (r-1) and the Koszul differential applied to
1398:   homogeneous polynomials of degree (r-1).

1400: .seealso: `PetscDTPKDEvalJet()`, `PetscDTPTrimmedSize()`
1401: @*/
1402: PetscErrorCode PetscDTPTrimmedEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt formDegree, PetscInt jetDegree, PetscReal p[])
1403: {
1404:   PetscFunctionBegin;
1405:   PetscCall(PetscDTPTrimmedEvalJet_Internal(dim, npoints, points, degree, formDegree, jetDegree, p));
1406:   PetscFunctionReturn(PETSC_SUCCESS);
1407: }

1409: /* solve the symmetric tridiagonal eigenvalue system, writing the eigenvalues into eigs and the eigenvectors into V
1410:  * with lds n; diag and subdiag are overwritten */
1411: static PetscErrorCode PetscDTSymmetricTridiagonalEigensolve(PetscInt n, PetscReal diag[], PetscReal subdiag[], PetscReal eigs[], PetscScalar V[])
1412: {
1413:   char          jobz   = 'V'; /* eigenvalues and eigenvectors */
1414:   char          range  = 'A'; /* all eigenvalues will be found */
1415:   PetscReal     VL     = 0.;  /* ignored because range is 'A' */
1416:   PetscReal     VU     = 0.;  /* ignored because range is 'A' */
1417:   PetscBLASInt  IL     = 0;   /* ignored because range is 'A' */
1418:   PetscBLASInt  IU     = 0;   /* ignored because range is 'A' */
1419:   PetscReal     abstol = 0.;  /* unused */
1420:   PetscBLASInt  bn, bm, ldz;  /* bm will equal bn on exit */
1421:   PetscBLASInt *isuppz;
1422:   PetscBLASInt  lwork, liwork;
1423:   PetscReal     workquery;
1424:   PetscBLASInt  iworkquery;
1425:   PetscBLASInt *iwork;
1426:   PetscBLASInt  info;
1427:   PetscReal    *work = NULL;

1429:   PetscFunctionBegin;
1430: #if !defined(PETSCDTGAUSSIANQUADRATURE_EIG)
1431:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
1432: #endif
1433:   PetscCall(PetscBLASIntCast(n, &bn));
1434:   PetscCall(PetscBLASIntCast(n, &ldz));
1435: #if !defined(PETSC_MISSING_LAPACK_STEGR)
1436:   PetscCall(PetscMalloc1(2 * n, &isuppz));
1437:   lwork  = -1;
1438:   liwork = -1;
1439:   PetscCallBLAS("LAPACKstegr", LAPACKstegr_(&jobz, &range, &bn, diag, subdiag, &VL, &VU, &IL, &IU, &abstol, &bm, eigs, V, &ldz, isuppz, &workquery, &lwork, &iworkquery, &liwork, &info));
1440:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEGR error");
1441:   lwork  = (PetscBLASInt)workquery;
1442:   liwork = (PetscBLASInt)iworkquery;
1443:   PetscCall(PetscMalloc2(lwork, &work, liwork, &iwork));
1444:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
1445:   PetscCallBLAS("LAPACKstegr", LAPACKstegr_(&jobz, &range, &bn, diag, subdiag, &VL, &VU, &IL, &IU, &abstol, &bm, eigs, V, &ldz, isuppz, work, &lwork, iwork, &liwork, &info));
1446:   PetscCall(PetscFPTrapPop());
1447:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEGR error");
1448:   PetscCall(PetscFree2(work, iwork));
1449:   PetscCall(PetscFree(isuppz));
1450: #elif !defined(PETSC_MISSING_LAPACK_STEQR)
1451:   jobz = 'I'; /* Compute eigenvalues and eigenvectors of the
1452:                  tridiagonal matrix.  Z is initialized to the identity
1453:                  matrix. */
1454:   PetscCall(PetscMalloc1(PetscMax(1, 2 * n - 2), &work));
1455:   PetscCallBLAS("LAPACKsteqr", LAPACKsteqr_("I", &bn, diag, subdiag, V, &ldz, work, &info));
1456:   PetscCall(PetscFPTrapPop());
1457:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEQR error");
1458:   PetscCall(PetscFree(work));
1459:   PetscCall(PetscArraycpy(eigs, diag, n));
1460: #endif
1461:   PetscFunctionReturn(PETSC_SUCCESS);
1462: }

1464: /* Formula for the weights at the endpoints (-1 and 1) of Gauss-Lobatto-Jacobi
1465:  * quadrature rules on the interval [-1, 1] */
1466: static PetscErrorCode PetscDTGaussLobattoJacobiEndweights_Internal(PetscInt n, PetscReal alpha, PetscReal beta, PetscReal *leftw, PetscReal *rightw)
1467: {
1468:   PetscReal twoab1;
1469:   PetscInt  m = n - 2;
1470:   PetscReal a = alpha + 1.;
1471:   PetscReal b = beta + 1.;
1472:   PetscReal gra, grb;

1474:   PetscFunctionBegin;
1475:   twoab1 = PetscPowReal(2., a + b - 1.);
1476: #if defined(PETSC_HAVE_LGAMMA)
1477:   grb = PetscExpReal(2. * PetscLGamma(b + 1.) + PetscLGamma(m + 1.) + PetscLGamma(m + a + 1.) - (PetscLGamma(m + b + 1) + PetscLGamma(m + a + b + 1.)));
1478:   gra = PetscExpReal(2. * PetscLGamma(a + 1.) + PetscLGamma(m + 1.) + PetscLGamma(m + b + 1.) - (PetscLGamma(m + a + 1) + PetscLGamma(m + a + b + 1.)));
1479: #else
1480:   {
1481:     PetscInt alphai = (PetscInt)alpha;
1482:     PetscInt betai  = (PetscInt)beta;

1484:     if ((PetscReal)alphai == alpha && (PetscReal)betai == beta) {
1485:       PetscReal binom1, binom2;

1487:       PetscCall(PetscDTBinomial(m + b, b, &binom1));
1488:       PetscCall(PetscDTBinomial(m + a + b, b, &binom2));
1489:       grb = 1. / (binom1 * binom2);
1490:       PetscCall(PetscDTBinomial(m + a, a, &binom1));
1491:       PetscCall(PetscDTBinomial(m + a + b, a, &binom2));
1492:       gra = 1. / (binom1 * binom2);
1493:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "lgamma() - math routine is unavailable.");
1494:   }
1495: #endif
1496:   *leftw  = twoab1 * grb / b;
1497:   *rightw = twoab1 * gra / a;
1498:   PetscFunctionReturn(PETSC_SUCCESS);
1499: }

1501: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
1502:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
1503: static inline PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
1504: {
1505:   PetscReal pn1, pn2;
1506:   PetscReal cnm1, cnm1x, cnm2;
1507:   PetscInt  k;

1509:   PetscFunctionBegin;
1510:   if (!n) {
1511:     *P = 1.0;
1512:     PetscFunctionReturn(PETSC_SUCCESS);
1513:   }
1514:   PetscDTJacobiRecurrence_Internal(1, a, b, cnm1, cnm1x, cnm2);
1515:   pn2 = 1.;
1516:   pn1 = cnm1 + cnm1x * x;
1517:   if (n == 1) {
1518:     *P = pn1;
1519:     PetscFunctionReturn(PETSC_SUCCESS);
1520:   }
1521:   *P = 0.0;
1522:   for (k = 2; k < n + 1; ++k) {
1523:     PetscDTJacobiRecurrence_Internal(k, a, b, cnm1, cnm1x, cnm2);

1525:     *P  = (cnm1 + cnm1x * x) * pn1 - cnm2 * pn2;
1526:     pn2 = pn1;
1527:     pn1 = *P;
1528:   }
1529:   PetscFunctionReturn(PETSC_SUCCESS);
1530: }

1532: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
1533: static inline PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscInt k, PetscReal *P)
1534: {
1535:   PetscReal nP;
1536:   PetscInt  i;

1538:   PetscFunctionBegin;
1539:   *P = 0.0;
1540:   if (k > n) PetscFunctionReturn(PETSC_SUCCESS);
1541:   PetscCall(PetscDTComputeJacobi(a + k, b + k, n - k, x, &nP));
1542:   for (i = 0; i < k; i++) nP *= (a + b + n + 1. + i) * 0.5;
1543:   *P = nP;
1544:   PetscFunctionReturn(PETSC_SUCCESS);
1545: }

1547: static PetscErrorCode PetscDTGaussJacobiQuadrature_Newton_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal x[], PetscReal w[])
1548: {
1549:   PetscInt  maxIter = 100;
1550:   PetscReal eps     = PetscExpReal(0.75 * PetscLogReal(PETSC_MACHINE_EPSILON));
1551:   PetscReal a1, a6, gf;
1552:   PetscInt  k;

1554:   PetscFunctionBegin;

1556:   a1 = PetscPowReal(2.0, a + b + 1);
1557: #if defined(PETSC_HAVE_LGAMMA)
1558:   {
1559:     PetscReal a2, a3, a4, a5;
1560:     a2 = PetscLGamma(a + npoints + 1);
1561:     a3 = PetscLGamma(b + npoints + 1);
1562:     a4 = PetscLGamma(a + b + npoints + 1);
1563:     a5 = PetscLGamma(npoints + 1);
1564:     gf = PetscExpReal(a2 + a3 - (a4 + a5));
1565:   }
1566: #else
1567:   {
1568:     PetscInt ia, ib;

1570:     ia = (PetscInt)a;
1571:     ib = (PetscInt)b;
1572:     gf = 1.;
1573:     if (ia == a && ia >= 0) { /* compute ratio of rising factorals wrt a */
1574:       for (k = 0; k < ia; k++) gf *= (npoints + 1. + k) / (npoints + b + 1. + k);
1575:     } else if (b == b && ib >= 0) { /* compute ratio of rising factorials wrt b */
1576:       for (k = 0; k < ib; k++) gf *= (npoints + 1. + k) / (npoints + a + 1. + k);
1577:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "lgamma() - math routine is unavailable.");
1578:   }
1579: #endif

1581:   a6 = a1 * gf;
1582:   /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
1583:    Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
1584:   for (k = 0; k < npoints; ++k) {
1585:     PetscReal r = PetscCosReal(PETSC_PI * (1. - (4. * k + 3. + 2. * b) / (4. * npoints + 2. * (a + b + 1.)))), dP;
1586:     PetscInt  j;

1588:     if (k > 0) r = 0.5 * (r + x[k - 1]);
1589:     for (j = 0; j < maxIter; ++j) {
1590:       PetscReal s = 0.0, delta, f, fp;
1591:       PetscInt  i;

1593:       for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
1594:       PetscCall(PetscDTComputeJacobi(a, b, npoints, r, &f));
1595:       PetscCall(PetscDTComputeJacobiDerivative(a, b, npoints, r, 1, &fp));
1596:       delta = f / (fp - f * s);
1597:       r     = r - delta;
1598:       if (PetscAbsReal(delta) < eps) break;
1599:     }
1600:     x[k] = r;
1601:     PetscCall(PetscDTComputeJacobiDerivative(a, b, npoints, x[k], 1, &dP));
1602:     w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
1603:   }
1604:   PetscFunctionReturn(PETSC_SUCCESS);
1605: }

1607: /* Compute the diagonals of the Jacobi matrix used in Golub & Welsch algorithms for Gauss-Jacobi
1608:  * quadrature weight calculations on [-1,1] for exponents (1. + x)^a (1.-x)^b */
1609: static PetscErrorCode PetscDTJacobiMatrix_Internal(PetscInt nPoints, PetscReal a, PetscReal b, PetscReal *d, PetscReal *s)
1610: {
1611:   PetscInt i;

1613:   PetscFunctionBegin;
1614:   for (i = 0; i < nPoints; i++) {
1615:     PetscReal A, B, C;

1617:     PetscDTJacobiRecurrence_Internal(i + 1, a, b, A, B, C);
1618:     d[i] = -A / B;
1619:     if (i) s[i - 1] *= C / B;
1620:     if (i < nPoints - 1) s[i] = 1. / B;
1621:   }
1622:   PetscFunctionReturn(PETSC_SUCCESS);
1623: }

1625: static PetscErrorCode PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
1626: {
1627:   PetscReal mu0;
1628:   PetscReal ga, gb, gab;
1629:   PetscInt  i;

1631:   PetscFunctionBegin;
1632:   PetscCall(PetscCitationsRegister(GolubWelschCitation, &GolubWelschCite));

1634: #if defined(PETSC_HAVE_TGAMMA)
1635:   ga  = PetscTGamma(a + 1);
1636:   gb  = PetscTGamma(b + 1);
1637:   gab = PetscTGamma(a + b + 2);
1638: #else
1639:   {
1640:     PetscInt ia, ib;

1642:     ia = (PetscInt)a;
1643:     ib = (PetscInt)b;
1644:     if (ia == a && ib == b && ia + 1 > 0 && ib + 1 > 0 && ia + ib + 2 > 0) { /* All gamma(x) terms are (x-1)! terms */
1645:       PetscCall(PetscDTFactorial(ia, &ga));
1646:       PetscCall(PetscDTFactorial(ib, &gb));
1647:       PetscCall(PetscDTFactorial(ia + ib + 1, &gab));
1648:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "tgamma() - math routine is unavailable.");
1649:   }
1650: #endif
1651:   mu0 = PetscPowReal(2., a + b + 1.) * ga * gb / gab;

1653: #if defined(PETSCDTGAUSSIANQUADRATURE_EIG)
1654:   {
1655:     PetscReal   *diag, *subdiag;
1656:     PetscScalar *V;

1658:     PetscCall(PetscMalloc2(npoints, &diag, npoints, &subdiag));
1659:     PetscCall(PetscMalloc1(npoints * npoints, &V));
1660:     PetscCall(PetscDTJacobiMatrix_Internal(npoints, a, b, diag, subdiag));
1661:     for (i = 0; i < npoints - 1; i++) subdiag[i] = PetscSqrtReal(subdiag[i]);
1662:     PetscCall(PetscDTSymmetricTridiagonalEigensolve(npoints, diag, subdiag, x, V));
1663:     for (i = 0; i < npoints; i++) w[i] = PetscSqr(PetscRealPart(V[i * npoints])) * mu0;
1664:     PetscCall(PetscFree(V));
1665:     PetscCall(PetscFree2(diag, subdiag));
1666:   }
1667: #else
1668:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
1669: #endif
1670:   { /* As of March 2, 2020, The Sun Performance Library breaks the LAPACK contract for xstegr and xsteqr: the
1671:        eigenvalues are not guaranteed to be in ascending order.  So we heave a passive aggressive sigh and check that
1672:        the eigenvalues are sorted */
1673:     PetscBool sorted;

1675:     PetscCall(PetscSortedReal(npoints, x, &sorted));
1676:     if (!sorted) {
1677:       PetscInt  *order, i;
1678:       PetscReal *tmp;

1680:       PetscCall(PetscMalloc2(npoints, &order, npoints, &tmp));
1681:       for (i = 0; i < npoints; i++) order[i] = i;
1682:       PetscCall(PetscSortRealWithPermutation(npoints, x, order));
1683:       PetscCall(PetscArraycpy(tmp, x, npoints));
1684:       for (i = 0; i < npoints; i++) x[i] = tmp[order[i]];
1685:       PetscCall(PetscArraycpy(tmp, w, npoints));
1686:       for (i = 0; i < npoints; i++) w[i] = tmp[order[i]];
1687:       PetscCall(PetscFree2(order, tmp));
1688:     }
1689:   }
1690:   PetscFunctionReturn(PETSC_SUCCESS);
1691: }

1693: static PetscErrorCode PetscDTGaussJacobiQuadrature_Internal(PetscInt npoints, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
1694: {
1695:   PetscFunctionBegin;
1696:   PetscCheck(npoints >= 1, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of points must be positive");
1697:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
1698:   PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
1699:   PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");

1701:   if (newton) PetscCall(PetscDTGaussJacobiQuadrature_Newton_Internal(npoints, alpha, beta, x, w));
1702:   else PetscCall(PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(npoints, alpha, beta, x, w));
1703:   if (alpha == beta) { /* symmetrize */
1704:     PetscInt i;
1705:     for (i = 0; i < (npoints + 1) / 2; i++) {
1706:       PetscInt  j  = npoints - 1 - i;
1707:       PetscReal xi = x[i];
1708:       PetscReal xj = x[j];
1709:       PetscReal wi = w[i];
1710:       PetscReal wj = w[j];

1712:       x[i] = (xi - xj) / 2.;
1713:       x[j] = (xj - xi) / 2.;
1714:       w[i] = w[j] = (wi + wj) / 2.;
1715:     }
1716:   }
1717:   PetscFunctionReturn(PETSC_SUCCESS);
1718: }

1720: /*@
1721:   PetscDTGaussJacobiQuadrature - quadrature for the interval [a, b] with the weight function
1722:   $(x-a)^\alpha (x-b)^\beta$.

1724:   Not Collective

1726:   Input Parameters:
1727: + npoints - the number of points in the quadrature rule
1728: . a       - the left endpoint of the interval
1729: . b       - the right endpoint of the interval
1730: . alpha   - the left exponent
1731: - beta    - the right exponent

1733:   Output Parameters:
1734: + x - array of length `npoints`, the locations of the quadrature points
1735: - w - array of length `npoints`, the weights of the quadrature points

1737:   Level: intermediate

1739:   Note:
1740:   This quadrature rule is exact for polynomials up to degree 2*npoints - 1.

1742: .seealso: `PetscDTGaussQuadrature()`
1743: @*/
1744: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
1745: {
1746:   PetscInt i;

1748:   PetscFunctionBegin;
1749:   PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal));
1750:   if (a != -1. || b != 1.) { /* shift */
1751:     for (i = 0; i < npoints; i++) {
1752:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1753:       w[i] *= (b - a) / 2.;
1754:     }
1755:   }
1756:   PetscFunctionReturn(PETSC_SUCCESS);
1757: }

1759: static PetscErrorCode PetscDTGaussLobattoJacobiQuadrature_Internal(PetscInt npoints, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
1760: {
1761:   PetscInt i;

1763:   PetscFunctionBegin;
1764:   PetscCheck(npoints >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of points must be positive");
1765:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
1766:   PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
1767:   PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");

1769:   x[0]           = -1.;
1770:   x[npoints - 1] = 1.;
1771:   if (npoints > 2) PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints - 2, alpha + 1., beta + 1., &x[1], &w[1], newton));
1772:   for (i = 1; i < npoints - 1; i++) w[i] /= (1. - x[i] * x[i]);
1773:   PetscCall(PetscDTGaussLobattoJacobiEndweights_Internal(npoints, alpha, beta, &w[0], &w[npoints - 1]));
1774:   PetscFunctionReturn(PETSC_SUCCESS);
1775: }

1777: /*@
1778:   PetscDTGaussLobattoJacobiQuadrature - quadrature for the interval [a, b] with the weight function
1779:   $(x-a)^\alpha (x-b)^\beta$, with endpoints a and b included as quadrature points.

1781:   Not Collective

1783:   Input Parameters:
1784: + npoints - the number of points in the quadrature rule
1785: . a       - the left endpoint of the interval
1786: . b       - the right endpoint of the interval
1787: . alpha   - the left exponent
1788: - beta    - the right exponent

1790:   Output Parameters:
1791: + x - array of length `npoints`, the locations of the quadrature points
1792: - w - array of length `npoints`, the weights of the quadrature points

1794:   Level: intermediate

1796:   Note:
1797:   This quadrature rule is exact for polynomials up to degree 2*npoints - 3.

1799: .seealso: `PetscDTGaussJacobiQuadrature()`
1800: @*/
1801: PetscErrorCode PetscDTGaussLobattoJacobiQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
1802: {
1803:   PetscInt i;

1805:   PetscFunctionBegin;
1806:   PetscCall(PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal));
1807:   if (a != -1. || b != 1.) { /* shift */
1808:     for (i = 0; i < npoints; i++) {
1809:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1810:       w[i] *= (b - a) / 2.;
1811:     }
1812:   }
1813:   PetscFunctionReturn(PETSC_SUCCESS);
1814: }

1816: /*@
1817:   PetscDTGaussQuadrature - create Gauss-Legendre quadrature

1819:   Not Collective

1821:   Input Parameters:
1822: + npoints - number of points
1823: . a       - left end of interval (often-1)
1824: - b       - right end of interval (often +1)

1826:   Output Parameters:
1827: + x - quadrature points
1828: - w - quadrature weights

1830:   Level: intermediate

1832:   References:
1833: .  * - Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 1969.

1835: .seealso: `PetscDTLegendreEval()`, `PetscDTGaussJacobiQuadrature()`
1836: @*/
1837: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
1838: {
1839:   PetscInt i;

1841:   PetscFunctionBegin;
1842:   PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints, 0., 0., x, w, PetscDTGaussQuadratureNewton_Internal));
1843:   if (a != -1. || b != 1.) { /* shift */
1844:     for (i = 0; i < npoints; i++) {
1845:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1846:       w[i] *= (b - a) / 2.;
1847:     }
1848:   }
1849:   PetscFunctionReturn(PETSC_SUCCESS);
1850: }

1852: /*@C
1853:   PetscDTGaussLobattoLegendreQuadrature - creates a set of the locations and weights of the Gauss-Lobatto-Legendre
1854:   nodes of a given size on the domain [-1,1]

1856:   Not Collective

1858:   Input Parameters:
1859: + npoints - number of grid nodes
1860: - type    - `PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA` or `PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON`

1862:   Output Parameters:
1863: + x - quadrature points
1864: - w - quadrature weights

1866:   Level: intermediate

1868:   Notes:
1869:   For n > 30  the Newton approach computes duplicate (incorrect) values for some nodes because the initial guess is apparently not
1870:   close enough to the desired solution

1872:   These are useful for implementing spectral methods based on Gauss-Lobatto-Legendre (GLL) nodes

1874:   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

1876: .seealso: `PetscDTGaussQuadrature()`, `PetscGaussLobattoLegendreCreateType`

1878: @*/
1879: PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt npoints, PetscGaussLobattoLegendreCreateType type, PetscReal *x, PetscReal *w)
1880: {
1881:   PetscBool newton;

1883:   PetscFunctionBegin;
1884:   PetscCheck(npoints >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must provide at least 2 grid points per element");
1885:   newton = (PetscBool)(type == PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON);
1886:   PetscCall(PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, 0., 0., x, w, newton));
1887:   PetscFunctionReturn(PETSC_SUCCESS);
1888: }

1890: /*@
1891:   PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

1893:   Not Collective

1895:   Input Parameters:
1896: + dim     - The spatial dimension
1897: . Nc      - The number of components
1898: . npoints - number of points in one dimension
1899: . a       - left end of interval (often-1)
1900: - b       - right end of interval (often +1)

1902:   Output Parameter:
1903: . q - A `PetscQuadrature` object

1905:   Level: intermediate

1907: .seealso: `PetscDTGaussQuadrature()`, `PetscDTLegendreEval()`
1908: @*/
1909: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
1910: {
1911:   DMPolytopeType ct;
1912:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints * PetscSqr(npoints) : PetscSqr(npoints) : npoints;
1913:   PetscReal     *x, *w, *xw, *ww;

1915:   PetscFunctionBegin;
1916:   PetscCall(PetscMalloc1(totpoints * dim, &x));
1917:   PetscCall(PetscMalloc1(totpoints * Nc, &w));
1918:   /* Set up the Golub-Welsch system */
1919:   switch (dim) {
1920:   case 0:
1921:     ct = DM_POLYTOPE_POINT;
1922:     PetscCall(PetscFree(x));
1923:     PetscCall(PetscFree(w));
1924:     PetscCall(PetscMalloc1(1, &x));
1925:     PetscCall(PetscMalloc1(Nc, &w));
1926:     totpoints = 1;
1927:     x[0]      = 0.0;
1928:     for (PetscInt c = 0; c < Nc; ++c) w[c] = 1.0;
1929:     break;
1930:   case 1:
1931:     ct = DM_POLYTOPE_SEGMENT;
1932:     PetscCall(PetscMalloc1(npoints, &ww));
1933:     PetscCall(PetscDTGaussQuadrature(npoints, a, b, x, ww));
1934:     for (PetscInt i = 0; i < npoints; ++i)
1935:       for (PetscInt c = 0; c < Nc; ++c) w[i * Nc + c] = ww[i];
1936:     PetscCall(PetscFree(ww));
1937:     break;
1938:   case 2:
1939:     ct = DM_POLYTOPE_QUADRILATERAL;
1940:     PetscCall(PetscMalloc2(npoints, &xw, npoints, &ww));
1941:     PetscCall(PetscDTGaussQuadrature(npoints, a, b, xw, ww));
1942:     for (PetscInt i = 0; i < npoints; ++i) {
1943:       for (PetscInt j = 0; j < npoints; ++j) {
1944:         x[(i * npoints + j) * dim + 0] = xw[i];
1945:         x[(i * npoints + j) * dim + 1] = xw[j];
1946:         for (PetscInt c = 0; c < Nc; ++c) w[(i * npoints + j) * Nc + c] = ww[i] * ww[j];
1947:       }
1948:     }
1949:     PetscCall(PetscFree2(xw, ww));
1950:     break;
1951:   case 3:
1952:     ct = DM_POLYTOPE_HEXAHEDRON;
1953:     PetscCall(PetscMalloc2(npoints, &xw, npoints, &ww));
1954:     PetscCall(PetscDTGaussQuadrature(npoints, a, b, xw, ww));
1955:     for (PetscInt i = 0; i < npoints; ++i) {
1956:       for (PetscInt j = 0; j < npoints; ++j) {
1957:         for (PetscInt k = 0; k < npoints; ++k) {
1958:           x[((i * npoints + j) * npoints + k) * dim + 0] = xw[i];
1959:           x[((i * npoints + j) * npoints + k) * dim + 1] = xw[j];
1960:           x[((i * npoints + j) * npoints + k) * dim + 2] = xw[k];
1961:           for (PetscInt c = 0; c < Nc; ++c) w[((i * npoints + j) * npoints + k) * Nc + c] = ww[i] * ww[j] * ww[k];
1962:         }
1963:       }
1964:     }
1965:     PetscCall(PetscFree2(xw, ww));
1966:     break;
1967:   default:
1968:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %" PetscInt_FMT, dim);
1969:   }
1970:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
1971:   PetscCall(PetscQuadratureSetCellType(*q, ct));
1972:   PetscCall(PetscQuadratureSetOrder(*q, 2 * npoints - 1));
1973:   PetscCall(PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w));
1974:   PetscCall(PetscObjectChangeTypeName((PetscObject)*q, "GaussTensor"));
1975:   PetscFunctionReturn(PETSC_SUCCESS);
1976: }

1978: /*@
1979:   PetscDTStroudConicalQuadrature - create Stroud conical quadrature for a simplex

1981:   Not Collective

1983:   Input Parameters:
1984: + dim     - The simplex dimension
1985: . Nc      - The number of components
1986: . npoints - The number of points in one dimension
1987: . a       - left end of interval (often-1)
1988: - b       - right end of interval (often +1)

1990:   Output Parameter:
1991: . q - A `PetscQuadrature` object

1993:   Level: intermediate

1995:   Note:
1996:   For `dim` == 1, this is Gauss-Legendre quadrature

1998:   References:
1999: . * - Karniadakis and Sherwin.  FIAT

2001: .seealso: `PetscDTGaussTensorQuadrature()`, `PetscDTGaussQuadrature()`
2002: @*/
2003: PetscErrorCode PetscDTStroudConicalQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
2004: {
2005:   DMPolytopeType ct;
2006:   PetscInt       totpoints;
2007:   PetscReal     *p1, *w1;
2008:   PetscReal     *x, *w;

2010:   PetscFunctionBegin;
2011:   PetscCheck(!(a != -1.0) && !(b != 1.0), PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
2012:   switch (dim) {
2013:   case 0:
2014:     ct = DM_POLYTOPE_POINT;
2015:     break;
2016:   case 1:
2017:     ct = DM_POLYTOPE_SEGMENT;
2018:     break;
2019:   case 2:
2020:     ct = DM_POLYTOPE_TRIANGLE;
2021:     break;
2022:   case 3:
2023:     ct = DM_POLYTOPE_TETRAHEDRON;
2024:     break;
2025:   default:
2026:     ct = DM_POLYTOPE_UNKNOWN;
2027:   }
2028:   totpoints = 1;
2029:   for (PetscInt i = 0; i < dim; ++i) totpoints *= npoints;
2030:   PetscCall(PetscMalloc1(totpoints * dim, &x));
2031:   PetscCall(PetscMalloc1(totpoints * Nc, &w));
2032:   PetscCall(PetscMalloc2(npoints, &p1, npoints, &w1));
2033:   for (PetscInt i = 0; i < totpoints * Nc; ++i) w[i] = 1.;
2034:   for (PetscInt i = 0, totprev = 1, totrem = totpoints / npoints; i < dim; ++i) {
2035:     PetscReal mul;

2037:     mul = PetscPowReal(2., -i);
2038:     PetscCall(PetscDTGaussJacobiQuadrature(npoints, -1., 1., i, 0.0, p1, w1));
2039:     for (PetscInt pt = 0, l = 0; l < totprev; l++) {
2040:       for (PetscInt j = 0; j < npoints; j++) {
2041:         for (PetscInt m = 0; m < totrem; m++, pt++) {
2042:           for (PetscInt k = 0; k < i; k++) x[pt * dim + k] = (x[pt * dim + k] + 1.) * (1. - p1[j]) * 0.5 - 1.;
2043:           x[pt * dim + i] = p1[j];
2044:           for (PetscInt c = 0; c < Nc; c++) w[pt * Nc + c] *= mul * w1[j];
2045:         }
2046:       }
2047:     }
2048:     totprev *= npoints;
2049:     totrem /= npoints;
2050:   }
2051:   PetscCall(PetscFree2(p1, w1));
2052:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
2053:   PetscCall(PetscQuadratureSetCellType(*q, ct));
2054:   PetscCall(PetscQuadratureSetOrder(*q, 2 * npoints - 1));
2055:   PetscCall(PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w));
2056:   PetscCall(PetscObjectChangeTypeName((PetscObject)*q, "StroudConical"));
2057:   PetscFunctionReturn(PETSC_SUCCESS);
2058: }

2060: static PetscBool MinSymTriQuadCite       = PETSC_FALSE;
2061: const char       MinSymTriQuadCitation[] = "@article{WitherdenVincent2015,\n"
2062:                                            "  title = {On the identification of symmetric quadrature rules for finite element methods},\n"
2063:                                            "  journal = {Computers & Mathematics with Applications},\n"
2064:                                            "  volume = {69},\n"
2065:                                            "  number = {10},\n"
2066:                                            "  pages = {1232-1241},\n"
2067:                                            "  year = {2015},\n"
2068:                                            "  issn = {0898-1221},\n"
2069:                                            "  doi = {10.1016/j.camwa.2015.03.017},\n"
2070:                                            "  url = {https://www.sciencedirect.com/science/article/pii/S0898122115001224},\n"
2071:                                            "  author = {F.D. Witherden and P.E. Vincent},\n"
2072:                                            "}\n";

2074: #include "petscdttriquadrules.h"

2076: static PetscBool MinSymTetQuadCite       = PETSC_FALSE;
2077: const char       MinSymTetQuadCitation[] = "@article{JaskowiecSukumar2021\n"
2078:                                            "  author = {Jaskowiec, Jan and Sukumar, N.},\n"
2079:                                            "  title = {High-order symmetric cubature rules for tetrahedra and pyramids},\n"
2080:                                            "  journal = {International Journal for Numerical Methods in Engineering},\n"
2081:                                            "  volume = {122},\n"
2082:                                            "  number = {1},\n"
2083:                                            "  pages = {148-171},\n"
2084:                                            "  doi = {10.1002/nme.6528},\n"
2085:                                            "  url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.6528},\n"
2086:                                            "  eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.6528},\n"
2087:                                            "  year = {2021}\n"
2088:                                            "}\n";

2090: #include "petscdttetquadrules.h"

2092: // https://en.wikipedia.org/wiki/Partition_(number_theory)
2093: static PetscErrorCode PetscDTPartitionNumber(PetscInt n, PetscInt *p)
2094: {
2095:   // sequence A000041 in the OEIS
2096:   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};
2097:   PetscInt       tabulated_max = PETSC_STATIC_ARRAY_LENGTH(partition) - 1;

2099:   PetscFunctionBegin;
2100:   PetscCheck(n >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Partition number not defined for negative number %" PetscInt_FMT, n);
2101:   // not implementing the pentagonal number recurrence, we don't need partition numbers for n that high
2102:   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);
2103:   *p = partition[n];
2104:   PetscFunctionReturn(PETSC_SUCCESS);
2105: }

2107: /*@
2108:   PetscDTSimplexQuadrature - Create a quadrature rule for a simplex that exactly integrates polynomials up to a given degree.

2110:   Not Collective

2112:   Input Parameters:
2113: + dim    - The spatial dimension of the simplex (1 = segment, 2 = triangle, 3 = tetrahedron)
2114: . degree - The largest polynomial degree that is required to be integrated exactly
2115: - type   - left end of interval (often-1)

2117:   Output Parameter:
2118: . quad - A `PetscQuadrature` object for integration over the biunit simplex
2119:             (defined by the bounds $x_i >= -1$ and $\sum_i x_i <= 2 - d$) that is exact for
2120:             polynomials up to the given degree

2122:   Level: intermediate

2124: .seealso: `PetscDTSimplexQuadratureType`, `PetscDTGaussQuadrature()`, `PetscDTStroudCononicalQuadrature()`, `PetscQuadrature`
2125: @*/
2126: PetscErrorCode PetscDTSimplexQuadrature(PetscInt dim, PetscInt degree, PetscDTSimplexQuadratureType type, PetscQuadrature *quad)
2127: {
2128:   PetscDTSimplexQuadratureType orig_type = type;

2130:   PetscFunctionBegin;
2131:   PetscCheck(dim >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Negative dimension %" PetscInt_FMT, dim);
2132:   PetscCheck(degree >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Negative degree %" PetscInt_FMT, degree);
2133:   if (type == PETSCDTSIMPLEXQUAD_DEFAULT) type = PETSCDTSIMPLEXQUAD_MINSYM;
2134:   if (type == PETSCDTSIMPLEXQUAD_CONIC || dim < 2) {
2135:     PetscInt points_per_dim = (degree + 2) / 2; // ceil((degree + 1) / 2);
2136:     PetscCall(PetscDTStroudConicalQuadrature(dim, 1, points_per_dim, -1, 1, quad));
2137:   } else {
2138:     DMPolytopeType    ct;
2139:     PetscInt          n    = dim + 1;
2140:     PetscInt          fact = 1;
2141:     PetscInt         *part, *perm;
2142:     PetscInt          p = 0;
2143:     PetscInt          max_degree;
2144:     const PetscInt   *nodes_per_type     = NULL;
2145:     const PetscInt   *all_num_full_nodes = NULL;
2146:     const PetscReal **weights_list       = NULL;
2147:     const PetscReal **compact_nodes_list = NULL;
2148:     const char       *citation           = NULL;
2149:     PetscBool        *cited              = NULL;

2151:     switch (dim) {
2152:     case 0:
2153:       ct = DM_POLYTOPE_POINT;
2154:       break;
2155:     case 1:
2156:       ct = DM_POLYTOPE_SEGMENT;
2157:       break;
2158:     case 2:
2159:       ct = DM_POLYTOPE_TRIANGLE;
2160:       break;
2161:     case 3:
2162:       ct = DM_POLYTOPE_TETRAHEDRON;
2163:       break;
2164:     default:
2165:       ct = DM_POLYTOPE_UNKNOWN;
2166:     }
2167:     switch (dim) {
2168:     case 2:
2169:       cited              = &MinSymTriQuadCite;
2170:       citation           = MinSymTriQuadCitation;
2171:       max_degree         = PetscDTWVTriQuad_max_degree;
2172:       nodes_per_type     = PetscDTWVTriQuad_num_orbits;
2173:       all_num_full_nodes = PetscDTWVTriQuad_num_nodes;
2174:       weights_list       = PetscDTWVTriQuad_weights;
2175:       compact_nodes_list = PetscDTWVTriQuad_orbits;
2176:       break;
2177:     case 3:
2178:       cited              = &MinSymTetQuadCite;
2179:       citation           = MinSymTetQuadCitation;
2180:       max_degree         = PetscDTJSTetQuad_max_degree;
2181:       nodes_per_type     = PetscDTJSTetQuad_num_orbits;
2182:       all_num_full_nodes = PetscDTJSTetQuad_num_nodes;
2183:       weights_list       = PetscDTJSTetQuad_weights;
2184:       compact_nodes_list = PetscDTJSTetQuad_orbits;
2185:       break;
2186:     default:
2187:       max_degree = -1;
2188:       break;
2189:     }

2191:     if (degree > max_degree) {
2192:       if (orig_type == PETSCDTSIMPLEXQUAD_DEFAULT) {
2193:         // fall back to conic
2194:         PetscCall(PetscDTSimplexQuadrature(dim, degree, PETSCDTSIMPLEXQUAD_CONIC, quad));
2195:         PetscFunctionReturn(PETSC_SUCCESS);
2196:       } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Minimal symmetric quadrature for dim %" PetscInt_FMT ", degree %" PetscInt_FMT " unsupported", dim, degree);
2197:     }

2199:     PetscCall(PetscCitationsRegister(citation, cited));

2201:     PetscCall(PetscDTPartitionNumber(n, &p));
2202:     for (PetscInt d = 2; d <= n; d++) fact *= d;

2204:     PetscInt         num_full_nodes      = all_num_full_nodes[degree];
2205:     const PetscReal *all_compact_nodes   = compact_nodes_list[degree];
2206:     const PetscReal *all_compact_weights = weights_list[degree];
2207:     nodes_per_type                       = &nodes_per_type[p * degree];

2209:     PetscReal      *points;
2210:     PetscReal      *counts;
2211:     PetscReal      *weights;
2212:     PetscReal      *bary_to_biunit; // row-major transformation of barycentric coordinate to biunit
2213:     PetscQuadrature q;

2215:     // compute the transformation
2216:     PetscCall(PetscMalloc1(n * dim, &bary_to_biunit));
2217:     for (PetscInt d = 0; d < dim; d++) {
2218:       for (PetscInt b = 0; b < n; b++) bary_to_biunit[d * n + b] = (d == b) ? 1.0 : -1.0;
2219:     }

2221:     PetscCall(PetscMalloc3(n, &part, n, &perm, n, &counts));
2222:     PetscCall(PetscCalloc1(num_full_nodes * dim, &points));
2223:     PetscCall(PetscMalloc1(num_full_nodes, &weights));

2225:     // (0, 0, ...) is the first partition lexicographically
2226:     PetscCall(PetscArrayzero(part, n));
2227:     PetscCall(PetscArrayzero(counts, n));
2228:     counts[0] = n;

2230:     // for each partition
2231:     for (PetscInt s = 0, node_offset = 0; s < p; s++) {
2232:       PetscInt num_compact_coords = part[n - 1] + 1;

2234:       const PetscReal *compact_nodes   = all_compact_nodes;
2235:       const PetscReal *compact_weights = all_compact_weights;
2236:       all_compact_nodes += num_compact_coords * nodes_per_type[s];
2237:       all_compact_weights += nodes_per_type[s];

2239:       // for every permutation of the vertices
2240:       for (PetscInt f = 0; f < fact; f++) {
2241:         PetscCall(PetscDTEnumPerm(n, f, perm, NULL));

2243:         // check if it is a valid permutation
2244:         PetscInt digit;
2245:         for (digit = 1; digit < n; digit++) {
2246:           // skip permutations that would duplicate a node because it has a smaller symmetry group
2247:           if (part[digit - 1] == part[digit] && perm[digit - 1] > perm[digit]) break;
2248:         }
2249:         if (digit < n) continue;

2251:         // create full nodes from this permutation of the compact nodes
2252:         PetscReal *full_nodes   = &points[node_offset * dim];
2253:         PetscReal *full_weights = &weights[node_offset];

2255:         PetscCall(PetscArraycpy(full_weights, compact_weights, nodes_per_type[s]));
2256:         for (PetscInt b = 0; b < n; b++) {
2257:           for (PetscInt d = 0; d < dim; d++) {
2258:             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]];
2259:           }
2260:         }
2261:         node_offset += nodes_per_type[s];
2262:       }

2264:       if (s < p - 1) { // Generate the next partition
2265:         /* A partition is described by the number of coordinates that are in
2266:          * each set of duplicates (counts) and redundantly by mapping each
2267:          * index to its set of duplicates (part)
2268:          *
2269:          * Counts should always be in nonincreasing order
2270:          *
2271:          * We want to generate the partitions lexically by part, which means
2272:          * finding the last index where count > 1 and reducing by 1.
2273:          *
2274:          * For the new counts beyond that index, we eagerly assign the remaining
2275:          * capacity of the partition to smaller indices (ensures lexical ordering),
2276:          * while respecting the nonincreasing invariant of the counts
2277:          */
2278:         PetscInt last_digit            = part[n - 1];
2279:         PetscInt last_digit_with_extra = last_digit;
2280:         while (counts[last_digit_with_extra] == 1) last_digit_with_extra--;
2281:         PetscInt limit               = --counts[last_digit_with_extra];
2282:         PetscInt total_to_distribute = last_digit - last_digit_with_extra + 1;
2283:         for (PetscInt digit = last_digit_with_extra + 1; digit < n; digit++) {
2284:           counts[digit] = PetscMin(limit, total_to_distribute);
2285:           total_to_distribute -= PetscMin(limit, total_to_distribute);
2286:         }
2287:         for (PetscInt digit = 0, offset = 0; digit < n; digit++) {
2288:           PetscInt count = counts[digit];
2289:           for (PetscInt c = 0; c < count; c++) part[offset++] = digit;
2290:         }
2291:       }
2292:     }
2293:     PetscCall(PetscFree3(part, perm, counts));
2294:     PetscCall(PetscFree(bary_to_biunit));
2295:     PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, &q));
2296:     PetscCall(PetscQuadratureSetCellType(q, ct));
2297:     PetscCall(PetscQuadratureSetOrder(q, degree));
2298:     PetscCall(PetscQuadratureSetData(q, dim, 1, num_full_nodes, points, weights));
2299:     *quad = q;
2300:   }
2301:   PetscFunctionReturn(PETSC_SUCCESS);
2302: }

2304: /*@
2305:   PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell

2307:   Not Collective

2309:   Input Parameters:
2310: + dim   - The cell dimension
2311: . level - The number of points in one dimension, 2^l
2312: . a     - left end of interval (often-1)
2313: - b     - right end of interval (often +1)

2315:   Output Parameter:
2316: . q - A `PetscQuadrature` object

2318:   Level: intermediate

2320: .seealso: `PetscDTGaussTensorQuadrature()`, `PetscQuadrature`
2321: @*/
2322: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
2323: {
2324:   DMPolytopeType  ct;
2325:   const PetscInt  p     = 16;                        /* Digits of precision in the evaluation */
2326:   const PetscReal alpha = (b - a) / 2.;              /* Half-width of the integration interval */
2327:   const PetscReal beta  = (b + a) / 2.;              /* Center of the integration interval */
2328:   const PetscReal h     = PetscPowReal(2.0, -level); /* Step size, length between x_k */
2329:   PetscReal       xk;                                /* Quadrature point x_k on reference domain [-1, 1] */
2330:   PetscReal       wk = 0.5 * PETSC_PI;               /* Quadrature weight at x_k */
2331:   PetscReal      *x, *w;
2332:   PetscInt        K, k, npoints;

2334:   PetscFunctionBegin;
2335:   PetscCheck(dim <= 1, PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %" PetscInt_FMT " not yet implemented", dim);
2336:   PetscCheck(level, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
2337:   switch (dim) {
2338:   case 0:
2339:     ct = DM_POLYTOPE_POINT;
2340:     break;
2341:   case 1:
2342:     ct = DM_POLYTOPE_SEGMENT;
2343:     break;
2344:   case 2:
2345:     ct = DM_POLYTOPE_QUADRILATERAL;
2346:     break;
2347:   case 3:
2348:     ct = DM_POLYTOPE_HEXAHEDRON;
2349:     break;
2350:   default:
2351:     ct = DM_POLYTOPE_UNKNOWN;
2352:   }
2353:   /* Find K such that the weights are < 32 digits of precision */
2354:   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)));
2355:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
2356:   PetscCall(PetscQuadratureSetCellType(*q, ct));
2357:   PetscCall(PetscQuadratureSetOrder(*q, 2 * K + 1));
2358:   npoints = 2 * K - 1;
2359:   PetscCall(PetscMalloc1(npoints * dim, &x));
2360:   PetscCall(PetscMalloc1(npoints, &w));
2361:   /* Center term */
2362:   x[0] = beta;
2363:   w[0] = 0.5 * alpha * PETSC_PI;
2364:   for (k = 1; k < K; ++k) {
2365:     wk           = 0.5 * alpha * h * PETSC_PI * PetscCoshReal(k * h) / PetscSqr(PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2366:     xk           = PetscTanhReal(0.5 * PETSC_PI * PetscSinhReal(k * h));
2367:     x[2 * k - 1] = -alpha * xk + beta;
2368:     w[2 * k - 1] = wk;
2369:     x[2 * k + 0] = alpha * xk + beta;
2370:     w[2 * k + 0] = wk;
2371:   }
2372:   PetscCall(PetscQuadratureSetData(*q, dim, 1, npoints, x, w));
2373:   PetscFunctionReturn(PETSC_SUCCESS);
2374: }

2376: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2377: {
2378:   const PetscInt  p     = 16;           /* Digits of precision in the evaluation */
2379:   const PetscReal alpha = (b - a) / 2.; /* Half-width of the integration interval */
2380:   const PetscReal beta  = (b + a) / 2.; /* Center of the integration interval */
2381:   PetscReal       h     = 1.0;          /* Step size, length between x_k */
2382:   PetscInt        l     = 0;            /* Level of refinement, h = 2^{-l} */
2383:   PetscReal       osum  = 0.0;          /* Integral on last level */
2384:   PetscReal       psum  = 0.0;          /* Integral on the level before the last level */
2385:   PetscReal       sum;                  /* Integral on current level */
2386:   PetscReal       yk;                   /* Quadrature point 1 - x_k on reference domain [-1, 1] */
2387:   PetscReal       lx, rx;               /* Quadrature points to the left and right of 0 on the real domain [a, b] */
2388:   PetscReal       wk;                   /* Quadrature weight at x_k */
2389:   PetscReal       lval, rval;           /* Terms in the quadature sum to the left and right of 0 */
2390:   PetscInt        d;                    /* Digits of precision in the integral */

2392:   PetscFunctionBegin;
2393:   PetscCheck(digits > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
2394:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
2395:   /* Center term */
2396:   func(&beta, ctx, &lval);
2397:   sum = 0.5 * alpha * PETSC_PI * lval;
2398:   /* */
2399:   do {
2400:     PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
2401:     PetscInt  k = 1;

2403:     ++l;
2404:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %" PetscInt_FMT " sum: %15.15f\n", l, sum); */
2405:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
2406:     psum = osum;
2407:     osum = sum;
2408:     h *= 0.5;
2409:     sum *= 0.5;
2410:     do {
2411:       wk = 0.5 * h * PETSC_PI * PetscCoshReal(k * h) / PetscSqr(PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2412:       yk = 1.0 / (PetscExpReal(0.5 * PETSC_PI * PetscSinhReal(k * h)) * PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2413:       lx = -alpha * (1.0 - yk) + beta;
2414:       rx = alpha * (1.0 - yk) + beta;
2415:       func(&lx, ctx, &lval);
2416:       func(&rx, ctx, &rval);
2417:       lterm   = alpha * wk * lval;
2418:       maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
2419:       sum += lterm;
2420:       rterm   = alpha * wk * rval;
2421:       maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
2422:       sum += rterm;
2423:       ++k;
2424:       /* Only need to evaluate every other point on refined levels */
2425:       if (l != 1) ++k;
2426:     } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */

2428:     d1 = PetscLog10Real(PetscAbsReal(sum - osum));
2429:     d2 = PetscLog10Real(PetscAbsReal(sum - psum));
2430:     d3 = PetscLog10Real(maxTerm) - p;
2431:     if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
2432:     else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
2433:     d = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1) / d2, 2 * d1), d3), d4)));
2434:   } while (d < digits && l < 12);
2435:   *sol = sum;
2436:   PetscCall(PetscFPTrapPop());
2437:   PetscFunctionReturn(PETSC_SUCCESS);
2438: }

2440: #if defined(PETSC_HAVE_MPFR)
2441: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2442: {
2443:   const PetscInt safetyFactor = 2; /* Calculate abcissa until 2*p digits */
2444:   PetscInt       l            = 0; /* Level of refinement, h = 2^{-l} */
2445:   mpfr_t         alpha;            /* Half-width of the integration interval */
2446:   mpfr_t         beta;             /* Center of the integration interval */
2447:   mpfr_t         h;                /* Step size, length between x_k */
2448:   mpfr_t         osum;             /* Integral on last level */
2449:   mpfr_t         psum;             /* Integral on the level before the last level */
2450:   mpfr_t         sum;              /* Integral on current level */
2451:   mpfr_t         yk;               /* Quadrature point 1 - x_k on reference domain [-1, 1] */
2452:   mpfr_t         lx, rx;           /* Quadrature points to the left and right of 0 on the real domain [a, b] */
2453:   mpfr_t         wk;               /* Quadrature weight at x_k */
2454:   PetscReal      lval, rval, rtmp; /* Terms in the quadature sum to the left and right of 0 */
2455:   PetscInt       d;                /* Digits of precision in the integral */
2456:   mpfr_t         pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;

2458:   PetscFunctionBegin;
2459:   PetscCheck(digits > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
2460:   /* Create high precision storage */
2461:   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);
2462:   /* Initialization */
2463:   mpfr_set_d(alpha, 0.5 * (b - a), MPFR_RNDN);
2464:   mpfr_set_d(beta, 0.5 * (b + a), MPFR_RNDN);
2465:   mpfr_set_d(osum, 0.0, MPFR_RNDN);
2466:   mpfr_set_d(psum, 0.0, MPFR_RNDN);
2467:   mpfr_set_d(h, 1.0, MPFR_RNDN);
2468:   mpfr_const_pi(pi2, MPFR_RNDN);
2469:   mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
2470:   /* Center term */
2471:   rtmp = 0.5 * (b + a);
2472:   func(&rtmp, ctx, &lval);
2473:   mpfr_set(sum, pi2, MPFR_RNDN);
2474:   mpfr_mul(sum, sum, alpha, MPFR_RNDN);
2475:   mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
2476:   /* */
2477:   do {
2478:     PetscReal d1, d2, d3, d4;
2479:     PetscInt  k = 1;

2481:     ++l;
2482:     mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
2483:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %" PetscInt_FMT " sum: %15.15f\n", l, sum); */
2484:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
2485:     mpfr_set(psum, osum, MPFR_RNDN);
2486:     mpfr_set(osum, sum, MPFR_RNDN);
2487:     mpfr_mul_d(h, h, 0.5, MPFR_RNDN);
2488:     mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
2489:     do {
2490:       mpfr_set_si(kh, k, MPFR_RNDN);
2491:       mpfr_mul(kh, kh, h, MPFR_RNDN);
2492:       /* Weight */
2493:       mpfr_set(wk, h, MPFR_RNDN);
2494:       mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
2495:       mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
2496:       mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
2497:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
2498:       mpfr_sqr(tmp, tmp, MPFR_RNDN);
2499:       mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
2500:       mpfr_div(wk, wk, tmp, MPFR_RNDN);
2501:       /* Abscissa */
2502:       mpfr_set_d(yk, 1.0, MPFR_RNDZ);
2503:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
2504:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
2505:       mpfr_exp(tmp, msinh, MPFR_RNDN);
2506:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
2507:       /* Quadrature points */
2508:       mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
2509:       mpfr_mul(lx, lx, alpha, MPFR_RNDU);
2510:       mpfr_add(lx, lx, beta, MPFR_RNDU);
2511:       mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
2512:       mpfr_mul(rx, rx, alpha, MPFR_RNDD);
2513:       mpfr_add(rx, rx, beta, MPFR_RNDD);
2514:       /* Evaluation */
2515:       rtmp = mpfr_get_d(lx, MPFR_RNDU);
2516:       func(&rtmp, ctx, &lval);
2517:       rtmp = mpfr_get_d(rx, MPFR_RNDD);
2518:       func(&rtmp, ctx, &rval);
2519:       /* Update */
2520:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
2521:       mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
2522:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
2523:       mpfr_abs(tmp, tmp, MPFR_RNDN);
2524:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
2525:       mpfr_set(curTerm, tmp, MPFR_RNDN);
2526:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
2527:       mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
2528:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
2529:       mpfr_abs(tmp, tmp, MPFR_RNDN);
2530:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
2531:       mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
2532:       ++k;
2533:       /* Only need to evaluate every other point on refined levels */
2534:       if (l != 1) ++k;
2535:       mpfr_log10(tmp, wk, MPFR_RNDN);
2536:       mpfr_abs(tmp, tmp, MPFR_RNDN);
2537:     } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor * digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
2538:     mpfr_sub(tmp, sum, osum, MPFR_RNDN);
2539:     mpfr_abs(tmp, tmp, MPFR_RNDN);
2540:     mpfr_log10(tmp, tmp, MPFR_RNDN);
2541:     d1 = mpfr_get_d(tmp, MPFR_RNDN);
2542:     mpfr_sub(tmp, sum, psum, MPFR_RNDN);
2543:     mpfr_abs(tmp, tmp, MPFR_RNDN);
2544:     mpfr_log10(tmp, tmp, MPFR_RNDN);
2545:     d2 = mpfr_get_d(tmp, MPFR_RNDN);
2546:     mpfr_log10(tmp, maxTerm, MPFR_RNDN);
2547:     d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
2548:     mpfr_log10(tmp, curTerm, MPFR_RNDN);
2549:     d4 = mpfr_get_d(tmp, MPFR_RNDN);
2550:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1) / d2, 2 * d1), d3), d4)));
2551:   } while (d < digits && l < 8);
2552:   *sol = mpfr_get_d(sum, MPFR_RNDN);
2553:   /* Cleanup */
2554:   mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
2555:   PetscFunctionReturn(PETSC_SUCCESS);
2556: }
2557: #else

2559: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2560: {
2561:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
2562: }
2563: #endif

2565: /*@
2566:   PetscDTTensorQuadratureCreate - create the tensor product quadrature from two lower-dimensional quadratures

2568:   Not Collective

2570:   Input Parameters:
2571: + q1 - The first quadrature
2572: - q2 - The second quadrature

2574:   Output Parameter:
2575: . q - A `PetscQuadrature` object

2577:   Level: intermediate

2579: .seealso: `PetscQuadrature`, `PetscDTGaussTensorQuadrature()`
2580: @*/
2581: PetscErrorCode PetscDTTensorQuadratureCreate(PetscQuadrature q1, PetscQuadrature q2, PetscQuadrature *q)
2582: {
2583:   DMPolytopeType   ct1, ct2, ct;
2584:   const PetscReal *x1, *w1, *x2, *w2;
2585:   PetscReal       *x, *w;
2586:   PetscInt         dim1, Nc1, Np1, order1, qa, d1;
2587:   PetscInt         dim2, Nc2, Np2, order2, qb, d2;
2588:   PetscInt         dim, Nc, Np, order, qc, d;

2590:   PetscFunctionBegin;
2593:   PetscAssertPointer(q, 3);
2594:   PetscCall(PetscQuadratureGetOrder(q1, &order1));
2595:   PetscCall(PetscQuadratureGetOrder(q2, &order2));
2596:   PetscCheck(order1 == order2, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Order1 %" PetscInt_FMT " != %" PetscInt_FMT " Order2", order1, order2);
2597:   PetscCall(PetscQuadratureGetData(q1, &dim1, &Nc1, &Np1, &x1, &w1));
2598:   PetscCall(PetscQuadratureGetCellType(q1, &ct1));
2599:   PetscCall(PetscQuadratureGetData(q2, &dim2, &Nc2, &Np2, &x2, &w2));
2600:   PetscCall(PetscQuadratureGetCellType(q2, &ct2));
2601:   PetscCheck(Nc1 == Nc2, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "NumComp1 %" PetscInt_FMT " != %" PetscInt_FMT " NumComp2", Nc1, Nc2);

2603:   switch (ct1) {
2604:   case DM_POLYTOPE_POINT:
2605:     ct = ct2;
2606:     break;
2607:   case DM_POLYTOPE_SEGMENT:
2608:     switch (ct2) {
2609:     case DM_POLYTOPE_POINT:
2610:       ct = ct1;
2611:       break;
2612:     case DM_POLYTOPE_SEGMENT:
2613:       ct = DM_POLYTOPE_QUADRILATERAL;
2614:       break;
2615:     case DM_POLYTOPE_TRIANGLE:
2616:       ct = DM_POLYTOPE_TRI_PRISM;
2617:       break;
2618:     case DM_POLYTOPE_QUADRILATERAL:
2619:       ct = DM_POLYTOPE_HEXAHEDRON;
2620:       break;
2621:     case DM_POLYTOPE_TETRAHEDRON:
2622:       ct = DM_POLYTOPE_UNKNOWN;
2623:       break;
2624:     case DM_POLYTOPE_HEXAHEDRON:
2625:       ct = DM_POLYTOPE_UNKNOWN;
2626:       break;
2627:     default:
2628:       ct = DM_POLYTOPE_UNKNOWN;
2629:     }
2630:     break;
2631:   case DM_POLYTOPE_TRIANGLE:
2632:     switch (ct2) {
2633:     case DM_POLYTOPE_POINT:
2634:       ct = ct1;
2635:       break;
2636:     case DM_POLYTOPE_SEGMENT:
2637:       ct = DM_POLYTOPE_TRI_PRISM;
2638:       break;
2639:     case DM_POLYTOPE_TRIANGLE:
2640:       ct = DM_POLYTOPE_UNKNOWN;
2641:       break;
2642:     case DM_POLYTOPE_QUADRILATERAL:
2643:       ct = DM_POLYTOPE_UNKNOWN;
2644:       break;
2645:     case DM_POLYTOPE_TETRAHEDRON:
2646:       ct = DM_POLYTOPE_UNKNOWN;
2647:       break;
2648:     case DM_POLYTOPE_HEXAHEDRON:
2649:       ct = DM_POLYTOPE_UNKNOWN;
2650:       break;
2651:     default:
2652:       ct = DM_POLYTOPE_UNKNOWN;
2653:     }
2654:     break;
2655:   case DM_POLYTOPE_QUADRILATERAL:
2656:     switch (ct2) {
2657:     case DM_POLYTOPE_POINT:
2658:       ct = ct1;
2659:       break;
2660:     case DM_POLYTOPE_SEGMENT:
2661:       ct = DM_POLYTOPE_HEXAHEDRON;
2662:       break;
2663:     case DM_POLYTOPE_TRIANGLE:
2664:       ct = DM_POLYTOPE_UNKNOWN;
2665:       break;
2666:     case DM_POLYTOPE_QUADRILATERAL:
2667:       ct = DM_POLYTOPE_UNKNOWN;
2668:       break;
2669:     case DM_POLYTOPE_TETRAHEDRON:
2670:       ct = DM_POLYTOPE_UNKNOWN;
2671:       break;
2672:     case DM_POLYTOPE_HEXAHEDRON:
2673:       ct = DM_POLYTOPE_UNKNOWN;
2674:       break;
2675:     default:
2676:       ct = DM_POLYTOPE_UNKNOWN;
2677:     }
2678:     break;
2679:   case DM_POLYTOPE_TETRAHEDRON:
2680:     switch (ct2) {
2681:     case DM_POLYTOPE_POINT:
2682:       ct = ct1;
2683:       break;
2684:     case DM_POLYTOPE_SEGMENT:
2685:       ct = DM_POLYTOPE_UNKNOWN;
2686:       break;
2687:     case DM_POLYTOPE_TRIANGLE:
2688:       ct = DM_POLYTOPE_UNKNOWN;
2689:       break;
2690:     case DM_POLYTOPE_QUADRILATERAL:
2691:       ct = DM_POLYTOPE_UNKNOWN;
2692:       break;
2693:     case DM_POLYTOPE_TETRAHEDRON:
2694:       ct = DM_POLYTOPE_UNKNOWN;
2695:       break;
2696:     case DM_POLYTOPE_HEXAHEDRON:
2697:       ct = DM_POLYTOPE_UNKNOWN;
2698:       break;
2699:     default:
2700:       ct = DM_POLYTOPE_UNKNOWN;
2701:     }
2702:     break;
2703:   case DM_POLYTOPE_HEXAHEDRON:
2704:     switch (ct2) {
2705:     case DM_POLYTOPE_POINT:
2706:       ct = ct1;
2707:       break;
2708:     case DM_POLYTOPE_SEGMENT:
2709:       ct = DM_POLYTOPE_UNKNOWN;
2710:       break;
2711:     case DM_POLYTOPE_TRIANGLE:
2712:       ct = DM_POLYTOPE_UNKNOWN;
2713:       break;
2714:     case DM_POLYTOPE_QUADRILATERAL:
2715:       ct = DM_POLYTOPE_UNKNOWN;
2716:       break;
2717:     case DM_POLYTOPE_TETRAHEDRON:
2718:       ct = DM_POLYTOPE_UNKNOWN;
2719:       break;
2720:     case DM_POLYTOPE_HEXAHEDRON:
2721:       ct = DM_POLYTOPE_UNKNOWN;
2722:       break;
2723:     default:
2724:       ct = DM_POLYTOPE_UNKNOWN;
2725:     }
2726:     break;
2727:   default:
2728:     ct = DM_POLYTOPE_UNKNOWN;
2729:   }
2730:   dim   = dim1 + dim2;
2731:   Nc    = Nc1;
2732:   Np    = Np1 * Np2;
2733:   order = order1;
2734:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
2735:   PetscCall(PetscQuadratureSetCellType(*q, ct));
2736:   PetscCall(PetscQuadratureSetOrder(*q, order));
2737:   PetscCall(PetscMalloc1(Np * dim, &x));
2738:   PetscCall(PetscMalloc1(Np, &w));
2739:   for (qa = 0, qc = 0; qa < Np1; ++qa) {
2740:     for (qb = 0; qb < Np2; ++qb, ++qc) {
2741:       for (d1 = 0, d = 0; d1 < dim1; ++d1, ++d) x[qc * dim + d] = x1[qa * dim1 + d1];
2742:       for (d2 = 0; d2 < dim2; ++d2, ++d) x[qc * dim + d] = x2[qb * dim2 + d2];
2743:       w[qc] = w1[qa] * w2[qb];
2744:     }
2745:   }
2746:   PetscCall(PetscQuadratureSetData(*q, dim, Nc, Np, x, w));
2747:   PetscFunctionReturn(PETSC_SUCCESS);
2748: }

2750: /* Overwrites A. Can only handle full-rank problems with m>=n
2751:    A in column-major format
2752:    Ainv in row-major format
2753:    tau has length m
2754:    worksize must be >= max(1,n)
2755:  */
2756: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m, PetscInt mstride, PetscInt n, PetscReal *A_in, PetscReal *Ainv_out, PetscScalar *tau, PetscInt worksize, PetscScalar *work)
2757: {
2758:   PetscBLASInt M, N, K, lda, ldb, ldwork, info;
2759:   PetscScalar *A, *Ainv, *R, *Q, Alpha;

2761:   PetscFunctionBegin;
2762: #if defined(PETSC_USE_COMPLEX)
2763:   {
2764:     PetscInt i, j;
2765:     PetscCall(PetscMalloc2(m * n, &A, m * n, &Ainv));
2766:     for (j = 0; j < n; j++) {
2767:       for (i = 0; i < m; i++) A[i + m * j] = A_in[i + mstride * j];
2768:     }
2769:     mstride = m;
2770:   }
2771: #else
2772:   A    = A_in;
2773:   Ainv = Ainv_out;
2774: #endif

2776:   PetscCall(PetscBLASIntCast(m, &M));
2777:   PetscCall(PetscBLASIntCast(n, &N));
2778:   PetscCall(PetscBLASIntCast(mstride, &lda));
2779:   PetscCall(PetscBLASIntCast(worksize, &ldwork));
2780:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
2781:   PetscCallBLAS("LAPACKgeqrf", LAPACKgeqrf_(&M, &N, A, &lda, tau, work, &ldwork, &info));
2782:   PetscCall(PetscFPTrapPop());
2783:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "xGEQRF error");
2784:   R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

2786:   /* Extract an explicit representation of Q */
2787:   Q = Ainv;
2788:   PetscCall(PetscArraycpy(Q, A, mstride * n));
2789:   K = N; /* full rank */
2790:   PetscCallBLAS("LAPACKorgqr", LAPACKorgqr_(&M, &N, &K, Q, &lda, tau, work, &ldwork, &info));
2791:   PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "xORGQR/xUNGQR error");

2793:   /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
2794:   Alpha = 1.0;
2795:   ldb   = lda;
2796:   PetscCallBLAS("BLAStrsm", BLAStrsm_("Right", "Upper", "ConjugateTranspose", "NotUnitTriangular", &M, &N, &Alpha, R, &lda, Q, &ldb));
2797:   /* Ainv is Q, overwritten with inverse */

2799: #if defined(PETSC_USE_COMPLEX)
2800:   {
2801:     PetscInt i;
2802:     for (i = 0; i < m * n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
2803:     PetscCall(PetscFree2(A, Ainv));
2804:   }
2805: #endif
2806:   PetscFunctionReturn(PETSC_SUCCESS);
2807: }

2809: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
2810: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval, const PetscReal *x, PetscInt ndegree, const PetscInt *degrees, PetscBool Transpose, PetscReal *B)
2811: {
2812:   PetscReal *Bv;
2813:   PetscInt   i, j;

2815:   PetscFunctionBegin;
2816:   PetscCall(PetscMalloc1((ninterval + 1) * ndegree, &Bv));
2817:   /* Point evaluation of L_p on all the source vertices */
2818:   PetscCall(PetscDTLegendreEval(ninterval + 1, x, ndegree, degrees, Bv, NULL, NULL));
2819:   /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
2820:   for (i = 0; i < ninterval; i++) {
2821:     for (j = 0; j < ndegree; j++) {
2822:       if (Transpose) B[i + ninterval * j] = Bv[(i + 1) * ndegree + j] - Bv[i * ndegree + j];
2823:       else B[i * ndegree + j] = Bv[(i + 1) * ndegree + j] - Bv[i * ndegree + j];
2824:     }
2825:   }
2826:   PetscCall(PetscFree(Bv));
2827:   PetscFunctionReturn(PETSC_SUCCESS);
2828: }

2830: /*@
2831:   PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

2833:   Not Collective

2835:   Input Parameters:
2836: + degree  - degree of reconstruction polynomial
2837: . nsource - number of source intervals
2838: . sourcex - sorted coordinates of source cell boundaries (length nsource+1)
2839: . ntarget - number of target intervals
2840: - targetx - sorted coordinates of target cell boundaries (length ntarget+1)

2842:   Output Parameter:
2843: . R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]

2845:   Level: advanced

2847: .seealso: `PetscDTLegendreEval()`
2848: @*/
2849: PetscErrorCode PetscDTReconstructPoly(PetscInt degree, PetscInt nsource, const PetscReal *sourcex, PetscInt ntarget, const PetscReal *targetx, PetscReal *R)
2850: {
2851:   PetscInt     i, j, k, *bdegrees, worksize;
2852:   PetscReal    xmin, xmax, center, hscale, *sourcey, *targety, *Bsource, *Bsinv, *Btarget;
2853:   PetscScalar *tau, *work;

2855:   PetscFunctionBegin;
2856:   PetscAssertPointer(sourcex, 3);
2857:   PetscAssertPointer(targetx, 5);
2858:   PetscAssertPointer(R, 6);
2859:   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);
2860:   if (PetscDefined(USE_DEBUG)) {
2861:     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]);
2862:     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]);
2863:   }
2864:   xmin     = PetscMin(sourcex[0], targetx[0]);
2865:   xmax     = PetscMax(sourcex[nsource], targetx[ntarget]);
2866:   center   = (xmin + xmax) / 2;
2867:   hscale   = (xmax - xmin) / 2;
2868:   worksize = nsource;
2869:   PetscCall(PetscMalloc4(degree + 1, &bdegrees, nsource + 1, &sourcey, nsource * (degree + 1), &Bsource, worksize, &work));
2870:   PetscCall(PetscMalloc4(nsource, &tau, nsource * (degree + 1), &Bsinv, ntarget + 1, &targety, ntarget * (degree + 1), &Btarget));
2871:   for (i = 0; i <= nsource; i++) sourcey[i] = (sourcex[i] - center) / hscale;
2872:   for (i = 0; i <= degree; i++) bdegrees[i] = i + 1;
2873:   PetscCall(PetscDTLegendreIntegrate(nsource, sourcey, degree + 1, bdegrees, PETSC_TRUE, Bsource));
2874:   PetscCall(PetscDTPseudoInverseQR(nsource, nsource, degree + 1, Bsource, Bsinv, tau, nsource, work));
2875:   for (i = 0; i <= ntarget; i++) targety[i] = (targetx[i] - center) / hscale;
2876:   PetscCall(PetscDTLegendreIntegrate(ntarget, targety, degree + 1, bdegrees, PETSC_FALSE, Btarget));
2877:   for (i = 0; i < ntarget; i++) {
2878:     PetscReal rowsum = 0;
2879:     for (j = 0; j < nsource; j++) {
2880:       PetscReal sum = 0;
2881:       for (k = 0; k < degree + 1; k++) sum += Btarget[i * (degree + 1) + k] * Bsinv[k * nsource + j];
2882:       R[i * nsource + j] = sum;
2883:       rowsum += sum;
2884:     }
2885:     for (j = 0; j < nsource; j++) R[i * nsource + j] /= rowsum; /* normalize each row */
2886:   }
2887:   PetscCall(PetscFree4(bdegrees, sourcey, Bsource, work));
2888:   PetscCall(PetscFree4(tau, Bsinv, targety, Btarget));
2889:   PetscFunctionReturn(PETSC_SUCCESS);
2890: }

2892: /*@C
2893:   PetscGaussLobattoLegendreIntegrate - Compute the L2 integral of a function on the GLL points

2895:   Not Collective

2897:   Input Parameters:
2898: + n       - the number of GLL nodes
2899: . nodes   - the GLL nodes
2900: . weights - the GLL weights
2901: - f       - the function values at the nodes

2903:   Output Parameter:
2904: . in - the value of the integral

2906:   Level: beginner

2908: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`
2909: @*/
2910: PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt n, PetscReal *nodes, PetscReal *weights, const PetscReal *f, PetscReal *in)
2911: {
2912:   PetscInt i;

2914:   PetscFunctionBegin;
2915:   *in = 0.;
2916:   for (i = 0; i < n; i++) *in += f[i] * f[i] * weights[i];
2917:   PetscFunctionReturn(PETSC_SUCCESS);
2918: }

2920: /*@C
2921:   PetscGaussLobattoLegendreElementLaplacianCreate - computes the Laplacian for a single 1d GLL element

2923:   Not Collective

2925:   Input Parameters:
2926: + n       - the number of GLL nodes
2927: . nodes   - the GLL nodes
2928: - weights - the GLL weights

2930:   Output Parameter:
2931: . AA - the stiffness element

2933:   Level: beginner

2935:   Notes:
2936:   Destroy this with `PetscGaussLobattoLegendreElementLaplacianDestroy()`

2938:   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)

2940: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianDestroy()`
2941: @*/
2942: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
2943: {
2944:   PetscReal      **A;
2945:   const PetscReal *gllnodes = nodes;
2946:   const PetscInt   p        = n - 1;
2947:   PetscReal        z0, z1, z2 = -1, x, Lpj, Lpr;
2948:   PetscInt         i, j, nn, r;

2950:   PetscFunctionBegin;
2951:   PetscCall(PetscMalloc1(n, &A));
2952:   PetscCall(PetscMalloc1(n * n, &A[0]));
2953:   for (i = 1; i < n; i++) A[i] = A[i - 1] + n;

2955:   for (j = 1; j < p; j++) {
2956:     x  = gllnodes[j];
2957:     z0 = 1.;
2958:     z1 = x;
2959:     for (nn = 1; nn < p; nn++) {
2960:       z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2961:       z0 = z1;
2962:       z1 = z2;
2963:     }
2964:     Lpj = z2;
2965:     for (r = 1; r < p; r++) {
2966:       if (r == j) {
2967:         A[j][j] = 2. / (3. * (1. - gllnodes[j] * gllnodes[j]) * Lpj * Lpj);
2968:       } else {
2969:         x  = gllnodes[r];
2970:         z0 = 1.;
2971:         z1 = x;
2972:         for (nn = 1; nn < p; nn++) {
2973:           z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2974:           z0 = z1;
2975:           z1 = z2;
2976:         }
2977:         Lpr     = z2;
2978:         A[r][j] = 4. / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * Lpr * (gllnodes[j] - gllnodes[r]) * (gllnodes[j] - gllnodes[r]));
2979:       }
2980:     }
2981:   }
2982:   for (j = 1; j < p + 1; j++) {
2983:     x  = gllnodes[j];
2984:     z0 = 1.;
2985:     z1 = x;
2986:     for (nn = 1; nn < p; nn++) {
2987:       z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2988:       z0 = z1;
2989:       z1 = z2;
2990:     }
2991:     Lpj     = z2;
2992:     A[j][0] = 4. * PetscPowRealInt(-1., p) / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * (1. + gllnodes[j]) * (1. + gllnodes[j]));
2993:     A[0][j] = A[j][0];
2994:   }
2995:   for (j = 0; j < p; j++) {
2996:     x  = gllnodes[j];
2997:     z0 = 1.;
2998:     z1 = x;
2999:     for (nn = 1; nn < p; nn++) {
3000:       z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
3001:       z0 = z1;
3002:       z1 = z2;
3003:     }
3004:     Lpj = z2;

3006:     A[p][j] = 4. / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * (1. - gllnodes[j]) * (1. - gllnodes[j]));
3007:     A[j][p] = A[p][j];
3008:   }
3009:   A[0][0] = 0.5 + (((PetscReal)p) * (((PetscReal)p) + 1.) - 2.) / 6.;
3010:   A[p][p] = A[0][0];
3011:   *AA     = A;
3012:   PetscFunctionReturn(PETSC_SUCCESS);
3013: }

3015: /*@C
3016:   PetscGaussLobattoLegendreElementLaplacianDestroy - frees the Laplacian for a single 1d GLL element created with `PetscGaussLobattoLegendreElementLaplacianCreate()`

3018:   Not Collective

3020:   Input Parameters:
3021: + n       - the number of GLL nodes
3022: . nodes   - the GLL nodes
3023: . weights - the GLL weightss
3024: - AA      - the stiffness element

3026:   Level: beginner

3028: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`
3029: @*/
3030: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3031: {
3032:   PetscFunctionBegin;
3033:   PetscCall(PetscFree((*AA)[0]));
3034:   PetscCall(PetscFree(*AA));
3035:   *AA = NULL;
3036:   PetscFunctionReturn(PETSC_SUCCESS);
3037: }

3039: /*@C
3040:   PetscGaussLobattoLegendreElementGradientCreate - computes the gradient for a single 1d GLL element

3042:   Not Collective

3044:   Input Parameters:
3045: + n       - the number of GLL nodes
3046: . nodes   - the GLL nodes
3047: - weights - the GLL weights

3049:   Output Parameters:
3050: + AA  - the stiffness element
3051: - AAT - the transpose of AA (pass in `NULL` if you do not need this array)

3053:   Level: beginner

3055:   Notes:
3056:   Destroy this with `PetscGaussLobattoLegendreElementGradientDestroy()`

3058:   You can access entries in these arrays with AA[i][j] but in memory it is stored in contiguous memory, row oriented

3060: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianDestroy()`, `PetscGaussLobattoLegendreElementGradientDestroy()`
3061: @*/
3062: PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA, PetscReal ***AAT)
3063: {
3064:   PetscReal      **A, **AT = NULL;
3065:   const PetscReal *gllnodes = nodes;
3066:   const PetscInt   p        = n - 1;
3067:   PetscReal        Li, Lj, d0;
3068:   PetscInt         i, j;

3070:   PetscFunctionBegin;
3071:   PetscCall(PetscMalloc1(n, &A));
3072:   PetscCall(PetscMalloc1(n * n, &A[0]));
3073:   for (i = 1; i < n; i++) A[i] = A[i - 1] + n;

3075:   if (AAT) {
3076:     PetscCall(PetscMalloc1(n, &AT));
3077:     PetscCall(PetscMalloc1(n * n, &AT[0]));
3078:     for (i = 1; i < n; i++) AT[i] = AT[i - 1] + n;
3079:   }

3081:   if (n == 1) A[0][0] = 0.;
3082:   d0 = (PetscReal)p * ((PetscReal)p + 1.) / 4.;
3083:   for (i = 0; i < n; i++) {
3084:     for (j = 0; j < n; j++) {
3085:       A[i][j] = 0.;
3086:       PetscCall(PetscDTComputeJacobi(0., 0., p, gllnodes[i], &Li));
3087:       PetscCall(PetscDTComputeJacobi(0., 0., p, gllnodes[j], &Lj));
3088:       if (i != j) A[i][j] = Li / (Lj * (gllnodes[i] - gllnodes[j]));
3089:       if ((j == i) && (i == 0)) A[i][j] = -d0;
3090:       if (j == i && i == p) A[i][j] = d0;
3091:       if (AT) AT[j][i] = A[i][j];
3092:     }
3093:   }
3094:   if (AAT) *AAT = AT;
3095:   *AA = A;
3096:   PetscFunctionReturn(PETSC_SUCCESS);
3097: }

3099: /*@C
3100:   PetscGaussLobattoLegendreElementGradientDestroy - frees the gradient for a single 1d GLL element obtained with `PetscGaussLobattoLegendreElementGradientCreate()`

3102:   Not Collective

3104:   Input Parameters:
3105: + n       - the number of GLL nodes
3106: . nodes   - the GLL nodes
3107: . weights - the GLL weights
3108: . AA      - the stiffness element
3109: - AAT     - the transpose of the element

3111:   Level: beginner

3113: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`, `PetscGaussLobattoLegendreElementAdvectionCreate()`
3114: @*/
3115: PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA, PetscReal ***AAT)
3116: {
3117:   PetscFunctionBegin;
3118:   PetscCall(PetscFree((*AA)[0]));
3119:   PetscCall(PetscFree(*AA));
3120:   *AA = NULL;
3121:   if (AAT) {
3122:     PetscCall(PetscFree((*AAT)[0]));
3123:     PetscCall(PetscFree(*AAT));
3124:     *AAT = NULL;
3125:   }
3126:   PetscFunctionReturn(PETSC_SUCCESS);
3127: }

3129: /*@C
3130:   PetscGaussLobattoLegendreElementAdvectionCreate - computes the advection operator for a single 1d GLL element

3132:   Not Collective

3134:   Input Parameters:
3135: + n       - the number of GLL nodes
3136: . nodes   - the GLL nodes
3137: - weights - the GLL weightss

3139:   Output Parameter:
3140: . AA - the stiffness element

3142:   Level: beginner

3144:   Notes:
3145:   Destroy this with `PetscGaussLobattoLegendreElementAdvectionDestroy()`

3147:   This is the same as the Gradient operator multiplied by the diagonal mass matrix

3149:   You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented

3151: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`, `PetscGaussLobattoLegendreElementAdvectionDestroy()`
3152: @*/
3153: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3154: {
3155:   PetscReal      **D;
3156:   const PetscReal *gllweights = weights;
3157:   const PetscInt   glln       = n;
3158:   PetscInt         i, j;

3160:   PetscFunctionBegin;
3161:   PetscCall(PetscGaussLobattoLegendreElementGradientCreate(n, nodes, weights, &D, NULL));
3162:   for (i = 0; i < glln; i++) {
3163:     for (j = 0; j < glln; j++) D[i][j] = gllweights[i] * D[i][j];
3164:   }
3165:   *AA = D;
3166:   PetscFunctionReturn(PETSC_SUCCESS);
3167: }

3169: /*@C
3170:   PetscGaussLobattoLegendreElementAdvectionDestroy - frees the advection stiffness for a single 1d GLL element created with `PetscGaussLobattoLegendreElementAdvectionCreate()`

3172:   Not Collective

3174:   Input Parameters:
3175: + n       - the number of GLL nodes
3176: . nodes   - the GLL nodes
3177: . weights - the GLL weights
3178: - AA      - advection

3180:   Level: beginner

3182: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementAdvectionCreate()`
3183: @*/
3184: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3185: {
3186:   PetscFunctionBegin;
3187:   PetscCall(PetscFree((*AA)[0]));
3188:   PetscCall(PetscFree(*AA));
3189:   *AA = NULL;
3190:   PetscFunctionReturn(PETSC_SUCCESS);
3191: }

3193: PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3194: {
3195:   PetscReal      **A;
3196:   const PetscReal *gllweights = weights;
3197:   const PetscInt   glln       = n;
3198:   PetscInt         i, j;

3200:   PetscFunctionBegin;
3201:   PetscCall(PetscMalloc1(glln, &A));
3202:   PetscCall(PetscMalloc1(glln * glln, &A[0]));
3203:   for (i = 1; i < glln; i++) A[i] = A[i - 1] + glln;
3204:   if (glln == 1) A[0][0] = 0.;
3205:   for (i = 0; i < glln; i++) {
3206:     for (j = 0; j < glln; j++) {
3207:       A[i][j] = 0.;
3208:       if (j == i) A[i][j] = gllweights[i];
3209:     }
3210:   }
3211:   *AA = A;
3212:   PetscFunctionReturn(PETSC_SUCCESS);
3213: }

3215: PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
3216: {
3217:   PetscFunctionBegin;
3218:   PetscCall(PetscFree((*AA)[0]));
3219:   PetscCall(PetscFree(*AA));
3220:   *AA = NULL;
3221:   PetscFunctionReturn(PETSC_SUCCESS);
3222: }

3224: /*@
3225:   PetscDTIndexToBary - convert an index into a barycentric coordinate.

3227:   Input Parameters:
3228: + 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)
3229: . sum   - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
3230: - index - the index to convert: should be >= 0 and < Binomial(len - 1 + sum, sum)

3232:   Output Parameter:
3233: . coord - will be filled with the barycentric coordinate

3235:   Level: beginner

3237:   Note:
3238:   The indices map to barycentric coordinates in lexicographic order, where the first index is the
3239:   least significant and the last index is the most significant.

3241: .seealso: `PetscDTBaryToIndex()`
3242: @*/
3243: PetscErrorCode PetscDTIndexToBary(PetscInt len, PetscInt sum, PetscInt index, PetscInt coord[])
3244: {
3245:   PetscInt c, d, s, total, subtotal, nexttotal;

3247:   PetscFunctionBeginHot;
3248:   PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
3249:   PetscCheck(index >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
3250:   if (!len) {
3251:     if (!sum && !index) PetscFunctionReturn(PETSC_SUCCESS);
3252:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
3253:   }
3254:   for (c = 1, total = 1; c <= len; c++) {
3255:     /* total is the number of ways to have a tuple of length c with sum */
3256:     if (index < total) break;
3257:     total = (total * (sum + c)) / c;
3258:   }
3259:   PetscCheck(c <= len, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index out of range");
3260:   for (d = c; d < len; d++) coord[d] = 0;
3261:   for (s = 0, subtotal = 1, nexttotal = 1; c > 0;) {
3262:     /* subtotal is the number of ways to have a tuple of length c with sum s */
3263:     /* nexttotal is the number of ways to have a tuple of length c-1 with sum s */
3264:     if ((index + subtotal) >= total) {
3265:       coord[--c] = sum - s;
3266:       index -= (total - subtotal);
3267:       sum       = s;
3268:       total     = nexttotal;
3269:       subtotal  = 1;
3270:       nexttotal = 1;
3271:       s         = 0;
3272:     } else {
3273:       subtotal  = (subtotal * (c + s)) / (s + 1);
3274:       nexttotal = (nexttotal * (c - 1 + s)) / (s + 1);
3275:       s++;
3276:     }
3277:   }
3278:   PetscFunctionReturn(PETSC_SUCCESS);
3279: }

3281: /*@
3282:   PetscDTBaryToIndex - convert a barycentric coordinate to an index

3284:   Input Parameters:
3285: + 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)
3286: . sum   - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
3287: - coord - a barycentric coordinate with the given length and sum

3289:   Output Parameter:
3290: . index - the unique index for the coordinate, >= 0 and < Binomial(len - 1 + sum, sum)

3292:   Level: beginner

3294:   Note:
3295:   The indices map to barycentric coordinates in lexicographic order, where the first index is the
3296:   least significant and the last index is the most significant.

3298: .seealso: `PetscDTIndexToBary`
3299: @*/
3300: PetscErrorCode PetscDTBaryToIndex(PetscInt len, PetscInt sum, const PetscInt coord[], PetscInt *index)
3301: {
3302:   PetscInt c;
3303:   PetscInt i;
3304:   PetscInt total;

3306:   PetscFunctionBeginHot;
3307:   PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
3308:   if (!len) {
3309:     if (!sum) {
3310:       *index = 0;
3311:       PetscFunctionReturn(PETSC_SUCCESS);
3312:     }
3313:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
3314:   }
3315:   for (c = 1, total = 1; c < len; c++) total = (total * (sum + c)) / c;
3316:   i = total - 1;
3317:   c = len - 1;
3318:   sum -= coord[c];
3319:   while (sum > 0) {
3320:     PetscInt subtotal;
3321:     PetscInt s;

3323:     for (s = 1, subtotal = 1; s < sum; s++) subtotal = (subtotal * (c + s)) / s;
3324:     i -= subtotal;
3325:     sum -= coord[--c];
3326:   }
3327:   *index = i;
3328:   PetscFunctionReturn(PETSC_SUCCESS);
3329: }

3331: /*@
3332:   PetscQuadratureComputePermutations - Compute permutations of quadrature points corresponding to domain orientations

3334:   Input Parameter:
3335: . quad - The `PetscQuadrature`

3337:   Output Parameters:
3338: + Np   - The number of domain orientations
3339: - perm - An array of `IS` permutations, one for ech orientation,

3341:   Level: developer

3343: .seealso: `PetscQuadratureSetCellType()`, `PetscQuadrature`
3344: @*/
3345: PetscErrorCode PetscQuadratureComputePermutations(PetscQuadrature quad, PetscInt *Np, IS *perm[])
3346: {
3347:   DMPolytopeType   ct;
3348:   const PetscReal *xq, *wq;
3349:   PetscInt         dim, qdim, d, Na, o, Nq, q, qp;

3351:   PetscFunctionBegin;
3352:   PetscCall(PetscQuadratureGetData(quad, &qdim, NULL, &Nq, &xq, &wq));
3353:   PetscCall(PetscQuadratureGetCellType(quad, &ct));
3354:   dim = DMPolytopeTypeGetDim(ct);
3355:   Na  = DMPolytopeTypeGetNumArrangments(ct);
3356:   PetscCall(PetscMalloc1(Na, perm));
3357:   if (Np) *Np = Na;
3358:   Na /= 2;
3359:   for (o = -Na; o < Na; ++o) {
3360:     DM        refdm;
3361:     PetscInt *idx;
3362:     PetscReal xi0[3] = {-1., -1., -1.}, v0[3], J[9], detJ, txq[3];
3363:     PetscBool flg;

3365:     PetscCall(DMPlexCreateReferenceCell(PETSC_COMM_SELF, ct, &refdm));
3366:     PetscCall(DMPlexOrientPoint(refdm, 0, o));
3367:     PetscCall(DMPlexComputeCellGeometryFEM(refdm, 0, NULL, v0, J, NULL, &detJ));
3368:     PetscCall(PetscMalloc1(Nq, &idx));
3369:     for (q = 0; q < Nq; ++q) {
3370:       CoordinatesRefToReal(dim, dim, xi0, v0, J, &xq[q * dim], txq);
3371:       for (qp = 0; qp < Nq; ++qp) {
3372:         PetscReal diff = 0.;

3374:         for (d = 0; d < dim; ++d) diff += PetscAbsReal(txq[d] - xq[qp * dim + d]);
3375:         if (diff < PETSC_SMALL) break;
3376:       }
3377:       PetscCheck(qp < Nq, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Transformed quad point %" PetscInt_FMT " does not match another quad point", q);
3378:       idx[q] = qp;
3379:     }
3380:     PetscCall(DMDestroy(&refdm));
3381:     PetscCall(ISCreateGeneral(PETSC_COMM_SELF, Nq, idx, PETSC_OWN_POINTER, &(*perm)[o + Na]));
3382:     PetscCall(ISGetInfo((*perm)[o + Na], IS_PERMUTATION, IS_LOCAL, PETSC_TRUE, &flg));
3383:     PetscCheck(flg, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Ordering for orientation %" PetscInt_FMT " was not a permutation", o);
3384:     PetscCall(ISSetPermutation((*perm)[o + Na]));
3385:   }
3386:   if (!Na) (*perm)[0] = NULL;
3387:   PetscFunctionReturn(PETSC_SUCCESS);
3388: }

3390: /*@
3391:   PetscDTCreateDefaultQuadrature - Create default quadrature for a given cell

3393:   Not collective

3395:   Input Parameters:
3396: + ct     - The integration domain
3397: - qorder - The desired quadrature order

3399:   Output Parameters:
3400: + q  - The cell quadrature
3401: - fq - The face quadrature

3403:   Level: developer

3405: .seealso: `PetscFECreateDefault()`, `PetscDTGaussTensorQuadrature()`, `PetscDTSimplexQuadrature()`, `PetscDTTensorQuadratureCreate()`
3406: @*/
3407: PetscErrorCode PetscDTCreateDefaultQuadrature(DMPolytopeType ct, PetscInt qorder, PetscQuadrature *q, PetscQuadrature *fq)
3408: {
3409:   const PetscInt quadPointsPerEdge = PetscMax(qorder + 1, 1);
3410:   const PetscInt dim               = DMPolytopeTypeGetDim(ct);

3412:   PetscFunctionBegin;
3413:   switch (ct) {
3414:   case DM_POLYTOPE_SEGMENT:
3415:   case DM_POLYTOPE_POINT_PRISM_TENSOR:
3416:   case DM_POLYTOPE_QUADRILATERAL:
3417:   case DM_POLYTOPE_SEG_PRISM_TENSOR:
3418:   case DM_POLYTOPE_HEXAHEDRON:
3419:   case DM_POLYTOPE_QUAD_PRISM_TENSOR:
3420:     PetscCall(PetscDTGaussTensorQuadrature(dim, 1, quadPointsPerEdge, -1.0, 1.0, q));
3421:     PetscCall(PetscDTGaussTensorQuadrature(dim - 1, 1, quadPointsPerEdge, -1.0, 1.0, fq));
3422:     break;
3423:   case DM_POLYTOPE_TRIANGLE:
3424:   case DM_POLYTOPE_TETRAHEDRON:
3425:     PetscCall(PetscDTSimplexQuadrature(dim, 2 * qorder, PETSCDTSIMPLEXQUAD_DEFAULT, q));
3426:     PetscCall(PetscDTSimplexQuadrature(dim - 1, 2 * qorder, PETSCDTSIMPLEXQUAD_DEFAULT, fq));
3427:     break;
3428:   case DM_POLYTOPE_TRI_PRISM:
3429:   case DM_POLYTOPE_TRI_PRISM_TENSOR: {
3430:     PetscQuadrature q1, q2;

3432:     // TODO: this should be able to use symmetric rules, but doing so causes tests to fail
3433:     PetscCall(PetscDTSimplexQuadrature(2, 2 * qorder, PETSCDTSIMPLEXQUAD_CONIC, &q1));
3434:     PetscCall(PetscDTGaussTensorQuadrature(1, 1, quadPointsPerEdge, -1.0, 1.0, &q2));
3435:     PetscCall(PetscDTTensorQuadratureCreate(q1, q2, q));
3436:     PetscCall(PetscQuadratureDestroy(&q2));
3437:     *fq = q1;
3438:     /* TODO Need separate quadratures for each face */
3439:   } break;
3440:   default:
3441:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "No quadrature for celltype %s", DMPolytopeTypes[PetscMin(ct, DM_POLYTOPE_UNKNOWN)]);
3442:   }
3443:   PetscFunctionReturn(PETSC_SUCCESS);
3444: }