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 <petscviewer.h>
8: #include <petscdmplex.h>
9: #include <petscdmshell.h>
11: #if defined(PETSC_HAVE_MPFR)
12: #include <mpfr.h>
13: #endif
15: const char *const PetscDTNodeTypes_shifted[] = {"default", "gaussjacobi", "equispaced", "tanhsinh", "PETSCDTNODES_", NULL};
16: const char *const *const PetscDTNodeTypes = PetscDTNodeTypes_shifted + 1;
18: const char *const PetscDTSimplexQuadratureTypes_shifted[] = {"default", "conic", "minsym", "PETSCDTSIMPLEXQUAD_", NULL};
19: const char *const *const PetscDTSimplexQuadratureTypes = PetscDTSimplexQuadratureTypes_shifted + 1;
21: static PetscBool GolubWelschCite = PETSC_FALSE;
22: const char GolubWelschCitation[] = "@article{GolubWelsch1969,\n"
23: " author = {Golub and Welsch},\n"
24: " title = {Calculation of Quadrature Rules},\n"
25: " journal = {Math. Comp.},\n"
26: " volume = {23},\n"
27: " number = {106},\n"
28: " pages = {221--230},\n"
29: " year = {1969}\n}\n";
31: /* Numerical tests in src/dm/dt/tests/ex1.c show that when computing the nodes and weights of Gauss-Jacobi
32: quadrature rules:
34: - in double precision, Newton's method and Golub & Welsch both work for moderate degrees (< 100),
35: - in single precision, Newton's method starts producing incorrect roots around n = 15, but
36: the weights from Golub & Welsch become a problem before then: they produces errors
37: in computing the Jacobi-polynomial Gram matrix around n = 6.
39: So we default to Newton's method (required fewer dependencies) */
40: PetscBool PetscDTGaussQuadratureNewton_Internal = PETSC_TRUE;
42: PetscClassId PETSCQUADRATURE_CLASSID = 0;
44: /*@
45: PetscQuadratureCreate - Create a `PetscQuadrature` object
47: Collective
49: Input Parameter:
50: . comm - The communicator for the `PetscQuadrature` object
52: Output Parameter:
53: . q - The PetscQuadrature object
55: Level: beginner
57: .seealso: `PetscQuadrature`, `Petscquadraturedestroy()`, `PetscQuadratureGetData()`
58: @*/
59: PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
60: {
61: PetscFunctionBegin;
63: PetscCall(DMInitializePackage());
64: PetscCall(PetscHeaderCreate(*q, PETSCQUADRATURE_CLASSID, "PetscQuadrature", "Quadrature", "DT", comm, PetscQuadratureDestroy, PetscQuadratureView));
65: (*q)->dim = -1;
66: (*q)->Nc = 1;
67: (*q)->order = -1;
68: (*q)->numPoints = 0;
69: (*q)->points = NULL;
70: (*q)->weights = NULL;
71: PetscFunctionReturn(PETSC_SUCCESS);
72: }
74: /*@
75: PetscQuadratureDuplicate - Create a deep copy of the `PetscQuadrature` object
77: Collective on q
79: Input Parameter:
80: . q - The `PetscQuadrature` object
82: Output Parameter:
83: . r - The new `PetscQuadrature` object
85: Level: beginner
87: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureDestroy()`, `PetscQuadratureGetData()`
88: @*/
89: PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
90: {
91: PetscInt order, dim, Nc, Nq;
92: const PetscReal *points, *weights;
93: PetscReal *p, *w;
95: PetscFunctionBegin;
97: PetscCall(PetscQuadratureCreate(PetscObjectComm((PetscObject)q), r));
98: PetscCall(PetscQuadratureGetOrder(q, &order));
99: PetscCall(PetscQuadratureSetOrder(*r, order));
100: PetscCall(PetscQuadratureGetData(q, &dim, &Nc, &Nq, &points, &weights));
101: PetscCall(PetscMalloc1(Nq * dim, &p));
102: PetscCall(PetscMalloc1(Nq * Nc, &w));
103: PetscCall(PetscArraycpy(p, points, Nq * dim));
104: PetscCall(PetscArraycpy(w, weights, Nc * Nq));
105: PetscCall(PetscQuadratureSetData(*r, dim, Nc, Nq, p, w));
106: PetscFunctionReturn(PETSC_SUCCESS);
107: }
109: /*@
110: PetscQuadratureDestroy - Destroys a `PetscQuadrature` object
112: Collective on q
114: Input Parameter:
115: . q - The `PetscQuadrature` object
117: Level: beginner
119: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureGetData()`
120: @*/
121: PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
122: {
123: PetscFunctionBegin;
124: if (!*q) PetscFunctionReturn(PETSC_SUCCESS);
126: if (--((PetscObject)(*q))->refct > 0) {
127: *q = NULL;
128: PetscFunctionReturn(PETSC_SUCCESS);
129: }
130: PetscCall(PetscFree((*q)->points));
131: PetscCall(PetscFree((*q)->weights));
132: PetscCall(PetscHeaderDestroy(q));
133: PetscFunctionReturn(PETSC_SUCCESS);
134: }
136: /*@
137: PetscQuadratureGetOrder - Return the order of the method in the `PetscQuadrature`
139: Not collective
141: Input Parameter:
142: . q - The `PetscQuadrature` object
144: Output Parameter:
145: . order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated
147: Level: intermediate
149: .seealso: `PetscQuadrature`, `PetscQuadratureSetOrder()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
150: @*/
151: PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature q, PetscInt *order)
152: {
153: PetscFunctionBegin;
156: *order = q->order;
157: PetscFunctionReturn(PETSC_SUCCESS);
158: }
160: /*@
161: PetscQuadratureSetOrder - Set the order of the method in the `PetscQuadrature`
163: Not collective
165: Input Parameters:
166: + q - The `PetscQuadrature` object
167: - order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated
169: Level: intermediate
171: .seealso: `PetscQuadrature`, `PetscQuadratureGetOrder()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
172: @*/
173: PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature q, PetscInt order)
174: {
175: PetscFunctionBegin;
177: q->order = order;
178: PetscFunctionReturn(PETSC_SUCCESS);
179: }
181: /*@
182: PetscQuadratureGetNumComponents - Return the number of components for functions to be integrated
184: Not collective
186: Input Parameter:
187: . q - The `PetscQuadrature` object
189: Output Parameter:
190: . Nc - The number of components
192: Note:
193: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.
195: Level: intermediate
197: .seealso: `PetscQuadrature`, `PetscQuadratureSetNumComponents()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
198: @*/
199: PetscErrorCode PetscQuadratureGetNumComponents(PetscQuadrature q, PetscInt *Nc)
200: {
201: PetscFunctionBegin;
204: *Nc = q->Nc;
205: PetscFunctionReturn(PETSC_SUCCESS);
206: }
208: /*@
209: PetscQuadratureSetNumComponents - Return the number of components for functions to be integrated
211: Not collective
213: Input Parameters:
214: + q - The PetscQuadrature object
215: - Nc - The number of components
217: Note:
218: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.
220: Level: intermediate
222: .seealso: `PetscQuadrature`, `PetscQuadratureGetNumComponents()`, `PetscQuadratureGetData()`, `PetscQuadratureSetData()`
223: @*/
224: PetscErrorCode PetscQuadratureSetNumComponents(PetscQuadrature q, PetscInt Nc)
225: {
226: PetscFunctionBegin;
228: q->Nc = Nc;
229: PetscFunctionReturn(PETSC_SUCCESS);
230: }
232: /*@C
233: PetscQuadratureGetData - Returns the data defining the `PetscQuadrature`
235: Not collective
237: Input Parameter:
238: . q - The `PetscQuadrature` object
240: Output Parameters:
241: + dim - The spatial dimension
242: . Nc - The number of components
243: . npoints - The number of quadrature points
244: . points - The coordinates of each quadrature point
245: - weights - The weight of each quadrature point
247: Level: intermediate
249: Fortran Note:
250: From Fortran you must call `PetscQuadratureRestoreData()` when you are done with the data
252: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureSetData()`
253: @*/
254: PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *Nc, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
255: {
256: PetscFunctionBegin;
258: if (dim) {
260: *dim = q->dim;
261: }
262: if (Nc) {
264: *Nc = q->Nc;
265: }
266: if (npoints) {
268: *npoints = q->numPoints;
269: }
270: if (points) {
272: *points = q->points;
273: }
274: if (weights) {
276: *weights = q->weights;
277: }
278: PetscFunctionReturn(PETSC_SUCCESS);
279: }
281: /*@
282: PetscQuadratureEqual - determine whether two quadratures are equivalent
284: Input Parameters:
285: + A - A `PetscQuadrature` object
286: - B - Another `PetscQuadrature` object
288: Output Parameters:
289: . equal - `PETSC_TRUE` if the quadratures are the same
291: Level: intermediate
293: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`
294: @*/
295: PetscErrorCode PetscQuadratureEqual(PetscQuadrature A, PetscQuadrature B, PetscBool *equal)
296: {
297: PetscFunctionBegin;
301: *equal = PETSC_FALSE;
302: if (A->dim != B->dim || A->Nc != B->Nc || A->order != B->order || A->numPoints != B->numPoints) PetscFunctionReturn(PETSC_SUCCESS);
303: for (PetscInt i = 0; i < A->numPoints * A->dim; i++) {
304: if (PetscAbsReal(A->points[i] - B->points[i]) > PETSC_SMALL) PetscFunctionReturn(PETSC_SUCCESS);
305: }
306: if (!A->weights && !B->weights) {
307: *equal = PETSC_TRUE;
308: PetscFunctionReturn(PETSC_SUCCESS);
309: }
310: if (A->weights && B->weights) {
311: for (PetscInt i = 0; i < A->numPoints; i++) {
312: if (PetscAbsReal(A->weights[i] - B->weights[i]) > PETSC_SMALL) PetscFunctionReturn(PETSC_SUCCESS);
313: }
314: *equal = PETSC_TRUE;
315: }
316: PetscFunctionReturn(PETSC_SUCCESS);
317: }
319: static PetscErrorCode PetscDTJacobianInverse_Internal(PetscInt m, PetscInt n, const PetscReal J[], PetscReal Jinv[])
320: {
321: PetscScalar *Js, *Jinvs;
322: PetscInt i, j, k;
323: PetscBLASInt bm, bn, info;
325: PetscFunctionBegin;
326: if (!m || !n) PetscFunctionReturn(PETSC_SUCCESS);
327: PetscCall(PetscBLASIntCast(m, &bm));
328: PetscCall(PetscBLASIntCast(n, &bn));
329: #if defined(PETSC_USE_COMPLEX)
330: PetscCall(PetscMalloc2(m * n, &Js, m * n, &Jinvs));
331: for (i = 0; i < m * n; i++) Js[i] = J[i];
332: #else
333: Js = (PetscReal *)J;
334: Jinvs = Jinv;
335: #endif
336: if (m == n) {
337: PetscBLASInt *pivots;
338: PetscScalar *W;
340: PetscCall(PetscMalloc2(m, &pivots, m, &W));
342: PetscCall(PetscArraycpy(Jinvs, Js, m * m));
343: PetscCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bm, &bm, Jinvs, &bm, pivots, &info));
344: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetrf %" PetscInt_FMT, (PetscInt)info);
345: PetscCallBLAS("LAPACKgetri", LAPACKgetri_(&bm, Jinvs, &bm, pivots, W, &bm, &info));
346: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetri %" PetscInt_FMT, (PetscInt)info);
347: PetscCall(PetscFree2(pivots, W));
348: } else if (m < n) {
349: PetscScalar *JJT;
350: PetscBLASInt *pivots;
351: PetscScalar *W;
353: PetscCall(PetscMalloc1(m * m, &JJT));
354: PetscCall(PetscMalloc2(m, &pivots, m, &W));
355: for (i = 0; i < m; i++) {
356: for (j = 0; j < m; j++) {
357: PetscScalar val = 0.;
359: for (k = 0; k < n; k++) val += Js[i * n + k] * Js[j * n + k];
360: JJT[i * m + j] = val;
361: }
362: }
364: PetscCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bm, &bm, JJT, &bm, pivots, &info));
365: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetrf %" PetscInt_FMT, (PetscInt)info);
366: PetscCallBLAS("LAPACKgetri", LAPACKgetri_(&bm, JJT, &bm, pivots, W, &bm, &info));
367: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetri %" PetscInt_FMT, (PetscInt)info);
368: for (i = 0; i < n; i++) {
369: for (j = 0; j < m; j++) {
370: PetscScalar val = 0.;
372: for (k = 0; k < m; k++) val += Js[k * n + i] * JJT[k * m + j];
373: Jinvs[i * m + j] = val;
374: }
375: }
376: PetscCall(PetscFree2(pivots, W));
377: PetscCall(PetscFree(JJT));
378: } else {
379: PetscScalar *JTJ;
380: PetscBLASInt *pivots;
381: PetscScalar *W;
383: PetscCall(PetscMalloc1(n * n, &JTJ));
384: PetscCall(PetscMalloc2(n, &pivots, n, &W));
385: for (i = 0; i < n; i++) {
386: for (j = 0; j < n; j++) {
387: PetscScalar val = 0.;
389: for (k = 0; k < m; k++) val += Js[k * n + i] * Js[k * n + j];
390: JTJ[i * n + j] = val;
391: }
392: }
394: PetscCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bn, &bn, JTJ, &bn, pivots, &info));
395: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetrf %" PetscInt_FMT, (PetscInt)info);
396: PetscCallBLAS("LAPACKgetri", LAPACKgetri_(&bn, JTJ, &bn, pivots, W, &bn, &info));
397: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "Error returned from LAPACKgetri %" PetscInt_FMT, (PetscInt)info);
398: for (i = 0; i < n; i++) {
399: for (j = 0; j < m; j++) {
400: PetscScalar val = 0.;
402: for (k = 0; k < n; k++) val += JTJ[i * n + k] * Js[j * n + k];
403: Jinvs[i * m + j] = val;
404: }
405: }
406: PetscCall(PetscFree2(pivots, W));
407: PetscCall(PetscFree(JTJ));
408: }
409: #if defined(PETSC_USE_COMPLEX)
410: for (i = 0; i < m * n; i++) Jinv[i] = PetscRealPart(Jinvs[i]);
411: PetscCall(PetscFree2(Js, Jinvs));
412: #endif
413: PetscFunctionReturn(PETSC_SUCCESS);
414: }
416: /*@
417: PetscQuadraturePushForward - Push forward a quadrature functional under an affine transformation.
419: Collecive on `PetscQuadrature`
421: Input Parameters:
422: + q - the quadrature functional
423: . imageDim - the dimension of the image of the transformation
424: . origin - a point in the original space
425: . originImage - the image of the origin under the transformation
426: . J - the Jacobian of the image: an [imageDim x dim] matrix in row major order
427: - 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]
429: Output Parameters:
430: . 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.
432: Level: intermediate
434: Note:
435: 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.
437: .seealso: `PetscQuadrature`, `PetscDTAltVPullback()`, `PetscDTAltVPullbackMatrix()`
438: @*/
439: PetscErrorCode PetscQuadraturePushForward(PetscQuadrature q, PetscInt imageDim, const PetscReal origin[], const PetscReal originImage[], const PetscReal J[], PetscInt formDegree, PetscQuadrature *Jinvstarq)
440: {
441: PetscInt dim, Nc, imageNc, formSize, Ncopies, imageFormSize, Npoints, pt, i, j, c;
442: const PetscReal *points;
443: const PetscReal *weights;
444: PetscReal *imagePoints, *imageWeights;
445: PetscReal *Jinv;
446: PetscReal *Jinvstar;
448: PetscFunctionBegin;
450: PetscCheck(imageDim >= PetscAbsInt(formDegree), PetscObjectComm((PetscObject)q), PETSC_ERR_ARG_INCOMP, "Cannot represent a %" PetscInt_FMT "-form in %" PetscInt_FMT " dimensions", PetscAbsInt(formDegree), imageDim);
451: PetscCall(PetscQuadratureGetData(q, &dim, &Nc, &Npoints, &points, &weights));
452: PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &formSize));
453: 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);
454: Ncopies = Nc / formSize;
455: PetscCall(PetscDTBinomialInt(imageDim, PetscAbsInt(formDegree), &imageFormSize));
456: imageNc = Ncopies * imageFormSize;
457: PetscCall(PetscMalloc1(Npoints * imageDim, &imagePoints));
458: PetscCall(PetscMalloc1(Npoints * imageNc, &imageWeights));
459: PetscCall(PetscMalloc2(imageDim * dim, &Jinv, formSize * imageFormSize, &Jinvstar));
460: PetscCall(PetscDTJacobianInverse_Internal(imageDim, dim, J, Jinv));
461: PetscCall(PetscDTAltVPullbackMatrix(imageDim, dim, Jinv, formDegree, Jinvstar));
462: for (pt = 0; pt < Npoints; pt++) {
463: const PetscReal *point = &points[pt * dim];
464: PetscReal *imagePoint = &imagePoints[pt * imageDim];
466: for (i = 0; i < imageDim; i++) {
467: PetscReal val = originImage[i];
469: for (j = 0; j < dim; j++) val += J[i * dim + j] * (point[j] - origin[j]);
470: imagePoint[i] = val;
471: }
472: for (c = 0; c < Ncopies; c++) {
473: const PetscReal *form = &weights[pt * Nc + c * formSize];
474: PetscReal *imageForm = &imageWeights[pt * imageNc + c * imageFormSize];
476: for (i = 0; i < imageFormSize; i++) {
477: PetscReal val = 0.;
479: for (j = 0; j < formSize; j++) val += Jinvstar[i * formSize + j] * form[j];
480: imageForm[i] = val;
481: }
482: }
483: }
484: PetscCall(PetscQuadratureCreate(PetscObjectComm((PetscObject)q), Jinvstarq));
485: PetscCall(PetscQuadratureSetData(*Jinvstarq, imageDim, imageNc, Npoints, imagePoints, imageWeights));
486: PetscCall(PetscFree2(Jinv, Jinvstar));
487: PetscFunctionReturn(PETSC_SUCCESS);
488: }
490: /*@C
491: PetscQuadratureSetData - Sets the data defining the quadrature
493: Not collective
495: Input Parameters:
496: + q - The `PetscQuadrature` object
497: . dim - The spatial dimension
498: . Nc - The number of components
499: . npoints - The number of quadrature points
500: . points - The coordinates of each quadrature point
501: - weights - The weight of each quadrature point
503: Level: intermediate
505: Note:
506: This routine owns the references to points and weights, so they must be allocated using `PetscMalloc()` and the user should not free them.
508: .seealso: `PetscQuadrature`, `PetscQuadratureCreate()`, `PetscQuadratureGetData()`
509: @*/
510: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt Nc, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
511: {
512: PetscFunctionBegin;
514: if (dim >= 0) q->dim = dim;
515: if (Nc >= 0) q->Nc = Nc;
516: if (npoints >= 0) q->numPoints = npoints;
517: if (points) {
519: q->points = points;
520: }
521: if (weights) {
523: q->weights = weights;
524: }
525: PetscFunctionReturn(PETSC_SUCCESS);
526: }
528: static PetscErrorCode PetscQuadratureView_Ascii(PetscQuadrature quad, PetscViewer v)
529: {
530: PetscInt q, d, c;
531: PetscViewerFormat format;
533: PetscFunctionBegin;
534: if (quad->Nc > 1) PetscCall(PetscViewerASCIIPrintf(v, "Quadrature of order %" PetscInt_FMT " on %" PetscInt_FMT " points (dim %" PetscInt_FMT ") with %" PetscInt_FMT " components\n", quad->order, quad->numPoints, quad->dim, quad->Nc));
535: else PetscCall(PetscViewerASCIIPrintf(v, "Quadrature of order %" PetscInt_FMT " on %" PetscInt_FMT " points (dim %" PetscInt_FMT ")\n", quad->order, quad->numPoints, quad->dim));
536: PetscCall(PetscViewerGetFormat(v, &format));
537: if (format != PETSC_VIEWER_ASCII_INFO_DETAIL) PetscFunctionReturn(PETSC_SUCCESS);
538: for (q = 0; q < quad->numPoints; ++q) {
539: PetscCall(PetscViewerASCIIPrintf(v, "p%" PetscInt_FMT " (", q));
540: PetscCall(PetscViewerASCIIUseTabs(v, PETSC_FALSE));
541: for (d = 0; d < quad->dim; ++d) {
542: if (d) PetscCall(PetscViewerASCIIPrintf(v, ", "));
543: PetscCall(PetscViewerASCIIPrintf(v, "%+g", (double)quad->points[q * quad->dim + d]));
544: }
545: PetscCall(PetscViewerASCIIPrintf(v, ") "));
546: if (quad->Nc > 1) PetscCall(PetscViewerASCIIPrintf(v, "w%" PetscInt_FMT " (", q));
547: for (c = 0; c < quad->Nc; ++c) {
548: if (c) PetscCall(PetscViewerASCIIPrintf(v, ", "));
549: PetscCall(PetscViewerASCIIPrintf(v, "%+g", (double)quad->weights[q * quad->Nc + c]));
550: }
551: if (quad->Nc > 1) PetscCall(PetscViewerASCIIPrintf(v, ")"));
552: PetscCall(PetscViewerASCIIPrintf(v, "\n"));
553: PetscCall(PetscViewerASCIIUseTabs(v, PETSC_TRUE));
554: }
555: PetscFunctionReturn(PETSC_SUCCESS);
556: }
558: /*@C
559: PetscQuadratureView - View a `PetscQuadrature` object
561: Collective on quad
563: Input Parameters:
564: + quad - The `PetscQuadrature` object
565: - viewer - The `PetscViewer` object
567: Level: beginner
569: .seealso: `PetscQuadrature`, `PetscViewer`, `PetscQuadratureCreate()`, `PetscQuadratureGetData()`
570: @*/
571: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
572: {
573: PetscBool iascii;
575: PetscFunctionBegin;
578: if (!viewer) PetscCall(PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject)quad), &viewer));
579: PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
580: PetscCall(PetscViewerASCIIPushTab(viewer));
581: if (iascii) PetscCall(PetscQuadratureView_Ascii(quad, viewer));
582: PetscCall(PetscViewerASCIIPopTab(viewer));
583: PetscFunctionReturn(PETSC_SUCCESS);
584: }
586: /*@C
587: PetscQuadratureExpandComposite - Return a quadrature over the composite element, which has the original quadrature in each subelement
589: Not collective
591: Input Parameters:
592: + q - The original `PetscQuadrature`
593: . numSubelements - The number of subelements the original element is divided into
594: . v0 - An array of the initial points for each subelement
595: - jac - An array of the Jacobian mappings from the reference to each subelement
597: Output Parameters:
598: . dim - The dimension
600: Note:
601: Together v0 and jac define an affine mapping from the original reference element to each subelement
603: Fortran Note:
604: Not available from Fortran
606: Level: intermediate
608: .seealso: `PetscQuadrature`, `PetscFECreate()`, `PetscSpaceGetDimension()`, `PetscDualSpaceGetDimension()`
609: @*/
610: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
611: {
612: const PetscReal *points, *weights;
613: PetscReal *pointsRef, *weightsRef;
614: PetscInt dim, Nc, order, npoints, npointsRef, c, p, cp, d, e;
616: PetscFunctionBegin;
621: PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, qref));
622: PetscCall(PetscQuadratureGetOrder(q, &order));
623: PetscCall(PetscQuadratureGetData(q, &dim, &Nc, &npoints, &points, &weights));
624: npointsRef = npoints * numSubelements;
625: PetscCall(PetscMalloc1(npointsRef * dim, &pointsRef));
626: PetscCall(PetscMalloc1(npointsRef * Nc, &weightsRef));
627: for (c = 0; c < numSubelements; ++c) {
628: for (p = 0; p < npoints; ++p) {
629: for (d = 0; d < dim; ++d) {
630: pointsRef[(c * npoints + p) * dim + d] = v0[c * dim + d];
631: for (e = 0; e < dim; ++e) pointsRef[(c * npoints + p) * dim + d] += jac[(c * dim + d) * dim + e] * (points[p * dim + e] + 1.0);
632: }
633: /* Could also use detJ here */
634: for (cp = 0; cp < Nc; ++cp) weightsRef[(c * npoints + p) * Nc + cp] = weights[p * Nc + cp] / numSubelements;
635: }
636: }
637: PetscCall(PetscQuadratureSetOrder(*qref, order));
638: PetscCall(PetscQuadratureSetData(*qref, dim, Nc, npointsRef, pointsRef, weightsRef));
639: PetscFunctionReturn(PETSC_SUCCESS);
640: }
642: /* Compute the coefficients for the Jacobi polynomial recurrence,
643: *
644: * J^{a,b}_n(x) = (cnm1 + cnm1x * x) * J^{a,b}_{n-1}(x) - cnm2 * J^{a,b}_{n-2}(x).
645: */
646: #define PetscDTJacobiRecurrence_Internal(n, a, b, cnm1, cnm1x, cnm2) \
647: do { \
648: PetscReal _a = (a); \
649: PetscReal _b = (b); \
650: PetscReal _n = (n); \
651: if (n == 1) { \
652: (cnm1) = (_a - _b) * 0.5; \
653: (cnm1x) = (_a + _b + 2.) * 0.5; \
654: (cnm2) = 0.; \
655: } else { \
656: PetscReal _2n = _n + _n; \
657: PetscReal _d = (_2n * (_n + _a + _b) * (_2n + _a + _b - 2)); \
658: PetscReal _n1 = (_2n + _a + _b - 1.) * (_a * _a - _b * _b); \
659: PetscReal _n1x = (_2n + _a + _b - 1.) * (_2n + _a + _b) * (_2n + _a + _b - 2); \
660: PetscReal _n2 = 2. * ((_n + _a - 1.) * (_n + _b - 1.) * (_2n + _a + _b)); \
661: (cnm1) = _n1 / _d; \
662: (cnm1x) = _n1x / _d; \
663: (cnm2) = _n2 / _d; \
664: } \
665: } while (0)
667: /*@
668: PetscDTJacobiNorm - Compute the weighted L2 norm of a Jacobi polynomial.
670: $\| P^{\alpha,\beta}_n \|_{\alpha,\beta}^2 = \int_{-1}^1 (1 + x)^{\alpha} (1 - x)^{\beta} P^{\alpha,\beta}_n (x)^2 dx.$
672: Input Parameters:
673: - alpha - the left exponent > -1
674: . beta - the right exponent > -1
675: + n - the polynomial degree
677: Output Parameter:
678: . norm - the weighted L2 norm
680: Level: beginner
682: .seealso: `PetscQuadrature`, `PetscDTJacobiEval()`
683: @*/
684: PetscErrorCode PetscDTJacobiNorm(PetscReal alpha, PetscReal beta, PetscInt n, PetscReal *norm)
685: {
686: PetscReal twoab1;
687: PetscReal gr;
689: PetscFunctionBegin;
690: PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Exponent alpha %g <= -1. invalid", (double)alpha);
691: PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Exponent beta %g <= -1. invalid", (double)beta);
692: PetscCheck(n >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "n %" PetscInt_FMT " < 0 invalid", n);
693: twoab1 = PetscPowReal(2., alpha + beta + 1.);
694: #if defined(PETSC_HAVE_LGAMMA)
695: if (!n) {
696: gr = PetscExpReal(PetscLGamma(alpha + 1.) + PetscLGamma(beta + 1.) - PetscLGamma(alpha + beta + 2.));
697: } else {
698: gr = PetscExpReal(PetscLGamma(n + alpha + 1.) + PetscLGamma(n + beta + 1.) - (PetscLGamma(n + 1.) + PetscLGamma(n + alpha + beta + 1.))) / (n + n + alpha + beta + 1.);
699: }
700: #else
701: {
702: PetscInt alphai = (PetscInt)alpha;
703: PetscInt betai = (PetscInt)beta;
704: PetscInt i;
706: gr = n ? (1. / (n + n + alpha + beta + 1.)) : 1.;
707: if ((PetscReal)alphai == alpha) {
708: if (!n) {
709: for (i = 0; i < alphai; i++) gr *= (i + 1.) / (beta + i + 1.);
710: gr /= (alpha + beta + 1.);
711: } else {
712: for (i = 0; i < alphai; i++) gr *= (n + i + 1.) / (n + beta + i + 1.);
713: }
714: } else if ((PetscReal)betai == beta) {
715: if (!n) {
716: for (i = 0; i < betai; i++) gr *= (i + 1.) / (alpha + i + 2.);
717: gr /= (alpha + beta + 1.);
718: } else {
719: for (i = 0; i < betai; i++) gr *= (n + i + 1.) / (n + alpha + i + 1.);
720: }
721: } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "lgamma() - math routine is unavailable.");
722: }
723: #endif
724: *norm = PetscSqrtReal(twoab1 * gr);
725: PetscFunctionReturn(PETSC_SUCCESS);
726: }
728: static PetscErrorCode PetscDTJacobiEval_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscInt k, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *p)
729: {
730: PetscReal ak, bk;
731: PetscReal abk1;
732: PetscInt i, l, maxdegree;
734: PetscFunctionBegin;
735: maxdegree = degrees[ndegree - 1] - k;
736: ak = a + k;
737: bk = b + k;
738: abk1 = a + b + k + 1.;
739: if (maxdegree < 0) {
740: for (i = 0; i < npoints; i++)
741: for (l = 0; l < ndegree; l++) p[i * ndegree + l] = 0.;
742: PetscFunctionReturn(PETSC_SUCCESS);
743: }
744: for (i = 0; i < npoints; i++) {
745: PetscReal pm1, pm2, x;
746: PetscReal cnm1, cnm1x, cnm2;
747: PetscInt j, m;
749: x = points[i];
750: pm2 = 1.;
751: PetscDTJacobiRecurrence_Internal(1, ak, bk, cnm1, cnm1x, cnm2);
752: pm1 = (cnm1 + cnm1x * x);
753: l = 0;
754: while (l < ndegree && degrees[l] - k < 0) p[l++] = 0.;
755: while (l < ndegree && degrees[l] - k == 0) {
756: p[l] = pm2;
757: for (m = 0; m < k; m++) p[l] *= (abk1 + m) * 0.5;
758: l++;
759: }
760: while (l < ndegree && degrees[l] - k == 1) {
761: p[l] = pm1;
762: for (m = 0; m < k; m++) p[l] *= (abk1 + 1 + m) * 0.5;
763: l++;
764: }
765: for (j = 2; j <= maxdegree; j++) {
766: PetscReal pp;
768: PetscDTJacobiRecurrence_Internal(j, ak, bk, cnm1, cnm1x, cnm2);
769: pp = (cnm1 + cnm1x * x) * pm1 - cnm2 * pm2;
770: pm2 = pm1;
771: pm1 = pp;
772: while (l < ndegree && degrees[l] - k == j) {
773: p[l] = pp;
774: for (m = 0; m < k; m++) p[l] *= (abk1 + j + m) * 0.5;
775: l++;
776: }
777: }
778: p += ndegree;
779: }
780: PetscFunctionReturn(PETSC_SUCCESS);
781: }
783: /*@
784: PetscDTJacobiEvalJet - Evaluate the jet (function and derivatives) of the Jacobi polynomials basis up to a given degree.
785: The Jacobi polynomials with indices $\alpha$ and $\beta$ are orthogonal with respect to the weighted inner product
786: $\langle f, g \rangle = \int_{-1}^1 (1+x)^{\alpha} (1-x)^{\beta} f(x) g(x) dx$.
788: Input Parameters:
789: + alpha - the left exponent of the weight
790: . beta - the right exponetn of the weight
791: . npoints - the number of points to evaluate the polynomials at
792: . points - [npoints] array of point coordinates
793: . degree - the maximm degree polynomial space to evaluate, (degree + 1) will be evaluated total.
794: - k - the maximum derivative to evaluate in the jet, (k + 1) will be evaluated total.
796: Output Parameters:
797: - p - an array containing the evaluations of the Jacobi polynomials's jets on the points. the size is (degree + 1) x
798: (k + 1) x npoints, which also describes the order of the dimensions of this three-dimensional array: the first
799: (slowest varying) dimension is polynomial degree; the second dimension is derivative order; the third (fastest
800: varying) dimension is the index of the evaluation point.
802: Level: advanced
804: .seealso: `PetscDTJacobiEval()`, `PetscDTPKDEvalJet()`
805: @*/
806: PetscErrorCode PetscDTJacobiEvalJet(PetscReal alpha, PetscReal beta, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
807: {
808: PetscInt i, j, l;
809: PetscInt *degrees;
810: PetscReal *psingle;
812: PetscFunctionBegin;
813: if (degree == 0) {
814: PetscInt zero = 0;
816: for (i = 0; i <= k; i++) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, 1, &zero, &p[i * npoints]));
817: PetscFunctionReturn(PETSC_SUCCESS);
818: }
819: PetscCall(PetscMalloc1(degree + 1, °rees));
820: PetscCall(PetscMalloc1((degree + 1) * npoints, &psingle));
821: for (i = 0; i <= degree; i++) degrees[i] = i;
822: for (i = 0; i <= k; i++) {
823: PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, degree + 1, degrees, psingle));
824: for (j = 0; j <= degree; j++) {
825: for (l = 0; l < npoints; l++) p[(j * (k + 1) + i) * npoints + l] = psingle[l * (degree + 1) + j];
826: }
827: }
828: PetscCall(PetscFree(psingle));
829: PetscCall(PetscFree(degrees));
830: PetscFunctionReturn(PETSC_SUCCESS);
831: }
833: /*@
834: PetscDTJacobiEval - evaluate Jacobi polynomials for the weight function $(1.+x)^{\alpha} (1.-x)^{\beta}$ at a set of points
835: at points
837: Not Collective
839: Input Parameters:
840: + npoints - number of spatial points to evaluate at
841: . alpha - the left exponent > -1
842: . beta - the right exponent > -1
843: . points - array of locations to evaluate at
844: . ndegree - number of basis degrees to evaluate
845: - degrees - sorted array of degrees to evaluate
847: Output Parameters:
848: + B - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension npoints*ndegrees, allocated by caller) (or NULL)
849: . D - row-oriented derivative evaluation matrix (or NULL)
850: - D2 - row-oriented second derivative evaluation matrix (or NULL)
852: Level: intermediate
854: .seealso: `PetscDTGaussQuadrature()`, `PetscDTLegendreEval()`
855: @*/
856: PetscErrorCode PetscDTJacobiEval(PetscInt npoints, PetscReal alpha, PetscReal beta, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *B, PetscReal *D, PetscReal *D2)
857: {
858: PetscFunctionBegin;
859: PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
860: PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");
861: if (!npoints || !ndegree) PetscFunctionReturn(PETSC_SUCCESS);
862: if (B) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 0, points, ndegree, degrees, B));
863: if (D) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 1, points, ndegree, degrees, D));
864: if (D2) PetscCall(PetscDTJacobiEval_Internal(npoints, alpha, beta, 2, points, ndegree, degrees, D2));
865: PetscFunctionReturn(PETSC_SUCCESS);
866: }
868: /*@
869: PetscDTLegendreEval - evaluate Legendre polynomials at points
871: Not Collective
873: Input Parameters:
874: + npoints - number of spatial points to evaluate at
875: . points - array of locations to evaluate at
876: . ndegree - number of basis degrees to evaluate
877: - degrees - sorted array of degrees to evaluate
879: Output Parameters:
880: + B - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension npoints*ndegrees, allocated by caller) (or NULL)
881: . D - row-oriented derivative evaluation matrix (or NULL)
882: - D2 - row-oriented second derivative evaluation matrix (or NULL)
884: Level: intermediate
886: .seealso: `PetscDTGaussQuadrature()`
887: @*/
888: PetscErrorCode PetscDTLegendreEval(PetscInt npoints, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *B, PetscReal *D, PetscReal *D2)
889: {
890: PetscFunctionBegin;
891: PetscCall(PetscDTJacobiEval(npoints, 0., 0., points, ndegree, degrees, B, D, D2));
892: PetscFunctionReturn(PETSC_SUCCESS);
893: }
895: /*@
896: 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)
898: Input Parameters:
899: + len - the desired length of the degree tuple
900: - index - the index to convert: should be >= 0
902: Output Parameter:
903: . degtup - will be filled with a tuple of degrees
905: Level: beginner
907: Note:
908: For two tuples x and y with the same degree sum, partial degree sums over the final elements of the tuples
909: acts as a tiebreaker. For example, (2, 1, 1) and (1, 2, 1) have the same degree sum, but the degree sum over the
910: last two elements is smaller for the former, so (2, 1, 1) < (1, 2, 1).
912: .seealso: `PetscDTGradedOrderToIndex()`
913: @*/
914: PetscErrorCode PetscDTIndexToGradedOrder(PetscInt len, PetscInt index, PetscInt degtup[])
915: {
916: PetscInt i, total;
917: PetscInt sum;
919: PetscFunctionBeginHot;
920: PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
921: PetscCheck(index >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
922: total = 1;
923: sum = 0;
924: while (index >= total) {
925: index -= total;
926: total = (total * (len + sum)) / (sum + 1);
927: sum++;
928: }
929: for (i = 0; i < len; i++) {
930: PetscInt c;
932: degtup[i] = sum;
933: for (c = 0, total = 1; c < sum; c++) {
934: /* going into the loop, total is the number of way to have a tuple of sum exactly c with length len - 1 - i */
935: if (index < total) break;
936: index -= total;
937: total = (total * (len - 1 - i + c)) / (c + 1);
938: degtup[i]--;
939: }
940: sum -= degtup[i];
941: }
942: PetscFunctionReturn(PETSC_SUCCESS);
943: }
945: /*@
946: PetscDTGradedOrderToIndex - convert a tuple into an index in a graded order, the inverse of `PetscDTIndexToGradedOrder()`.
948: Input Parameters:
949: + len - the length of the degree tuple
950: - degtup - tuple with this length
952: Output Parameter:
953: . index - index in graded order: >= 0
955: Level: Beginner
957: Note:
958: For two tuples x and y with the same degree sum, partial degree sums over the final elements of the tuples
959: acts as a tiebreaker. For example, (2, 1, 1) and (1, 2, 1) have the same degree sum, but the degree sum over the
960: last two elements is smaller for the former, so (2, 1, 1) < (1, 2, 1).
962: .seealso: `PetscDTIndexToGradedOrder()`
963: @*/
964: PetscErrorCode PetscDTGradedOrderToIndex(PetscInt len, const PetscInt degtup[], PetscInt *index)
965: {
966: PetscInt i, idx, sum, total;
968: PetscFunctionBeginHot;
969: PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
970: for (i = 0, sum = 0; i < len; i++) sum += degtup[i];
971: idx = 0;
972: total = 1;
973: for (i = 0; i < sum; i++) {
974: idx += total;
975: total = (total * (len + i)) / (i + 1);
976: }
977: for (i = 0; i < len - 1; i++) {
978: PetscInt c;
980: total = 1;
981: sum -= degtup[i];
982: for (c = 0; c < sum; c++) {
983: idx += total;
984: total = (total * (len - 1 - i + c)) / (c + 1);
985: }
986: }
987: *index = idx;
988: PetscFunctionReturn(PETSC_SUCCESS);
989: }
991: static PetscBool PKDCite = PETSC_FALSE;
992: const char PKDCitation[] = "@article{Kirby2010,\n"
993: " title={Singularity-free evaluation of collapsed-coordinate orthogonal polynomials},\n"
994: " author={Kirby, Robert C},\n"
995: " journal={ACM Transactions on Mathematical Software (TOMS)},\n"
996: " volume={37},\n"
997: " number={1},\n"
998: " pages={1--16},\n"
999: " year={2010},\n"
1000: " publisher={ACM New York, NY, USA}\n}\n";
1002: /*@
1003: PetscDTPKDEvalJet - Evaluate the jet (function and derivatives) of the Proriol-Koornwinder-Dubiner (PKD) basis for
1004: the space of polynomials up to a given degree. The PKD basis is L2-orthonormal on the biunit simplex (which is used
1005: as the reference element for finite elements in PETSc), which makes it a stable basis to use for evaluating
1006: polynomials in that domain.
1008: Input Parameters:
1009: + dim - the number of variables in the multivariate polynomials
1010: . npoints - the number of points to evaluate the polynomials at
1011: . points - [npoints x dim] array of point coordinates
1012: . 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.
1013: - k - the maximum order partial derivative to evaluate in the jet. There are (dim + k choose dim) partial derivatives
1014: in the jet. Choosing k = 0 means to evaluate just the function and no derivatives
1016: Output Parameters:
1017: - p - an array containing the evaluations of the PKD polynomials' jets on the points. The size is ((dim + degree)
1018: choose dim) x ((dim + k) choose dim) x npoints, which also describes the order of the dimensions of this
1019: three-dimensional array: the first (slowest varying) dimension is basis function index; the second dimension is jet
1020: index; the third (fastest varying) dimension is the index of the evaluation point.
1022: Level: advanced
1024: Notes:
1025: The ordering of the basis functions, and the ordering of the derivatives in the jet, both follow the graded
1026: ordering of `PetscDTIndexToGradedOrder()` and `PetscDTGradedOrderToIndex()`. For example, in 3D, the polynomial with
1027: 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);
1028: 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).
1030: The implementation uses Kirby's singularity-free evaluation algorithm, https://doi.org/10.1145/1644001.1644006.
1032: .seealso: `PetscDTGradedOrderToIndex()`, `PetscDTIndexToGradedOrder()`, `PetscDTJacobiEvalJet()`
1033: @*/
1034: PetscErrorCode PetscDTPKDEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
1035: {
1036: PetscInt degidx, kidx, d, pt;
1037: PetscInt Nk, Ndeg;
1038: PetscInt *ktup, *degtup;
1039: PetscReal *scales, initscale, scaleexp;
1041: PetscFunctionBegin;
1042: PetscCall(PetscCitationsRegister(PKDCitation, &PKDCite));
1043: PetscCall(PetscDTBinomialInt(dim + k, k, &Nk));
1044: PetscCall(PetscDTBinomialInt(degree + dim, degree, &Ndeg));
1045: PetscCall(PetscMalloc2(dim, °tup, dim, &ktup));
1046: PetscCall(PetscMalloc1(Ndeg, &scales));
1047: initscale = 1.;
1048: if (dim > 1) {
1049: PetscCall(PetscDTBinomial(dim, 2, &scaleexp));
1050: initscale = PetscPowReal(2., scaleexp * 0.5);
1051: }
1052: for (degidx = 0; degidx < Ndeg; degidx++) {
1053: PetscInt e, i;
1054: PetscInt m1idx = -1, m2idx = -1;
1055: PetscInt n;
1056: PetscInt degsum;
1057: PetscReal alpha;
1058: PetscReal cnm1, cnm1x, cnm2;
1059: PetscReal norm;
1061: PetscCall(PetscDTIndexToGradedOrder(dim, degidx, degtup));
1062: for (d = dim - 1; d >= 0; d--)
1063: if (degtup[d]) break;
1064: if (d < 0) { /* constant is 1 everywhere, all derivatives are zero */
1065: scales[degidx] = initscale;
1066: for (e = 0; e < dim; e++) {
1067: PetscCall(PetscDTJacobiNorm(e, 0., 0, &norm));
1068: scales[degidx] /= norm;
1069: }
1070: for (i = 0; i < npoints; i++) p[degidx * Nk * npoints + i] = 1.;
1071: for (i = 0; i < (Nk - 1) * npoints; i++) p[(degidx * Nk + 1) * npoints + i] = 0.;
1072: continue;
1073: }
1074: n = degtup[d];
1075: degtup[d]--;
1076: PetscCall(PetscDTGradedOrderToIndex(dim, degtup, &m1idx));
1077: if (degtup[d] > 0) {
1078: degtup[d]--;
1079: PetscCall(PetscDTGradedOrderToIndex(dim, degtup, &m2idx));
1080: degtup[d]++;
1081: }
1082: degtup[d]++;
1083: for (e = 0, degsum = 0; e < d; e++) degsum += degtup[e];
1084: alpha = 2 * degsum + d;
1085: PetscDTJacobiRecurrence_Internal(n, alpha, 0., cnm1, cnm1x, cnm2);
1087: scales[degidx] = initscale;
1088: for (e = 0, degsum = 0; e < dim; e++) {
1089: PetscInt f;
1090: PetscReal ealpha;
1091: PetscReal enorm;
1093: ealpha = 2 * degsum + e;
1094: for (f = 0; f < degsum; f++) scales[degidx] *= 2.;
1095: PetscCall(PetscDTJacobiNorm(ealpha, 0., degtup[e], &enorm));
1096: scales[degidx] /= enorm;
1097: degsum += degtup[e];
1098: }
1100: for (pt = 0; pt < npoints; pt++) {
1101: /* compute the multipliers */
1102: PetscReal thetanm1, thetanm1x, thetanm2;
1104: thetanm1x = dim - (d + 1) + 2. * points[pt * dim + d];
1105: for (e = d + 1; e < dim; e++) thetanm1x += points[pt * dim + e];
1106: thetanm1x *= 0.5;
1107: thetanm1 = (2. - (dim - (d + 1)));
1108: for (e = d + 1; e < dim; e++) thetanm1 -= points[pt * dim + e];
1109: thetanm1 *= 0.5;
1110: thetanm2 = thetanm1 * thetanm1;
1112: for (kidx = 0; kidx < Nk; kidx++) {
1113: PetscInt f;
1115: PetscCall(PetscDTIndexToGradedOrder(dim, kidx, ktup));
1116: /* first sum in the same derivative terms */
1117: p[(degidx * Nk + kidx) * npoints + pt] = (cnm1 * thetanm1 + cnm1x * thetanm1x) * p[(m1idx * Nk + kidx) * npoints + pt];
1118: if (m2idx >= 0) p[(degidx * Nk + kidx) * npoints + pt] -= cnm2 * thetanm2 * p[(m2idx * Nk + kidx) * npoints + pt];
1120: for (f = d; f < dim; f++) {
1121: PetscInt km1idx, mplty = ktup[f];
1123: if (!mplty) continue;
1124: ktup[f]--;
1125: PetscCall(PetscDTGradedOrderToIndex(dim, ktup, &km1idx));
1127: /* the derivative of cnm1x * thetanm1x wrt x variable f is 0.5 * cnm1x if f > d otherwise it is cnm1x */
1128: /* the derivative of cnm1 * thetanm1 wrt x variable f is 0 if f == d, otherwise it is -0.5 * cnm1 */
1129: /* the derivative of -cnm2 * thetanm2 wrt x variable f is 0 if f == d, otherwise it is cnm2 * thetanm1 */
1130: if (f > d) {
1131: PetscInt f2;
1133: p[(degidx * Nk + kidx) * npoints + pt] += mplty * 0.5 * (cnm1x - cnm1) * p[(m1idx * Nk + km1idx) * npoints + pt];
1134: if (m2idx >= 0) {
1135: p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm2 * thetanm1 * p[(m2idx * Nk + km1idx) * npoints + pt];
1136: /* second derivatives of -cnm2 * thetanm2 wrt x variable f,f2 is like - 0.5 * cnm2 */
1137: for (f2 = f; f2 < dim; f2++) {
1138: PetscInt km2idx, mplty2 = ktup[f2];
1139: PetscInt factor;
1141: if (!mplty2) continue;
1142: ktup[f2]--;
1143: PetscCall(PetscDTGradedOrderToIndex(dim, ktup, &km2idx));
1145: factor = mplty * mplty2;
1146: if (f == f2) factor /= 2;
1147: p[(degidx * Nk + kidx) * npoints + pt] -= 0.5 * factor * cnm2 * p[(m2idx * Nk + km2idx) * npoints + pt];
1148: ktup[f2]++;
1149: }
1150: }
1151: } else {
1152: p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm1x * p[(m1idx * Nk + km1idx) * npoints + pt];
1153: }
1154: ktup[f]++;
1155: }
1156: }
1157: }
1158: }
1159: for (degidx = 0; degidx < Ndeg; degidx++) {
1160: PetscReal scale = scales[degidx];
1161: PetscInt i;
1163: for (i = 0; i < Nk * npoints; i++) p[degidx * Nk * npoints + i] *= scale;
1164: }
1165: PetscCall(PetscFree(scales));
1166: PetscCall(PetscFree2(degtup, ktup));
1167: PetscFunctionReturn(PETSC_SUCCESS);
1168: }
1170: /*@
1171: PetscDTPTrimmedSize - The size of the trimmed polynomial space of k-forms with a given degree and form degree,
1172: which can be evaluated in `PetscDTPTrimmedEvalJet()`.
1174: Input Parameters:
1175: + dim - the number of variables in the multivariate polynomials
1176: . degree - the degree (sum of degrees on the variables in a monomial) of the trimmed polynomial space.
1177: - formDegree - the degree of the form
1179: Output Parameters:
1180: - size - The number ((dim + degree) choose (dim + formDegree)) x ((degree + formDegree - 1) choose (formDegree))
1182: Level: advanced
1184: .seealso: `PetscDTPTrimmedEvalJet()`
1185: @*/
1186: PetscErrorCode PetscDTPTrimmedSize(PetscInt dim, PetscInt degree, PetscInt formDegree, PetscInt *size)
1187: {
1188: PetscInt Nrk, Nbpt; // number of trimmed polynomials
1190: PetscFunctionBegin;
1191: formDegree = PetscAbsInt(formDegree);
1192: PetscCall(PetscDTBinomialInt(degree + dim, degree + formDegree, &Nbpt));
1193: PetscCall(PetscDTBinomialInt(degree + formDegree - 1, formDegree, &Nrk));
1194: Nbpt *= Nrk;
1195: *size = Nbpt;
1196: PetscFunctionReturn(PETSC_SUCCESS);
1197: }
1199: /* there was a reference implementation based on section 4.4 of Arnold, Falk & Winther (acta numerica, 2006), but it
1200: * was inferior to this implementation */
1201: static PetscErrorCode PetscDTPTrimmedEvalJet_Internal(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt formDegree, PetscInt jetDegree, PetscReal p[])
1202: {
1203: PetscInt formDegreeOrig = formDegree;
1204: PetscBool formNegative = (formDegreeOrig < 0) ? PETSC_TRUE : PETSC_FALSE;
1206: PetscFunctionBegin;
1207: formDegree = PetscAbsInt(formDegreeOrig);
1208: if (formDegree == 0) {
1209: PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree, jetDegree, p));
1210: PetscFunctionReturn(PETSC_SUCCESS);
1211: }
1212: if (formDegree == dim) {
1213: PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree - 1, jetDegree, p));
1214: PetscFunctionReturn(PETSC_SUCCESS);
1215: }
1216: PetscInt Nbpt;
1217: PetscCall(PetscDTPTrimmedSize(dim, degree, formDegree, &Nbpt));
1218: PetscInt Nf;
1219: PetscCall(PetscDTBinomialInt(dim, formDegree, &Nf));
1220: PetscInt Nk;
1221: PetscCall(PetscDTBinomialInt(dim + jetDegree, dim, &Nk));
1222: PetscCall(PetscArrayzero(p, Nbpt * Nf * Nk * npoints));
1224: PetscInt Nbpm1; // number of scalar polynomials up to degree - 1;
1225: PetscCall(PetscDTBinomialInt(dim + degree - 1, dim, &Nbpm1));
1226: PetscReal *p_scalar;
1227: PetscCall(PetscMalloc1(Nbpm1 * Nk * npoints, &p_scalar));
1228: PetscCall(PetscDTPKDEvalJet(dim, npoints, points, degree - 1, jetDegree, p_scalar));
1229: PetscInt total = 0;
1230: // First add the full polynomials up to degree - 1 into the basis: take the scalar
1231: // and copy one for each form component
1232: for (PetscInt i = 0; i < Nbpm1; i++) {
1233: const PetscReal *src = &p_scalar[i * Nk * npoints];
1234: for (PetscInt f = 0; f < Nf; f++) {
1235: PetscReal *dest = &p[(total++ * Nf + f) * Nk * npoints];
1236: PetscCall(PetscArraycpy(dest, src, Nk * npoints));
1237: }
1238: }
1239: PetscInt *form_atoms;
1240: PetscCall(PetscMalloc1(formDegree + 1, &form_atoms));
1241: // construct the interior product pattern
1242: PetscInt(*pattern)[3];
1243: PetscInt Nf1; // number of formDegree + 1 forms
1244: PetscCall(PetscDTBinomialInt(dim, formDegree + 1, &Nf1));
1245: PetscInt nnz = Nf1 * (formDegree + 1);
1246: PetscCall(PetscMalloc1(Nf1 * (formDegree + 1), &pattern));
1247: PetscCall(PetscDTAltVInteriorPattern(dim, formDegree + 1, pattern));
1248: PetscReal centroid = (1. - dim) / (dim + 1.);
1249: PetscInt *deriv;
1250: PetscCall(PetscMalloc1(dim, &deriv));
1251: for (PetscInt d = dim; d >= formDegree + 1; d--) {
1252: PetscInt Nfd1; // number of formDegree + 1 forms in dimension d that include dx_0
1253: // (equal to the number of formDegree forms in dimension d-1)
1254: PetscCall(PetscDTBinomialInt(d - 1, formDegree, &Nfd1));
1255: // The number of homogeneous (degree-1) scalar polynomials in d variables
1256: PetscInt Nh;
1257: PetscCall(PetscDTBinomialInt(d - 1 + degree - 1, d - 1, &Nh));
1258: const PetscReal *h_scalar = &p_scalar[(Nbpm1 - Nh) * Nk * npoints];
1259: for (PetscInt b = 0; b < Nh; b++) {
1260: const PetscReal *h_s = &h_scalar[b * Nk * npoints];
1261: for (PetscInt f = 0; f < Nfd1; f++) {
1262: // construct all formDegree+1 forms that start with dx_(dim - d) /\ ...
1263: form_atoms[0] = dim - d;
1264: PetscCall(PetscDTEnumSubset(d - 1, formDegree, f, &form_atoms[1]));
1265: for (PetscInt i = 0; i < formDegree; i++) form_atoms[1 + i] += form_atoms[0] + 1;
1266: PetscInt f_ind; // index of the resulting form
1267: PetscCall(PetscDTSubsetIndex(dim, formDegree + 1, form_atoms, &f_ind));
1268: PetscReal *p_f = &p[total++ * Nf * Nk * npoints];
1269: for (PetscInt nz = 0; nz < nnz; nz++) {
1270: PetscInt i = pattern[nz][0]; // formDegree component
1271: PetscInt j = pattern[nz][1]; // (formDegree + 1) component
1272: PetscInt v = pattern[nz][2]; // coordinate component
1273: PetscReal scale = v < 0 ? -1. : 1.;
1275: i = formNegative ? (Nf - 1 - i) : i;
1276: scale = (formNegative && (i & 1)) ? -scale : scale;
1277: v = v < 0 ? -(v + 1) : v;
1278: if (j != f_ind) continue;
1279: PetscReal *p_i = &p_f[i * Nk * npoints];
1280: for (PetscInt jet = 0; jet < Nk; jet++) {
1281: const PetscReal *h_jet = &h_s[jet * npoints];
1282: PetscReal *p_jet = &p_i[jet * npoints];
1284: for (PetscInt pt = 0; pt < npoints; pt++) p_jet[pt] += scale * h_jet[pt] * (points[pt * dim + v] - centroid);
1285: PetscCall(PetscDTIndexToGradedOrder(dim, jet, deriv));
1286: deriv[v]++;
1287: PetscReal mult = deriv[v];
1288: PetscInt l;
1289: PetscCall(PetscDTGradedOrderToIndex(dim, deriv, &l));
1290: if (l >= Nk) continue;
1291: p_jet = &p_i[l * npoints];
1292: for (PetscInt pt = 0; pt < npoints; pt++) p_jet[pt] += scale * mult * h_jet[pt];
1293: deriv[v]--;
1294: }
1295: }
1296: }
1297: }
1298: }
1299: PetscCheck(total == Nbpt, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Incorrectly counted P trimmed polynomials");
1300: PetscCall(PetscFree(deriv));
1301: PetscCall(PetscFree(pattern));
1302: PetscCall(PetscFree(form_atoms));
1303: PetscCall(PetscFree(p_scalar));
1304: PetscFunctionReturn(PETSC_SUCCESS);
1305: }
1307: /*@
1308: PetscDTPTrimmedEvalJet - Evaluate the jet (function and derivatives) of a basis of the trimmed polynomial k-forms up to
1309: a given degree.
1311: Input Parameters:
1312: + dim - the number of variables in the multivariate polynomials
1313: . npoints - the number of points to evaluate the polynomials at
1314: . points - [npoints x dim] array of point coordinates
1315: . degree - the degree (sum of degrees on the variables in a monomial) of the trimmed polynomial space to evaluate.
1316: There are ((dim + degree) choose (dim + formDegree)) x ((degree + formDegree - 1) choose (formDegree)) polynomials in this space.
1317: (You can use `PetscDTPTrimmedSize()` to compute this size.)
1318: . formDegree - the degree of the form
1319: - jetDegree - the maximum order partial derivative to evaluate in the jet. There are ((dim + jetDegree) choose dim) partial derivatives
1320: in the jet. Choosing jetDegree = 0 means to evaluate just the function and no derivatives
1322: Output Parameters:
1323: - p - an array containing the evaluations of the PKD polynomials' jets on the points. The size is
1324: `PetscDTPTrimmedSize()` x ((dim + formDegree) choose dim) x ((dim + k) choose dim) x npoints,
1325: which also describes the order of the dimensions of this
1326: four-dimensional array:
1327: the first (slowest varying) dimension is basis function index;
1328: the second dimension is component of the form;
1329: the third dimension is jet index;
1330: the fourth (fastest varying) dimension is the index of the evaluation point.
1332: Level: advanced
1334: Notes:
1335: The ordering of the basis functions is not graded, so the basis functions are not nested by degree like `PetscDTPKDEvalJet()`.
1336: The basis functions are not an L2-orthonormal basis on any particular domain.
1338: The implementation is based on the description of the trimmed polynomials up to degree r as
1339: the direct sum of polynomials up to degree (r-1) and the Koszul differential applied to
1340: homogeneous polynomials of degree (r-1).
1342: .seealso: `PetscDTPKDEvalJet()`, `PetscDTPTrimmedSize()`
1343: @*/
1344: PetscErrorCode PetscDTPTrimmedEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt formDegree, PetscInt jetDegree, PetscReal p[])
1345: {
1346: PetscFunctionBegin;
1347: PetscCall(PetscDTPTrimmedEvalJet_Internal(dim, npoints, points, degree, formDegree, jetDegree, p));
1348: PetscFunctionReturn(PETSC_SUCCESS);
1349: }
1351: /* solve the symmetric tridiagonal eigenvalue system, writing the eigenvalues into eigs and the eigenvectors into V
1352: * with lds n; diag and subdiag are overwritten */
1353: static PetscErrorCode PetscDTSymmetricTridiagonalEigensolve(PetscInt n, PetscReal diag[], PetscReal subdiag[], PetscReal eigs[], PetscScalar V[])
1354: {
1355: char jobz = 'V'; /* eigenvalues and eigenvectors */
1356: char range = 'A'; /* all eigenvalues will be found */
1357: PetscReal VL = 0.; /* ignored because range is 'A' */
1358: PetscReal VU = 0.; /* ignored because range is 'A' */
1359: PetscBLASInt IL = 0; /* ignored because range is 'A' */
1360: PetscBLASInt IU = 0; /* ignored because range is 'A' */
1361: PetscReal abstol = 0.; /* unused */
1362: PetscBLASInt bn, bm, ldz; /* bm will equal bn on exit */
1363: PetscBLASInt *isuppz;
1364: PetscBLASInt lwork, liwork;
1365: PetscReal workquery;
1366: PetscBLASInt iworkquery;
1367: PetscBLASInt *iwork;
1368: PetscBLASInt info;
1369: PetscReal *work = NULL;
1371: PetscFunctionBegin;
1372: #if !defined(PETSCDTGAUSSIANQUADRATURE_EIG)
1373: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
1374: #endif
1375: PetscCall(PetscBLASIntCast(n, &bn));
1376: PetscCall(PetscBLASIntCast(n, &ldz));
1377: #if !defined(PETSC_MISSING_LAPACK_STEGR)
1378: PetscCall(PetscMalloc1(2 * n, &isuppz));
1379: lwork = -1;
1380: liwork = -1;
1381: PetscCallBLAS("LAPACKstegr", LAPACKstegr_(&jobz, &range, &bn, diag, subdiag, &VL, &VU, &IL, &IU, &abstol, &bm, eigs, V, &ldz, isuppz, &workquery, &lwork, &iworkquery, &liwork, &info));
1382: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEGR error");
1383: lwork = (PetscBLASInt)workquery;
1384: liwork = (PetscBLASInt)iworkquery;
1385: PetscCall(PetscMalloc2(lwork, &work, liwork, &iwork));
1386: PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
1387: PetscCallBLAS("LAPACKstegr", LAPACKstegr_(&jobz, &range, &bn, diag, subdiag, &VL, &VU, &IL, &IU, &abstol, &bm, eigs, V, &ldz, isuppz, work, &lwork, iwork, &liwork, &info));
1388: PetscCall(PetscFPTrapPop());
1389: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEGR error");
1390: PetscCall(PetscFree2(work, iwork));
1391: PetscCall(PetscFree(isuppz));
1392: #elif !defined(PETSC_MISSING_LAPACK_STEQR)
1393: jobz = 'I'; /* Compute eigenvalues and eigenvectors of the
1394: tridiagonal matrix. Z is initialized to the identity
1395: matrix. */
1396: PetscCall(PetscMalloc1(PetscMax(1, 2 * n - 2), &work));
1397: PetscCallBLAS("LAPACKsteqr", LAPACKsteqr_("I", &bn, diag, subdiag, V, &ldz, work, &info));
1398: PetscCall(PetscFPTrapPop());
1399: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_PLIB, "xSTEQR error");
1400: PetscCall(PetscFree(work));
1401: PetscCall(PetscArraycpy(eigs, diag, n));
1402: #endif
1403: PetscFunctionReturn(PETSC_SUCCESS);
1404: }
1406: /* Formula for the weights at the endpoints (-1 and 1) of Gauss-Lobatto-Jacobi
1407: * quadrature rules on the interval [-1, 1] */
1408: static PetscErrorCode PetscDTGaussLobattoJacobiEndweights_Internal(PetscInt n, PetscReal alpha, PetscReal beta, PetscReal *leftw, PetscReal *rightw)
1409: {
1410: PetscReal twoab1;
1411: PetscInt m = n - 2;
1412: PetscReal a = alpha + 1.;
1413: PetscReal b = beta + 1.;
1414: PetscReal gra, grb;
1416: PetscFunctionBegin;
1417: twoab1 = PetscPowReal(2., a + b - 1.);
1418: #if defined(PETSC_HAVE_LGAMMA)
1419: grb = PetscExpReal(2. * PetscLGamma(b + 1.) + PetscLGamma(m + 1.) + PetscLGamma(m + a + 1.) - (PetscLGamma(m + b + 1) + PetscLGamma(m + a + b + 1.)));
1420: gra = PetscExpReal(2. * PetscLGamma(a + 1.) + PetscLGamma(m + 1.) + PetscLGamma(m + b + 1.) - (PetscLGamma(m + a + 1) + PetscLGamma(m + a + b + 1.)));
1421: #else
1422: {
1423: PetscInt alphai = (PetscInt)alpha;
1424: PetscInt betai = (PetscInt)beta;
1426: if ((PetscReal)alphai == alpha && (PetscReal)betai == beta) {
1427: PetscReal binom1, binom2;
1429: PetscCall(PetscDTBinomial(m + b, b, &binom1));
1430: PetscCall(PetscDTBinomial(m + a + b, b, &binom2));
1431: grb = 1. / (binom1 * binom2);
1432: PetscCall(PetscDTBinomial(m + a, a, &binom1));
1433: PetscCall(PetscDTBinomial(m + a + b, a, &binom2));
1434: gra = 1. / (binom1 * binom2);
1435: } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "lgamma() - math routine is unavailable.");
1436: }
1437: #endif
1438: *leftw = twoab1 * grb / b;
1439: *rightw = twoab1 * gra / a;
1440: PetscFunctionReturn(PETSC_SUCCESS);
1441: }
1443: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
1444: Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
1445: static inline PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
1446: {
1447: PetscReal pn1, pn2;
1448: PetscReal cnm1, cnm1x, cnm2;
1449: PetscInt k;
1451: PetscFunctionBegin;
1452: if (!n) {
1453: *P = 1.0;
1454: PetscFunctionReturn(PETSC_SUCCESS);
1455: }
1456: PetscDTJacobiRecurrence_Internal(1, a, b, cnm1, cnm1x, cnm2);
1457: pn2 = 1.;
1458: pn1 = cnm1 + cnm1x * x;
1459: if (n == 1) {
1460: *P = pn1;
1461: PetscFunctionReturn(PETSC_SUCCESS);
1462: }
1463: *P = 0.0;
1464: for (k = 2; k < n + 1; ++k) {
1465: PetscDTJacobiRecurrence_Internal(k, a, b, cnm1, cnm1x, cnm2);
1467: *P = (cnm1 + cnm1x * x) * pn1 - cnm2 * pn2;
1468: pn2 = pn1;
1469: pn1 = *P;
1470: }
1471: PetscFunctionReturn(PETSC_SUCCESS);
1472: }
1474: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
1475: static inline PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscInt k, PetscReal *P)
1476: {
1477: PetscReal nP;
1478: PetscInt i;
1480: PetscFunctionBegin;
1481: *P = 0.0;
1482: if (k > n) PetscFunctionReturn(PETSC_SUCCESS);
1483: PetscCall(PetscDTComputeJacobi(a + k, b + k, n - k, x, &nP));
1484: for (i = 0; i < k; i++) nP *= (a + b + n + 1. + i) * 0.5;
1485: *P = nP;
1486: PetscFunctionReturn(PETSC_SUCCESS);
1487: }
1489: static PetscErrorCode PetscDTGaussJacobiQuadrature_Newton_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal x[], PetscReal w[])
1490: {
1491: PetscInt maxIter = 100;
1492: PetscReal eps = PetscExpReal(0.75 * PetscLogReal(PETSC_MACHINE_EPSILON));
1493: PetscReal a1, a6, gf;
1494: PetscInt k;
1496: PetscFunctionBegin;
1498: a1 = PetscPowReal(2.0, a + b + 1);
1499: #if defined(PETSC_HAVE_LGAMMA)
1500: {
1501: PetscReal a2, a3, a4, a5;
1502: a2 = PetscLGamma(a + npoints + 1);
1503: a3 = PetscLGamma(b + npoints + 1);
1504: a4 = PetscLGamma(a + b + npoints + 1);
1505: a5 = PetscLGamma(npoints + 1);
1506: gf = PetscExpReal(a2 + a3 - (a4 + a5));
1507: }
1508: #else
1509: {
1510: PetscInt ia, ib;
1512: ia = (PetscInt)a;
1513: ib = (PetscInt)b;
1514: gf = 1.;
1515: if (ia == a && ia >= 0) { /* compute ratio of rising factorals wrt a */
1516: for (k = 0; k < ia; k++) gf *= (npoints + 1. + k) / (npoints + b + 1. + k);
1517: } else if (b == b && ib >= 0) { /* compute ratio of rising factorials wrt b */
1518: for (k = 0; k < ib; k++) gf *= (npoints + 1. + k) / (npoints + a + 1. + k);
1519: } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "lgamma() - math routine is unavailable.");
1520: }
1521: #endif
1523: a6 = a1 * gf;
1524: /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
1525: Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
1526: for (k = 0; k < npoints; ++k) {
1527: PetscReal r = PetscCosReal(PETSC_PI * (1. - (4. * k + 3. + 2. * b) / (4. * npoints + 2. * (a + b + 1.)))), dP;
1528: PetscInt j;
1530: if (k > 0) r = 0.5 * (r + x[k - 1]);
1531: for (j = 0; j < maxIter; ++j) {
1532: PetscReal s = 0.0, delta, f, fp;
1533: PetscInt i;
1535: for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
1536: PetscCall(PetscDTComputeJacobi(a, b, npoints, r, &f));
1537: PetscCall(PetscDTComputeJacobiDerivative(a, b, npoints, r, 1, &fp));
1538: delta = f / (fp - f * s);
1539: r = r - delta;
1540: if (PetscAbsReal(delta) < eps) break;
1541: }
1542: x[k] = r;
1543: PetscCall(PetscDTComputeJacobiDerivative(a, b, npoints, x[k], 1, &dP));
1544: w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
1545: }
1546: PetscFunctionReturn(PETSC_SUCCESS);
1547: }
1549: /* Compute the diagonals of the Jacobi matrix used in Golub & Welsch algorithms for Gauss-Jacobi
1550: * quadrature weight calculations on [-1,1] for exponents (1. + x)^a (1.-x)^b */
1551: static PetscErrorCode PetscDTJacobiMatrix_Internal(PetscInt nPoints, PetscReal a, PetscReal b, PetscReal *d, PetscReal *s)
1552: {
1553: PetscInt i;
1555: PetscFunctionBegin;
1556: for (i = 0; i < nPoints; i++) {
1557: PetscReal A, B, C;
1559: PetscDTJacobiRecurrence_Internal(i + 1, a, b, A, B, C);
1560: d[i] = -A / B;
1561: if (i) s[i - 1] *= C / B;
1562: if (i < nPoints - 1) s[i] = 1. / B;
1563: }
1564: PetscFunctionReturn(PETSC_SUCCESS);
1565: }
1567: static PetscErrorCode PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
1568: {
1569: PetscReal mu0;
1570: PetscReal ga, gb, gab;
1571: PetscInt i;
1573: PetscFunctionBegin;
1574: PetscCall(PetscCitationsRegister(GolubWelschCitation, &GolubWelschCite));
1576: #if defined(PETSC_HAVE_TGAMMA)
1577: ga = PetscTGamma(a + 1);
1578: gb = PetscTGamma(b + 1);
1579: gab = PetscTGamma(a + b + 2);
1580: #else
1581: {
1582: PetscInt ia, ib;
1584: ia = (PetscInt)a;
1585: ib = (PetscInt)b;
1586: if (ia == a && ib == b && ia + 1 > 0 && ib + 1 > 0 && ia + ib + 2 > 0) { /* All gamma(x) terms are (x-1)! terms */
1587: PetscCall(PetscDTFactorial(ia, &ga));
1588: PetscCall(PetscDTFactorial(ib, &gb));
1589: PetscCall(PetscDTFactorial(ia + ib + 1, &gb));
1590: } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "tgamma() - math routine is unavailable.");
1591: }
1592: #endif
1593: mu0 = PetscPowReal(2., a + b + 1.) * ga * gb / gab;
1595: #if defined(PETSCDTGAUSSIANQUADRATURE_EIG)
1596: {
1597: PetscReal *diag, *subdiag;
1598: PetscScalar *V;
1600: PetscCall(PetscMalloc2(npoints, &diag, npoints, &subdiag));
1601: PetscCall(PetscMalloc1(npoints * npoints, &V));
1602: PetscCall(PetscDTJacobiMatrix_Internal(npoints, a, b, diag, subdiag));
1603: for (i = 0; i < npoints - 1; i++) subdiag[i] = PetscSqrtReal(subdiag[i]);
1604: PetscCall(PetscDTSymmetricTridiagonalEigensolve(npoints, diag, subdiag, x, V));
1605: for (i = 0; i < npoints; i++) w[i] = PetscSqr(PetscRealPart(V[i * npoints])) * mu0;
1606: PetscCall(PetscFree(V));
1607: PetscCall(PetscFree2(diag, subdiag));
1608: }
1609: #else
1610: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
1611: #endif
1612: { /* As of March 2, 2020, The Sun Performance Library breaks the LAPACK contract for xstegr and xsteqr: the
1613: eigenvalues are not guaranteed to be in ascending order. So we heave a passive aggressive sigh and check that
1614: the eigenvalues are sorted */
1615: PetscBool sorted;
1617: PetscCall(PetscSortedReal(npoints, x, &sorted));
1618: if (!sorted) {
1619: PetscInt *order, i;
1620: PetscReal *tmp;
1622: PetscCall(PetscMalloc2(npoints, &order, npoints, &tmp));
1623: for (i = 0; i < npoints; i++) order[i] = i;
1624: PetscCall(PetscSortRealWithPermutation(npoints, x, order));
1625: PetscCall(PetscArraycpy(tmp, x, npoints));
1626: for (i = 0; i < npoints; i++) x[i] = tmp[order[i]];
1627: PetscCall(PetscArraycpy(tmp, w, npoints));
1628: for (i = 0; i < npoints; i++) w[i] = tmp[order[i]];
1629: PetscCall(PetscFree2(order, tmp));
1630: }
1631: }
1632: PetscFunctionReturn(PETSC_SUCCESS);
1633: }
1635: static PetscErrorCode PetscDTGaussJacobiQuadrature_Internal(PetscInt npoints, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
1636: {
1637: PetscFunctionBegin;
1638: PetscCheck(npoints >= 1, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of points must be positive");
1639: /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
1640: PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
1641: PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");
1643: if (newton) PetscCall(PetscDTGaussJacobiQuadrature_Newton_Internal(npoints, alpha, beta, x, w));
1644: else PetscCall(PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(npoints, alpha, beta, x, w));
1645: if (alpha == beta) { /* symmetrize */
1646: PetscInt i;
1647: for (i = 0; i < (npoints + 1) / 2; i++) {
1648: PetscInt j = npoints - 1 - i;
1649: PetscReal xi = x[i];
1650: PetscReal xj = x[j];
1651: PetscReal wi = w[i];
1652: PetscReal wj = w[j];
1654: x[i] = (xi - xj) / 2.;
1655: x[j] = (xj - xi) / 2.;
1656: w[i] = w[j] = (wi + wj) / 2.;
1657: }
1658: }
1659: PetscFunctionReturn(PETSC_SUCCESS);
1660: }
1662: /*@
1663: PetscDTGaussJacobiQuadrature - quadrature for the interval [a, b] with the weight function
1664: $(x-a)^\alpha (x-b)^\beta$.
1666: Not collective
1668: Input Parameters:
1669: + npoints - the number of points in the quadrature rule
1670: . a - the left endpoint of the interval
1671: . b - the right endpoint of the interval
1672: . alpha - the left exponent
1673: - beta - the right exponent
1675: Output Parameters:
1676: + x - array of length npoints, the locations of the quadrature points
1677: - w - array of length npoints, the weights of the quadrature points
1679: Level: intermediate
1681: Note:
1682: This quadrature rule is exact for polynomials up to degree 2*npoints - 1.
1684: .seealso: `PetscDTGaussQuadrature()`
1685: @*/
1686: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
1687: {
1688: PetscInt i;
1690: PetscFunctionBegin;
1691: PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal));
1692: if (a != -1. || b != 1.) { /* shift */
1693: for (i = 0; i < npoints; i++) {
1694: x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1695: w[i] *= (b - a) / 2.;
1696: }
1697: }
1698: PetscFunctionReturn(PETSC_SUCCESS);
1699: }
1701: static PetscErrorCode PetscDTGaussLobattoJacobiQuadrature_Internal(PetscInt npoints, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
1702: {
1703: PetscInt i;
1705: PetscFunctionBegin;
1706: PetscCheck(npoints >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Number of points must be positive");
1707: /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
1708: PetscCheck(alpha > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "alpha must be > -1.");
1709: PetscCheck(beta > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "beta must be > -1.");
1711: x[0] = -1.;
1712: x[npoints - 1] = 1.;
1713: if (npoints > 2) PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints - 2, alpha + 1., beta + 1., &x[1], &w[1], newton));
1714: for (i = 1; i < npoints - 1; i++) w[i] /= (1. - x[i] * x[i]);
1715: PetscCall(PetscDTGaussLobattoJacobiEndweights_Internal(npoints, alpha, beta, &w[0], &w[npoints - 1]));
1716: PetscFunctionReturn(PETSC_SUCCESS);
1717: }
1719: /*@
1720: PetscDTGaussLobattoJacobiQuadrature - quadrature for the interval [a, b] with the weight function
1721: $(x-a)^\alpha (x-b)^\beta$, with endpoints a and b included as quadrature points.
1723: Not collective
1725: Input Parameters:
1726: + npoints - the number of points in the quadrature rule
1727: . a - the left endpoint of the interval
1728: . b - the right endpoint of the interval
1729: . alpha - the left exponent
1730: - beta - the right exponent
1732: Output Parameters:
1733: + x - array of length npoints, the locations of the quadrature points
1734: - w - array of length npoints, the weights of the quadrature points
1736: Level: intermediate
1738: Note:
1739: This quadrature rule is exact for polynomials up to degree 2*npoints - 3.
1741: .seealso: `PetscDTGaussJacobiQuadrature()`
1742: @*/
1743: PetscErrorCode PetscDTGaussLobattoJacobiQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
1744: {
1745: PetscInt i;
1747: PetscFunctionBegin;
1748: PetscCall(PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal));
1749: if (a != -1. || b != 1.) { /* shift */
1750: for (i = 0; i < npoints; i++) {
1751: x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1752: w[i] *= (b - a) / 2.;
1753: }
1754: }
1755: PetscFunctionReturn(PETSC_SUCCESS);
1756: }
1758: /*@
1759: PetscDTGaussQuadrature - create Gauss-Legendre quadrature
1761: Not Collective
1763: Input Parameters:
1764: + npoints - number of points
1765: . a - left end of interval (often-1)
1766: - b - right end of interval (often +1)
1768: Output Parameters:
1769: + x - quadrature points
1770: - w - quadrature weights
1772: Level: intermediate
1774: References:
1775: . * - Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 1969.
1777: .seealso: `PetscDTLegendreEval()`, `PetscDTGaussJacobiQuadrature()`
1778: @*/
1779: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
1780: {
1781: PetscInt i;
1783: PetscFunctionBegin;
1784: PetscCall(PetscDTGaussJacobiQuadrature_Internal(npoints, 0., 0., x, w, PetscDTGaussQuadratureNewton_Internal));
1785: if (a != -1. || b != 1.) { /* shift */
1786: for (i = 0; i < npoints; i++) {
1787: x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
1788: w[i] *= (b - a) / 2.;
1789: }
1790: }
1791: PetscFunctionReturn(PETSC_SUCCESS);
1792: }
1794: /*@C
1795: PetscDTGaussLobattoLegendreQuadrature - creates a set of the locations and weights of the Gauss-Lobatto-Legendre
1796: nodes of a given size on the domain [-1,1]
1798: Not Collective
1800: Input Parameters:
1801: + n - number of grid nodes
1802: - type - `PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA` or `PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON`
1804: Output Parameters:
1805: + x - quadrature points
1806: - w - quadrature weights
1808: Level: intermediate
1810: Notes:
1811: For n > 30 the Newton approach computes duplicate (incorrect) values for some nodes because the initial guess is apparently not
1812: close enough to the desired solution
1814: These are useful for implementing spectral methods based on Gauss-Lobatto-Legendre (GLL) nodes
1816: 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
1818: .seealso: `PetscDTGaussQuadrature()`, `PetscGaussLobattoLegendreCreateType`
1820: @*/
1821: PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt npoints, PetscGaussLobattoLegendreCreateType type, PetscReal *x, PetscReal *w)
1822: {
1823: PetscBool newton;
1825: PetscFunctionBegin;
1826: PetscCheck(npoints >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must provide at least 2 grid points per element");
1827: newton = (PetscBool)(type == PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON);
1828: PetscCall(PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, 0., 0., x, w, newton));
1829: PetscFunctionReturn(PETSC_SUCCESS);
1830: }
1832: /*@
1833: PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature
1835: Not Collective
1837: Input Parameters:
1838: + dim - The spatial dimension
1839: . Nc - The number of components
1840: . npoints - number of points in one dimension
1841: . a - left end of interval (often-1)
1842: - b - right end of interval (often +1)
1844: Output Parameter:
1845: . q - A `PetscQuadrature` object
1847: Level: intermediate
1849: .seealso: `PetscDTGaussQuadrature()`, `PetscDTLegendreEval()`
1850: @*/
1851: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
1852: {
1853: PetscInt totpoints = dim > 1 ? dim > 2 ? npoints * PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k, c;
1854: PetscReal *x, *w, *xw, *ww;
1856: PetscFunctionBegin;
1857: PetscCall(PetscMalloc1(totpoints * dim, &x));
1858: PetscCall(PetscMalloc1(totpoints * Nc, &w));
1859: /* Set up the Golub-Welsch system */
1860: switch (dim) {
1861: case 0:
1862: PetscCall(PetscFree(x));
1863: PetscCall(PetscFree(w));
1864: PetscCall(PetscMalloc1(1, &x));
1865: PetscCall(PetscMalloc1(Nc, &w));
1866: x[0] = 0.0;
1867: for (c = 0; c < Nc; ++c) w[c] = 1.0;
1868: break;
1869: case 1:
1870: PetscCall(PetscMalloc1(npoints, &ww));
1871: PetscCall(PetscDTGaussQuadrature(npoints, a, b, x, ww));
1872: for (i = 0; i < npoints; ++i)
1873: for (c = 0; c < Nc; ++c) w[i * Nc + c] = ww[i];
1874: PetscCall(PetscFree(ww));
1875: break;
1876: case 2:
1877: PetscCall(PetscMalloc2(npoints, &xw, npoints, &ww));
1878: PetscCall(PetscDTGaussQuadrature(npoints, a, b, xw, ww));
1879: for (i = 0; i < npoints; ++i) {
1880: for (j = 0; j < npoints; ++j) {
1881: x[(i * npoints + j) * dim + 0] = xw[i];
1882: x[(i * npoints + j) * dim + 1] = xw[j];
1883: for (c = 0; c < Nc; ++c) w[(i * npoints + j) * Nc + c] = ww[i] * ww[j];
1884: }
1885: }
1886: PetscCall(PetscFree2(xw, ww));
1887: break;
1888: case 3:
1889: PetscCall(PetscMalloc2(npoints, &xw, npoints, &ww));
1890: PetscCall(PetscDTGaussQuadrature(npoints, a, b, xw, ww));
1891: for (i = 0; i < npoints; ++i) {
1892: for (j = 0; j < npoints; ++j) {
1893: for (k = 0; k < npoints; ++k) {
1894: x[((i * npoints + j) * npoints + k) * dim + 0] = xw[i];
1895: x[((i * npoints + j) * npoints + k) * dim + 1] = xw[j];
1896: x[((i * npoints + j) * npoints + k) * dim + 2] = xw[k];
1897: for (c = 0; c < Nc; ++c) w[((i * npoints + j) * npoints + k) * Nc + c] = ww[i] * ww[j] * ww[k];
1898: }
1899: }
1900: }
1901: PetscCall(PetscFree2(xw, ww));
1902: break;
1903: default:
1904: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %" PetscInt_FMT, dim);
1905: }
1906: PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
1907: PetscCall(PetscQuadratureSetOrder(*q, 2 * npoints - 1));
1908: PetscCall(PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w));
1909: PetscCall(PetscObjectChangeTypeName((PetscObject)*q, "GaussTensor"));
1910: PetscFunctionReturn(PETSC_SUCCESS);
1911: }
1913: /*@
1914: PetscDTStroudConicalQuadrature - create Stroud conical quadrature for a simplex
1916: Not Collective
1918: Input Parameters:
1919: + dim - The simplex dimension
1920: . Nc - The number of components
1921: . npoints - The number of points in one dimension
1922: . a - left end of interval (often-1)
1923: - b - right end of interval (often +1)
1925: Output Parameter:
1926: . q - A PetscQuadrature object
1928: Level: intermediate
1930: Note:
1931: For dim == 1, this is Gauss-Legendre quadrature
1933: References:
1934: . * - Karniadakis and Sherwin. FIAT
1936: .seealso: `PetscDTGaussTensorQuadrature()`, `PetscDTGaussQuadrature()`
1937: @*/
1938: PetscErrorCode PetscDTStroudConicalQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
1939: {
1940: PetscInt totprev, totrem;
1941: PetscInt totpoints;
1942: PetscReal *p1, *w1;
1943: PetscReal *x, *w;
1944: PetscInt i, j, k, l, m, pt, c;
1946: PetscFunctionBegin;
1947: PetscCheck(!(a != -1.0) && !(b != 1.0), PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
1948: totpoints = 1;
1949: for (i = 0, totpoints = 1; i < dim; i++) totpoints *= npoints;
1950: PetscCall(PetscMalloc1(totpoints * dim, &x));
1951: PetscCall(PetscMalloc1(totpoints * Nc, &w));
1952: PetscCall(PetscMalloc2(npoints, &p1, npoints, &w1));
1953: for (i = 0; i < totpoints * Nc; i++) w[i] = 1.;
1954: for (i = 0, totprev = 1, totrem = totpoints / npoints; i < dim; i++) {
1955: PetscReal mul;
1957: mul = PetscPowReal(2., -i);
1958: PetscCall(PetscDTGaussJacobiQuadrature(npoints, -1., 1., i, 0.0, p1, w1));
1959: for (pt = 0, l = 0; l < totprev; l++) {
1960: for (j = 0; j < npoints; j++) {
1961: for (m = 0; m < totrem; m++, pt++) {
1962: for (k = 0; k < i; k++) x[pt * dim + k] = (x[pt * dim + k] + 1.) * (1. - p1[j]) * 0.5 - 1.;
1963: x[pt * dim + i] = p1[j];
1964: for (c = 0; c < Nc; c++) w[pt * Nc + c] *= mul * w1[j];
1965: }
1966: }
1967: }
1968: totprev *= npoints;
1969: totrem /= npoints;
1970: }
1971: PetscCall(PetscFree2(p1, w1));
1972: PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
1973: PetscCall(PetscQuadratureSetOrder(*q, 2 * npoints - 1));
1974: PetscCall(PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w));
1975: PetscCall(PetscObjectChangeTypeName((PetscObject)*q, "StroudConical"));
1976: PetscFunctionReturn(PETSC_SUCCESS);
1977: }
1979: static PetscBool MinSymTriQuadCite = PETSC_FALSE;
1980: const char MinSymTriQuadCitation[] = "@article{WitherdenVincent2015,\n"
1981: " title = {On the identification of symmetric quadrature rules for finite element methods},\n"
1982: " journal = {Computers & Mathematics with Applications},\n"
1983: " volume = {69},\n"
1984: " number = {10},\n"
1985: " pages = {1232-1241},\n"
1986: " year = {2015},\n"
1987: " issn = {0898-1221},\n"
1988: " doi = {10.1016/j.camwa.2015.03.017},\n"
1989: " url = {https://www.sciencedirect.com/science/article/pii/S0898122115001224},\n"
1990: " author = {F.D. Witherden and P.E. Vincent},\n"
1991: "}\n";
1993: #include "petscdttriquadrules.h"
1995: static PetscBool MinSymTetQuadCite = PETSC_FALSE;
1996: const char MinSymTetQuadCitation[] = "@article{JaskowiecSukumar2021\n"
1997: " author = {Jaskowiec, Jan and Sukumar, N.},\n"
1998: " title = {High-order symmetric cubature rules for tetrahedra and pyramids},\n"
1999: " journal = {International Journal for Numerical Methods in Engineering},\n"
2000: " volume = {122},\n"
2001: " number = {1},\n"
2002: " pages = {148-171},\n"
2003: " doi = {10.1002/nme.6528},\n"
2004: " url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.6528},\n"
2005: " eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.6528},\n"
2006: " year = {2021}\n"
2007: "}\n";
2009: #include "petscdttetquadrules.h"
2011: // https://en.wikipedia.org/wiki/Partition_(number_theory)
2012: static PetscErrorCode PetscDTPartitionNumber(PetscInt n, PetscInt *p)
2013: {
2014: // sequence A000041 in the OEIS
2015: 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};
2016: PetscInt tabulated_max = PETSC_STATIC_ARRAY_LENGTH(partition) - 1;
2018: PetscFunctionBegin;
2019: PetscCheck(n >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Partition number not defined for negative number %" PetscInt_FMT, n);
2020: // not implementing the pentagonal number recurrence, we don't need partition numbers for n that high
2021: 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);
2022: *p = partition[n];
2023: PetscFunctionReturn(PETSC_SUCCESS);
2024: }
2026: /*@
2027: PetscDTSimplexQuadrature - Create a quadrature rule for a simplex that exactly integrates polynomials up to a given degree.
2029: Not Collective
2031: Input Parameters:
2032: + dim - The spatial dimension of the simplex (1 = segment, 2 = triangle, 3 = tetrahedron)
2033: . degree - The largest polynomial degree that is required to be integrated exactly
2034: - type - left end of interval (often-1)
2036: Output Parameter:
2037: . quad - A `PetscQuadrature` object for integration over the biunit simplex
2038: (defined by the bounds $x_i >= -1$ and $\sum_i x_i <= 2 - d$) that is exact for
2039: polynomials up to the given degree
2041: Level: intermediate
2043: .seealso: `PetscDTSimplexQuadratureType`, `PetscDTGaussQuadrature()`, `PetscDTStroudCononicalQuadrature()`, `PetscQuadrature`
2044: @*/
2045: PetscErrorCode PetscDTSimplexQuadrature(PetscInt dim, PetscInt degree, PetscDTSimplexQuadratureType type, PetscQuadrature *quad)
2046: {
2047: PetscDTSimplexQuadratureType orig_type = type;
2049: PetscFunctionBegin;
2050: PetscCheck(dim >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Negative dimension %" PetscInt_FMT, dim);
2051: PetscCheck(degree >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Negative degree %" PetscInt_FMT, degree);
2052: if (type == PETSCDTSIMPLEXQUAD_DEFAULT) type = PETSCDTSIMPLEXQUAD_MINSYM;
2053: if (type == PETSCDTSIMPLEXQUAD_CONIC || dim < 2) {
2054: PetscInt points_per_dim = (degree + 2) / 2; // ceil((degree + 1) / 2);
2055: PetscCall(PetscDTStroudConicalQuadrature(dim, 1, points_per_dim, -1, 1, quad));
2056: } else {
2057: PetscInt n = dim + 1;
2058: PetscInt fact = 1;
2059: PetscInt *part, *perm;
2060: PetscInt p = 0;
2061: PetscInt max_degree;
2062: const PetscInt *nodes_per_type = NULL;
2063: const PetscInt *all_num_full_nodes = NULL;
2064: const PetscReal **weights_list = NULL;
2065: const PetscReal **compact_nodes_list = NULL;
2066: const char *citation = NULL;
2067: PetscBool *cited = NULL;
2069: switch (dim) {
2070: case 2:
2071: cited = &MinSymTriQuadCite;
2072: citation = MinSymTriQuadCitation;
2073: max_degree = PetscDTWVTriQuad_max_degree;
2074: nodes_per_type = PetscDTWVTriQuad_num_orbits;
2075: all_num_full_nodes = PetscDTWVTriQuad_num_nodes;
2076: weights_list = PetscDTWVTriQuad_weights;
2077: compact_nodes_list = PetscDTWVTriQuad_orbits;
2078: break;
2079: case 3:
2080: cited = &MinSymTetQuadCite;
2081: citation = MinSymTetQuadCitation;
2082: max_degree = PetscDTJSTetQuad_max_degree;
2083: nodes_per_type = PetscDTJSTetQuad_num_orbits;
2084: all_num_full_nodes = PetscDTJSTetQuad_num_nodes;
2085: weights_list = PetscDTJSTetQuad_weights;
2086: compact_nodes_list = PetscDTJSTetQuad_orbits;
2087: break;
2088: default:
2089: max_degree = -1;
2090: break;
2091: }
2093: if (degree > max_degree) {
2094: if (orig_type == PETSCDTSIMPLEXQUAD_DEFAULT) {
2095: // fall back to conic
2096: PetscCall(PetscDTSimplexQuadrature(dim, degree, PETSCDTSIMPLEXQUAD_CONIC, quad));
2097: PetscFunctionReturn(PETSC_SUCCESS);
2098: } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Minimal symmetric quadrature for dim %" PetscInt_FMT ", degree %" PetscInt_FMT " unsupported", dim, degree);
2099: }
2101: PetscCall(PetscCitationsRegister(citation, cited));
2103: PetscCall(PetscDTPartitionNumber(n, &p));
2104: for (PetscInt d = 2; d <= n; d++) fact *= d;
2106: PetscInt num_full_nodes = all_num_full_nodes[degree];
2107: const PetscReal *all_compact_nodes = compact_nodes_list[degree];
2108: const PetscReal *all_compact_weights = weights_list[degree];
2109: nodes_per_type = &nodes_per_type[p * degree];
2111: PetscReal *points;
2112: PetscReal *counts;
2113: PetscReal *weights;
2114: PetscReal *bary_to_biunit; // row-major transformation of barycentric coordinate to biunit
2115: PetscQuadrature q;
2117: // compute the transformation
2118: PetscCall(PetscMalloc1(n * dim, &bary_to_biunit));
2119: for (PetscInt d = 0; d < dim; d++) {
2120: for (PetscInt b = 0; b < n; b++) bary_to_biunit[d * n + b] = (d == b) ? 1.0 : -1.0;
2121: }
2123: PetscCall(PetscMalloc3(n, &part, n, &perm, n, &counts));
2124: PetscCall(PetscCalloc1(num_full_nodes * dim, &points));
2125: PetscCall(PetscMalloc1(num_full_nodes, &weights));
2127: // (0, 0, ...) is the first partition lexicographically
2128: PetscCall(PetscArrayzero(part, n));
2129: PetscCall(PetscArrayzero(counts, n));
2130: counts[0] = n;
2132: // for each partition
2133: for (PetscInt s = 0, node_offset = 0; s < p; s++) {
2134: PetscInt num_compact_coords = part[n - 1] + 1;
2136: const PetscReal *compact_nodes = all_compact_nodes;
2137: const PetscReal *compact_weights = all_compact_weights;
2138: all_compact_nodes += num_compact_coords * nodes_per_type[s];
2139: all_compact_weights += nodes_per_type[s];
2141: // for every permutation of the vertices
2142: for (PetscInt f = 0; f < fact; f++) {
2143: PetscCall(PetscDTEnumPerm(n, f, perm, NULL));
2145: // check if it is a valid permutation
2146: PetscInt digit;
2147: for (digit = 1; digit < n; digit++) {
2148: // skip permutations that would duplicate a node because it has a smaller symmetry group
2149: if (part[digit - 1] == part[digit] && perm[digit - 1] > perm[digit]) break;
2150: }
2151: if (digit < n) continue;
2153: // create full nodes from this permutation of the compact nodes
2154: PetscReal *full_nodes = &points[node_offset * dim];
2155: PetscReal *full_weights = &weights[node_offset];
2157: PetscCall(PetscArraycpy(full_weights, compact_weights, nodes_per_type[s]));
2158: for (PetscInt b = 0; b < n; b++) {
2159: for (PetscInt d = 0; d < dim; d++) {
2160: 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]];
2161: }
2162: }
2163: node_offset += nodes_per_type[s];
2164: }
2166: if (s < p - 1) { // Generate the next partition
2167: /* A partition is described by the number of coordinates that are in
2168: * each set of duplicates (counts) and redundantly by mapping each
2169: * index to its set of duplicates (part)
2170: *
2171: * Counts should always be in nonincreasing order
2172: *
2173: * We want to generate the partitions lexically by part, which means
2174: * finding the last index where count > 1 and reducing by 1.
2175: *
2176: * For the new counts beyond that index, we eagerly assign the remaining
2177: * capacity of the partition to smaller indices (ensures lexical ordering),
2178: * while respecting the nonincreasing invariant of the counts
2179: */
2180: PetscInt last_digit = part[n - 1];
2181: PetscInt last_digit_with_extra = last_digit;
2182: while (counts[last_digit_with_extra] == 1) last_digit_with_extra--;
2183: PetscInt limit = --counts[last_digit_with_extra];
2184: PetscInt total_to_distribute = last_digit - last_digit_with_extra + 1;
2185: for (PetscInt digit = last_digit_with_extra + 1; digit < n; digit++) {
2186: counts[digit] = PetscMin(limit, total_to_distribute);
2187: total_to_distribute -= PetscMin(limit, total_to_distribute);
2188: }
2189: for (PetscInt digit = 0, offset = 0; digit < n; digit++) {
2190: PetscInt count = counts[digit];
2191: for (PetscInt c = 0; c < count; c++) part[offset++] = digit;
2192: }
2193: }
2194: }
2195: PetscCall(PetscFree3(part, perm, counts));
2196: PetscCall(PetscFree(bary_to_biunit));
2197: PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, &q));
2198: PetscCall(PetscQuadratureSetOrder(q, degree));
2199: PetscCall(PetscQuadratureSetData(q, dim, 1, num_full_nodes, points, weights));
2200: *quad = q;
2201: }
2202: PetscFunctionReturn(PETSC_SUCCESS);
2203: }
2205: /*@
2206: PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell
2208: Not Collective
2210: Input Parameters:
2211: + dim - The cell dimension
2212: . level - The number of points in one dimension, 2^l
2213: . a - left end of interval (often-1)
2214: - b - right end of interval (often +1)
2216: Output Parameter:
2217: . q - A `PetscQuadrature` object
2219: Level: intermediate
2221: .seealso: `PetscDTGaussTensorQuadrature()`, `PetscQuadrature`
2222: @*/
2223: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
2224: {
2225: const PetscInt p = 16; /* Digits of precision in the evaluation */
2226: const PetscReal alpha = (b - a) / 2.; /* Half-width of the integration interval */
2227: const PetscReal beta = (b + a) / 2.; /* Center of the integration interval */
2228: const PetscReal h = PetscPowReal(2.0, -level); /* Step size, length between x_k */
2229: PetscReal xk; /* Quadrature point x_k on reference domain [-1, 1] */
2230: PetscReal wk = 0.5 * PETSC_PI; /* Quadrature weight at x_k */
2231: PetscReal *x, *w;
2232: PetscInt K, k, npoints;
2234: PetscFunctionBegin;
2235: PetscCheck(dim <= 1, PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %" PetscInt_FMT " not yet implemented", dim);
2236: PetscCheck(level, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
2237: /* Find K such that the weights are < 32 digits of precision */
2238: 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)));
2239: PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
2240: PetscCall(PetscQuadratureSetOrder(*q, 2 * K + 1));
2241: npoints = 2 * K - 1;
2242: PetscCall(PetscMalloc1(npoints * dim, &x));
2243: PetscCall(PetscMalloc1(npoints, &w));
2244: /* Center term */
2245: x[0] = beta;
2246: w[0] = 0.5 * alpha * PETSC_PI;
2247: for (k = 1; k < K; ++k) {
2248: wk = 0.5 * alpha * h * PETSC_PI * PetscCoshReal(k * h) / PetscSqr(PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2249: xk = PetscTanhReal(0.5 * PETSC_PI * PetscSinhReal(k * h));
2250: x[2 * k - 1] = -alpha * xk + beta;
2251: w[2 * k - 1] = wk;
2252: x[2 * k + 0] = alpha * xk + beta;
2253: w[2 * k + 0] = wk;
2254: }
2255: PetscCall(PetscQuadratureSetData(*q, dim, 1, npoints, x, w));
2256: PetscFunctionReturn(PETSC_SUCCESS);
2257: }
2259: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2260: {
2261: const PetscInt p = 16; /* Digits of precision in the evaluation */
2262: const PetscReal alpha = (b - a) / 2.; /* Half-width of the integration interval */
2263: const PetscReal beta = (b + a) / 2.; /* Center of the integration interval */
2264: PetscReal h = 1.0; /* Step size, length between x_k */
2265: PetscInt l = 0; /* Level of refinement, h = 2^{-l} */
2266: PetscReal osum = 0.0; /* Integral on last level */
2267: PetscReal psum = 0.0; /* Integral on the level before the last level */
2268: PetscReal sum; /* Integral on current level */
2269: PetscReal yk; /* Quadrature point 1 - x_k on reference domain [-1, 1] */
2270: PetscReal lx, rx; /* Quadrature points to the left and right of 0 on the real domain [a, b] */
2271: PetscReal wk; /* Quadrature weight at x_k */
2272: PetscReal lval, rval; /* Terms in the quadature sum to the left and right of 0 */
2273: PetscInt d; /* Digits of precision in the integral */
2275: PetscFunctionBegin;
2276: PetscCheck(digits > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
2277: /* Center term */
2278: func(&beta, ctx, &lval);
2279: sum = 0.5 * alpha * PETSC_PI * lval;
2280: /* */
2281: do {
2282: PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
2283: PetscInt k = 1;
2285: ++l;
2286: /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %" PetscInt_FMT " sum: %15.15f\n", l, sum); */
2287: /* At each level of refinement, h --> h/2 and sum --> sum/2 */
2288: psum = osum;
2289: osum = sum;
2290: h *= 0.5;
2291: sum *= 0.5;
2292: do {
2293: wk = 0.5 * h * PETSC_PI * PetscCoshReal(k * h) / PetscSqr(PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2294: yk = 1.0 / (PetscExpReal(0.5 * PETSC_PI * PetscSinhReal(k * h)) * PetscCoshReal(0.5 * PETSC_PI * PetscSinhReal(k * h)));
2295: lx = -alpha * (1.0 - yk) + beta;
2296: rx = alpha * (1.0 - yk) + beta;
2297: func(&lx, ctx, &lval);
2298: func(&rx, ctx, &rval);
2299: lterm = alpha * wk * lval;
2300: maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
2301: sum += lterm;
2302: rterm = alpha * wk * rval;
2303: maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
2304: sum += rterm;
2305: ++k;
2306: /* Only need to evaluate every other point on refined levels */
2307: if (l != 1) ++k;
2308: } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */
2310: d1 = PetscLog10Real(PetscAbsReal(sum - osum));
2311: d2 = PetscLog10Real(PetscAbsReal(sum - psum));
2312: d3 = PetscLog10Real(maxTerm) - p;
2313: if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
2314: else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
2315: d = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1) / d2, 2 * d1), d3), d4)));
2316: } while (d < digits && l < 12);
2317: *sol = sum;
2319: PetscFunctionReturn(PETSC_SUCCESS);
2320: }
2322: #if defined(PETSC_HAVE_MPFR)
2323: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2324: {
2325: const PetscInt safetyFactor = 2; /* Calculate abcissa until 2*p digits */
2326: PetscInt l = 0; /* Level of refinement, h = 2^{-l} */
2327: mpfr_t alpha; /* Half-width of the integration interval */
2328: mpfr_t beta; /* Center of the integration interval */
2329: mpfr_t h; /* Step size, length between x_k */
2330: mpfr_t osum; /* Integral on last level */
2331: mpfr_t psum; /* Integral on the level before the last level */
2332: mpfr_t sum; /* Integral on current level */
2333: mpfr_t yk; /* Quadrature point 1 - x_k on reference domain [-1, 1] */
2334: mpfr_t lx, rx; /* Quadrature points to the left and right of 0 on the real domain [a, b] */
2335: mpfr_t wk; /* Quadrature weight at x_k */
2336: PetscReal lval, rval, rtmp; /* Terms in the quadature sum to the left and right of 0 */
2337: PetscInt d; /* Digits of precision in the integral */
2338: mpfr_t pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;
2340: PetscFunctionBegin;
2341: PetscCheck(digits > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
2342: /* Create high precision storage */
2343: 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);
2344: /* Initialization */
2345: mpfr_set_d(alpha, 0.5 * (b - a), MPFR_RNDN);
2346: mpfr_set_d(beta, 0.5 * (b + a), MPFR_RNDN);
2347: mpfr_set_d(osum, 0.0, MPFR_RNDN);
2348: mpfr_set_d(psum, 0.0, MPFR_RNDN);
2349: mpfr_set_d(h, 1.0, MPFR_RNDN);
2350: mpfr_const_pi(pi2, MPFR_RNDN);
2351: mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
2352: /* Center term */
2353: rtmp = 0.5 * (b + a);
2354: func(&rtmp, ctx, &lval);
2355: mpfr_set(sum, pi2, MPFR_RNDN);
2356: mpfr_mul(sum, sum, alpha, MPFR_RNDN);
2357: mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
2358: /* */
2359: do {
2360: PetscReal d1, d2, d3, d4;
2361: PetscInt k = 1;
2363: ++l;
2364: mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
2365: /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %" PetscInt_FMT " sum: %15.15f\n", l, sum); */
2366: /* At each level of refinement, h --> h/2 and sum --> sum/2 */
2367: mpfr_set(psum, osum, MPFR_RNDN);
2368: mpfr_set(osum, sum, MPFR_RNDN);
2369: mpfr_mul_d(h, h, 0.5, MPFR_RNDN);
2370: mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
2371: do {
2372: mpfr_set_si(kh, k, MPFR_RNDN);
2373: mpfr_mul(kh, kh, h, MPFR_RNDN);
2374: /* Weight */
2375: mpfr_set(wk, h, MPFR_RNDN);
2376: mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
2377: mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
2378: mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
2379: mpfr_cosh(tmp, msinh, MPFR_RNDN);
2380: mpfr_sqr(tmp, tmp, MPFR_RNDN);
2381: mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
2382: mpfr_div(wk, wk, tmp, MPFR_RNDN);
2383: /* Abscissa */
2384: mpfr_set_d(yk, 1.0, MPFR_RNDZ);
2385: mpfr_cosh(tmp, msinh, MPFR_RNDN);
2386: mpfr_div(yk, yk, tmp, MPFR_RNDZ);
2387: mpfr_exp(tmp, msinh, MPFR_RNDN);
2388: mpfr_div(yk, yk, tmp, MPFR_RNDZ);
2389: /* Quadrature points */
2390: mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
2391: mpfr_mul(lx, lx, alpha, MPFR_RNDU);
2392: mpfr_add(lx, lx, beta, MPFR_RNDU);
2393: mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
2394: mpfr_mul(rx, rx, alpha, MPFR_RNDD);
2395: mpfr_add(rx, rx, beta, MPFR_RNDD);
2396: /* Evaluation */
2397: rtmp = mpfr_get_d(lx, MPFR_RNDU);
2398: func(&rtmp, ctx, &lval);
2399: rtmp = mpfr_get_d(rx, MPFR_RNDD);
2400: func(&rtmp, ctx, &rval);
2401: /* Update */
2402: mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
2403: mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
2404: mpfr_add(sum, sum, tmp, MPFR_RNDN);
2405: mpfr_abs(tmp, tmp, MPFR_RNDN);
2406: mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
2407: mpfr_set(curTerm, tmp, MPFR_RNDN);
2408: mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
2409: mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
2410: mpfr_add(sum, sum, tmp, MPFR_RNDN);
2411: mpfr_abs(tmp, tmp, MPFR_RNDN);
2412: mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
2413: mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
2414: ++k;
2415: /* Only need to evaluate every other point on refined levels */
2416: if (l != 1) ++k;
2417: mpfr_log10(tmp, wk, MPFR_RNDN);
2418: mpfr_abs(tmp, tmp, MPFR_RNDN);
2419: } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor * digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
2420: mpfr_sub(tmp, sum, osum, MPFR_RNDN);
2421: mpfr_abs(tmp, tmp, MPFR_RNDN);
2422: mpfr_log10(tmp, tmp, MPFR_RNDN);
2423: d1 = mpfr_get_d(tmp, MPFR_RNDN);
2424: mpfr_sub(tmp, sum, psum, MPFR_RNDN);
2425: mpfr_abs(tmp, tmp, MPFR_RNDN);
2426: mpfr_log10(tmp, tmp, MPFR_RNDN);
2427: d2 = mpfr_get_d(tmp, MPFR_RNDN);
2428: mpfr_log10(tmp, maxTerm, MPFR_RNDN);
2429: d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
2430: mpfr_log10(tmp, curTerm, MPFR_RNDN);
2431: d4 = mpfr_get_d(tmp, MPFR_RNDN);
2432: d = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1) / d2, 2 * d1), d3), d4)));
2433: } while (d < digits && l < 8);
2434: *sol = mpfr_get_d(sum, MPFR_RNDN);
2435: /* Cleanup */
2436: mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
2437: PetscFunctionReturn(PETSC_SUCCESS);
2438: }
2439: #else
2441: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(const PetscReal[], void *, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, void *ctx, PetscReal *sol)
2442: {
2443: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
2444: }
2445: #endif
2447: /*@
2448: PetscDTTensorQuadratureCreate - create the tensor product quadrature from two lower-dimensional quadratures
2450: Not Collective
2452: Input Parameters:
2453: + q1 - The first quadrature
2454: - q2 - The second quadrature
2456: Output Parameter:
2457: . q - A `PetscQuadrature` object
2459: Level: intermediate
2461: .seealso: `PetscQuadrature`, `PetscDTGaussTensorQuadrature()`
2462: @*/
2463: PetscErrorCode PetscDTTensorQuadratureCreate(PetscQuadrature q1, PetscQuadrature q2, PetscQuadrature *q)
2464: {
2465: const PetscReal *x1, *w1, *x2, *w2;
2466: PetscReal *x, *w;
2467: PetscInt dim1, Nc1, Np1, order1, qa, d1;
2468: PetscInt dim2, Nc2, Np2, order2, qb, d2;
2469: PetscInt dim, Nc, Np, order, qc, d;
2471: PetscFunctionBegin;
2475: PetscCall(PetscQuadratureGetOrder(q1, &order1));
2476: PetscCall(PetscQuadratureGetOrder(q2, &order2));
2477: PetscCheck(order1 == order2, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Order1 %" PetscInt_FMT " != %" PetscInt_FMT " Order2", order1, order2);
2478: PetscCall(PetscQuadratureGetData(q1, &dim1, &Nc1, &Np1, &x1, &w1));
2479: PetscCall(PetscQuadratureGetData(q2, &dim2, &Nc2, &Np2, &x2, &w2));
2480: PetscCheck(Nc1 == Nc2, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "NumComp1 %" PetscInt_FMT " != %" PetscInt_FMT " NumComp2", Nc1, Nc2);
2482: dim = dim1 + dim2;
2483: Nc = Nc1;
2484: Np = Np1 * Np2;
2485: order = order1;
2486: PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, q));
2487: PetscCall(PetscQuadratureSetOrder(*q, order));
2488: PetscCall(PetscMalloc1(Np * dim, &x));
2489: PetscCall(PetscMalloc1(Np, &w));
2490: for (qa = 0, qc = 0; qa < Np1; ++qa) {
2491: for (qb = 0; qb < Np2; ++qb, ++qc) {
2492: for (d1 = 0, d = 0; d1 < dim1; ++d1, ++d) x[qc * dim + d] = x1[qa * dim1 + d1];
2493: for (d2 = 0; d2 < dim2; ++d2, ++d) x[qc * dim + d] = x2[qb * dim2 + d2];
2494: w[qc] = w1[qa] * w2[qb];
2495: }
2496: }
2497: PetscCall(PetscQuadratureSetData(*q, dim, Nc, Np, x, w));
2498: PetscFunctionReturn(PETSC_SUCCESS);
2499: }
2501: /* Overwrites A. Can only handle full-rank problems with m>=n
2502: A in column-major format
2503: Ainv in row-major format
2504: tau has length m
2505: worksize must be >= max(1,n)
2506: */
2507: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m, PetscInt mstride, PetscInt n, PetscReal *A_in, PetscReal *Ainv_out, PetscScalar *tau, PetscInt worksize, PetscScalar *work)
2508: {
2509: PetscBLASInt M, N, K, lda, ldb, ldwork, info;
2510: PetscScalar *A, *Ainv, *R, *Q, Alpha;
2512: PetscFunctionBegin;
2513: #if defined(PETSC_USE_COMPLEX)
2514: {
2515: PetscInt i, j;
2516: PetscCall(PetscMalloc2(m * n, &A, m * n, &Ainv));
2517: for (j = 0; j < n; j++) {
2518: for (i = 0; i < m; i++) A[i + m * j] = A_in[i + mstride * j];
2519: }
2520: mstride = m;
2521: }
2522: #else
2523: A = A_in;
2524: Ainv = Ainv_out;
2525: #endif
2527: PetscCall(PetscBLASIntCast(m, &M));
2528: PetscCall(PetscBLASIntCast(n, &N));
2529: PetscCall(PetscBLASIntCast(mstride, &lda));
2530: PetscCall(PetscBLASIntCast(worksize, &ldwork));
2531: PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
2532: PetscCallBLAS("LAPACKgeqrf", LAPACKgeqrf_(&M, &N, A, &lda, tau, work, &ldwork, &info));
2533: PetscCall(PetscFPTrapPop());
2534: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "xGEQRF error");
2535: R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */
2537: /* Extract an explicit representation of Q */
2538: Q = Ainv;
2539: PetscCall(PetscArraycpy(Q, A, mstride * n));
2540: K = N; /* full rank */
2541: PetscCallBLAS("LAPACKorgqr", LAPACKorgqr_(&M, &N, &K, Q, &lda, tau, work, &ldwork, &info));
2542: PetscCheck(!info, PETSC_COMM_SELF, PETSC_ERR_LIB, "xORGQR/xUNGQR error");
2544: /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
2545: Alpha = 1.0;
2546: ldb = lda;
2547: PetscCallBLAS("BLAStrsm", BLAStrsm_("Right", "Upper", "ConjugateTranspose", "NotUnitTriangular", &M, &N, &Alpha, R, &lda, Q, &ldb));
2548: /* Ainv is Q, overwritten with inverse */
2550: #if defined(PETSC_USE_COMPLEX)
2551: {
2552: PetscInt i;
2553: for (i = 0; i < m * n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
2554: PetscCall(PetscFree2(A, Ainv));
2555: }
2556: #endif
2557: PetscFunctionReturn(PETSC_SUCCESS);
2558: }
2560: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
2561: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval, const PetscReal *x, PetscInt ndegree, const PetscInt *degrees, PetscBool Transpose, PetscReal *B)
2562: {
2563: PetscReal *Bv;
2564: PetscInt i, j;
2566: PetscFunctionBegin;
2567: PetscCall(PetscMalloc1((ninterval + 1) * ndegree, &Bv));
2568: /* Point evaluation of L_p on all the source vertices */
2569: PetscCall(PetscDTLegendreEval(ninterval + 1, x, ndegree, degrees, Bv, NULL, NULL));
2570: /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
2571: for (i = 0; i < ninterval; i++) {
2572: for (j = 0; j < ndegree; j++) {
2573: if (Transpose) B[i + ninterval * j] = Bv[(i + 1) * ndegree + j] - Bv[i * ndegree + j];
2574: else B[i * ndegree + j] = Bv[(i + 1) * ndegree + j] - Bv[i * ndegree + j];
2575: }
2576: }
2577: PetscCall(PetscFree(Bv));
2578: PetscFunctionReturn(PETSC_SUCCESS);
2579: }
2581: /*@
2582: PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals
2584: Not Collective
2586: Input Parameters:
2587: + degree - degree of reconstruction polynomial
2588: . nsource - number of source intervals
2589: . sourcex - sorted coordinates of source cell boundaries (length nsource+1)
2590: . ntarget - number of target intervals
2591: - targetx - sorted coordinates of target cell boundaries (length ntarget+1)
2593: Output Parameter:
2594: . R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]
2596: Level: advanced
2598: .seealso: `PetscDTLegendreEval()`
2599: @*/
2600: PetscErrorCode PetscDTReconstructPoly(PetscInt degree, PetscInt nsource, const PetscReal *sourcex, PetscInt ntarget, const PetscReal *targetx, PetscReal *R)
2601: {
2602: PetscInt i, j, k, *bdegrees, worksize;
2603: PetscReal xmin, xmax, center, hscale, *sourcey, *targety, *Bsource, *Bsinv, *Btarget;
2604: PetscScalar *tau, *work;
2606: PetscFunctionBegin;
2610: 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);
2611: if (PetscDefined(USE_DEBUG)) {
2612: 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]);
2613: 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]);
2614: }
2615: xmin = PetscMin(sourcex[0], targetx[0]);
2616: xmax = PetscMax(sourcex[nsource], targetx[ntarget]);
2617: center = (xmin + xmax) / 2;
2618: hscale = (xmax - xmin) / 2;
2619: worksize = nsource;
2620: PetscCall(PetscMalloc4(degree + 1, &bdegrees, nsource + 1, &sourcey, nsource * (degree + 1), &Bsource, worksize, &work));
2621: PetscCall(PetscMalloc4(nsource, &tau, nsource * (degree + 1), &Bsinv, ntarget + 1, &targety, ntarget * (degree + 1), &Btarget));
2622: for (i = 0; i <= nsource; i++) sourcey[i] = (sourcex[i] - center) / hscale;
2623: for (i = 0; i <= degree; i++) bdegrees[i] = i + 1;
2624: PetscCall(PetscDTLegendreIntegrate(nsource, sourcey, degree + 1, bdegrees, PETSC_TRUE, Bsource));
2625: PetscCall(PetscDTPseudoInverseQR(nsource, nsource, degree + 1, Bsource, Bsinv, tau, nsource, work));
2626: for (i = 0; i <= ntarget; i++) targety[i] = (targetx[i] - center) / hscale;
2627: PetscCall(PetscDTLegendreIntegrate(ntarget, targety, degree + 1, bdegrees, PETSC_FALSE, Btarget));
2628: for (i = 0; i < ntarget; i++) {
2629: PetscReal rowsum = 0;
2630: for (j = 0; j < nsource; j++) {
2631: PetscReal sum = 0;
2632: for (k = 0; k < degree + 1; k++) sum += Btarget[i * (degree + 1) + k] * Bsinv[k * nsource + j];
2633: R[i * nsource + j] = sum;
2634: rowsum += sum;
2635: }
2636: for (j = 0; j < nsource; j++) R[i * nsource + j] /= rowsum; /* normalize each row */
2637: }
2638: PetscCall(PetscFree4(bdegrees, sourcey, Bsource, work));
2639: PetscCall(PetscFree4(tau, Bsinv, targety, Btarget));
2640: PetscFunctionReturn(PETSC_SUCCESS);
2641: }
2643: /*@C
2644: PetscGaussLobattoLegendreIntegrate - Compute the L2 integral of a function on the GLL points
2646: Not Collective
2648: Input Parameters:
2649: + n - the number of GLL nodes
2650: . nodes - the GLL nodes
2651: . weights - the GLL weights
2652: - f - the function values at the nodes
2654: Output Parameter:
2655: . in - the value of the integral
2657: Level: beginner
2659: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`
2660: @*/
2661: PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt n, PetscReal *nodes, PetscReal *weights, const PetscReal *f, PetscReal *in)
2662: {
2663: PetscInt i;
2665: PetscFunctionBegin;
2666: *in = 0.;
2667: for (i = 0; i < n; i++) *in += f[i] * f[i] * weights[i];
2668: PetscFunctionReturn(PETSC_SUCCESS);
2669: }
2671: /*@C
2672: PetscGaussLobattoLegendreElementLaplacianCreate - computes the Laplacian for a single 1d GLL element
2674: Not Collective
2676: Input Parameters:
2677: + n - the number of GLL nodes
2678: . nodes - the GLL nodes
2679: - weights - the GLL weights
2681: Output Parameter:
2682: . A - the stiffness element
2684: Level: beginner
2686: Notes:
2687: Destroy this with `PetscGaussLobattoLegendreElementLaplacianDestroy()`
2689: 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)
2691: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianDestroy()`
2692: @*/
2693: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
2694: {
2695: PetscReal **A;
2696: const PetscReal *gllnodes = nodes;
2697: const PetscInt p = n - 1;
2698: PetscReal z0, z1, z2 = -1, x, Lpj, Lpr;
2699: PetscInt i, j, nn, r;
2701: PetscFunctionBegin;
2702: PetscCall(PetscMalloc1(n, &A));
2703: PetscCall(PetscMalloc1(n * n, &A[0]));
2704: for (i = 1; i < n; i++) A[i] = A[i - 1] + n;
2706: for (j = 1; j < p; j++) {
2707: x = gllnodes[j];
2708: z0 = 1.;
2709: z1 = x;
2710: for (nn = 1; nn < p; nn++) {
2711: z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2712: z0 = z1;
2713: z1 = z2;
2714: }
2715: Lpj = z2;
2716: for (r = 1; r < p; r++) {
2717: if (r == j) {
2718: A[j][j] = 2. / (3. * (1. - gllnodes[j] * gllnodes[j]) * Lpj * Lpj);
2719: } else {
2720: x = gllnodes[r];
2721: z0 = 1.;
2722: z1 = x;
2723: for (nn = 1; nn < p; nn++) {
2724: z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2725: z0 = z1;
2726: z1 = z2;
2727: }
2728: Lpr = z2;
2729: A[r][j] = 4. / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * Lpr * (gllnodes[j] - gllnodes[r]) * (gllnodes[j] - gllnodes[r]));
2730: }
2731: }
2732: }
2733: for (j = 1; j < p + 1; j++) {
2734: x = gllnodes[j];
2735: z0 = 1.;
2736: z1 = x;
2737: for (nn = 1; nn < p; nn++) {
2738: z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2739: z0 = z1;
2740: z1 = z2;
2741: }
2742: Lpj = z2;
2743: A[j][0] = 4. * PetscPowRealInt(-1., p) / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * (1. + gllnodes[j]) * (1. + gllnodes[j]));
2744: A[0][j] = A[j][0];
2745: }
2746: for (j = 0; j < p; j++) {
2747: x = gllnodes[j];
2748: z0 = 1.;
2749: z1 = x;
2750: for (nn = 1; nn < p; nn++) {
2751: z2 = x * z1 * (2. * ((PetscReal)nn) + 1.) / (((PetscReal)nn) + 1.) - z0 * (((PetscReal)nn) / (((PetscReal)nn) + 1.));
2752: z0 = z1;
2753: z1 = z2;
2754: }
2755: Lpj = z2;
2757: A[p][j] = 4. / (((PetscReal)p) * (((PetscReal)p) + 1.) * Lpj * (1. - gllnodes[j]) * (1. - gllnodes[j]));
2758: A[j][p] = A[p][j];
2759: }
2760: A[0][0] = 0.5 + (((PetscReal)p) * (((PetscReal)p) + 1.) - 2.) / 6.;
2761: A[p][p] = A[0][0];
2762: *AA = A;
2763: PetscFunctionReturn(PETSC_SUCCESS);
2764: }
2766: /*@C
2767: PetscGaussLobattoLegendreElementLaplacianDestroy - frees the Laplacian for a single 1d GLL element created with `PetscGaussLobattoLegendreElementLaplacianCreate()`
2769: Not Collective
2771: Input Parameters:
2772: + n - the number of GLL nodes
2773: . nodes - the GLL nodes
2774: . weights - the GLL weightss
2775: - A - the stiffness element
2777: Level: beginner
2779: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`
2780: @*/
2781: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
2782: {
2783: PetscFunctionBegin;
2784: PetscCall(PetscFree((*AA)[0]));
2785: PetscCall(PetscFree(*AA));
2786: *AA = NULL;
2787: PetscFunctionReturn(PETSC_SUCCESS);
2788: }
2790: /*@C
2791: PetscGaussLobattoLegendreElementGradientCreate - computes the gradient for a single 1d GLL element
2793: Not Collective
2795: Input Parameter:
2796: + n - the number of GLL nodes
2797: . nodes - the GLL nodes
2798: . weights - the GLL weights
2800: Output Parameters:
2801: . AA - the stiffness element
2802: - AAT - the transpose of AA (pass in NULL if you do not need this array)
2804: Level: beginner
2806: Notes:
2807: Destroy this with `PetscGaussLobattoLegendreElementGradientDestroy()`
2809: You can access entries in these arrays with AA[i][j] but in memory it is stored in contiguous memory, row oriented
2811: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianDestroy()`, `PetscGaussLobattoLegendreElementGradientDestroy()`
2812: @*/
2813: PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA, PetscReal ***AAT)
2814: {
2815: PetscReal **A, **AT = NULL;
2816: const PetscReal *gllnodes = nodes;
2817: const PetscInt p = n - 1;
2818: PetscReal Li, Lj, d0;
2819: PetscInt i, j;
2821: PetscFunctionBegin;
2822: PetscCall(PetscMalloc1(n, &A));
2823: PetscCall(PetscMalloc1(n * n, &A[0]));
2824: for (i = 1; i < n; i++) A[i] = A[i - 1] + n;
2826: if (AAT) {
2827: PetscCall(PetscMalloc1(n, &AT));
2828: PetscCall(PetscMalloc1(n * n, &AT[0]));
2829: for (i = 1; i < n; i++) AT[i] = AT[i - 1] + n;
2830: }
2832: if (n == 1) A[0][0] = 0.;
2833: d0 = (PetscReal)p * ((PetscReal)p + 1.) / 4.;
2834: for (i = 0; i < n; i++) {
2835: for (j = 0; j < n; j++) {
2836: A[i][j] = 0.;
2837: PetscCall(PetscDTComputeJacobi(0., 0., p, gllnodes[i], &Li));
2838: PetscCall(PetscDTComputeJacobi(0., 0., p, gllnodes[j], &Lj));
2839: if (i != j) A[i][j] = Li / (Lj * (gllnodes[i] - gllnodes[j]));
2840: if ((j == i) && (i == 0)) A[i][j] = -d0;
2841: if (j == i && i == p) A[i][j] = d0;
2842: if (AT) AT[j][i] = A[i][j];
2843: }
2844: }
2845: if (AAT) *AAT = AT;
2846: *AA = A;
2847: PetscFunctionReturn(PETSC_SUCCESS);
2848: }
2850: /*@C
2851: PetscGaussLobattoLegendreElementGradientDestroy - frees the gradient for a single 1d GLL element obtained with `PetscGaussLobattoLegendreElementGradientCreate()`
2853: Not Collective
2855: Input Parameters:
2856: + n - the number of GLL nodes
2857: . nodes - the GLL nodes
2858: . weights - the GLL weights
2859: . AA - the stiffness element
2860: - AAT - the transpose of the element
2862: Level: beginner
2864: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`, `PetscGaussLobattoLegendreElementAdvectionCreate()`
2865: @*/
2866: PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA, PetscReal ***AAT)
2867: {
2868: PetscFunctionBegin;
2869: PetscCall(PetscFree((*AA)[0]));
2870: PetscCall(PetscFree(*AA));
2871: *AA = NULL;
2872: if (*AAT) {
2873: PetscCall(PetscFree((*AAT)[0]));
2874: PetscCall(PetscFree(*AAT));
2875: *AAT = NULL;
2876: }
2877: PetscFunctionReturn(PETSC_SUCCESS);
2878: }
2880: /*@C
2881: PetscGaussLobattoLegendreElementAdvectionCreate - computes the advection operator for a single 1d GLL element
2883: Not Collective
2885: Input Parameters:
2886: + n - the number of GLL nodes
2887: . nodes - the GLL nodes
2888: - weights - the GLL weightss
2890: Output Parameter:
2891: . AA - the stiffness element
2893: Level: beginner
2895: Notes:
2896: Destroy this with `PetscGaussLobattoLegendreElementAdvectionDestroy()`
2898: This is the same as the Gradient operator multiplied by the diagonal mass matrix
2900: You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented
2902: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementLaplacianCreate()`, `PetscGaussLobattoLegendreElementAdvectionDestroy()`
2903: @*/
2904: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
2905: {
2906: PetscReal **D;
2907: const PetscReal *gllweights = weights;
2908: const PetscInt glln = n;
2909: PetscInt i, j;
2911: PetscFunctionBegin;
2912: PetscCall(PetscGaussLobattoLegendreElementGradientCreate(n, nodes, weights, &D, NULL));
2913: for (i = 0; i < glln; i++) {
2914: for (j = 0; j < glln; j++) D[i][j] = gllweights[i] * D[i][j];
2915: }
2916: *AA = D;
2917: PetscFunctionReturn(PETSC_SUCCESS);
2918: }
2920: /*@C
2921: PetscGaussLobattoLegendreElementAdvectionDestroy - frees the advection stiffness for a single 1d GLL element created with `PetscGaussLobattoLegendreElementAdvectionCreate()`
2923: Not Collective
2925: Input Parameters:
2926: + n - the number of GLL nodes
2927: . nodes - the GLL nodes
2928: . weights - the GLL weights
2929: - A - advection
2931: Level: beginner
2933: .seealso: `PetscDTGaussLobattoLegendreQuadrature()`, `PetscGaussLobattoLegendreElementAdvectionCreate()`
2934: @*/
2935: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
2936: {
2937: PetscFunctionBegin;
2938: PetscCall(PetscFree((*AA)[0]));
2939: PetscCall(PetscFree(*AA));
2940: *AA = NULL;
2941: PetscFunctionReturn(PETSC_SUCCESS);
2942: }
2944: PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
2945: {
2946: PetscReal **A;
2947: const PetscReal *gllweights = weights;
2948: const PetscInt glln = n;
2949: PetscInt i, j;
2951: PetscFunctionBegin;
2952: PetscCall(PetscMalloc1(glln, &A));
2953: PetscCall(PetscMalloc1(glln * glln, &A[0]));
2954: for (i = 1; i < glln; i++) A[i] = A[i - 1] + glln;
2955: if (glln == 1) A[0][0] = 0.;
2956: for (i = 0; i < glln; i++) {
2957: for (j = 0; j < glln; j++) {
2958: A[i][j] = 0.;
2959: if (j == i) A[i][j] = gllweights[i];
2960: }
2961: }
2962: *AA = A;
2963: PetscFunctionReturn(PETSC_SUCCESS);
2964: }
2966: PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt n, PetscReal *nodes, PetscReal *weights, PetscReal ***AA)
2967: {
2968: PetscFunctionBegin;
2969: PetscCall(PetscFree((*AA)[0]));
2970: PetscCall(PetscFree(*AA));
2971: *AA = NULL;
2972: PetscFunctionReturn(PETSC_SUCCESS);
2973: }
2975: /*@
2976: PetscDTIndexToBary - convert an index into a barycentric coordinate.
2978: Input Parameters:
2979: + 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)
2980: . sum - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
2981: - index - the index to convert: should be >= 0 and < Binomial(len - 1 + sum, sum)
2983: Output Parameter:
2984: . coord - will be filled with the barycentric coordinate
2986: Level: beginner
2988: Note:
2989: The indices map to barycentric coordinates in lexicographic order, where the first index is the
2990: least significant and the last index is the most significant.
2992: .seealso: `PetscDTBaryToIndex()`
2993: @*/
2994: PetscErrorCode PetscDTIndexToBary(PetscInt len, PetscInt sum, PetscInt index, PetscInt coord[])
2995: {
2996: PetscInt c, d, s, total, subtotal, nexttotal;
2998: PetscFunctionBeginHot;
2999: PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
3000: PetscCheck(index >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
3001: if (!len) {
3002: if (!sum && !index) PetscFunctionReturn(PETSC_SUCCESS);
3003: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
3004: }
3005: for (c = 1, total = 1; c <= len; c++) {
3006: /* total is the number of ways to have a tuple of length c with sum */
3007: if (index < total) break;
3008: total = (total * (sum + c)) / c;
3009: }
3010: PetscCheck(c <= len, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index out of range");
3011: for (d = c; d < len; d++) coord[d] = 0;
3012: for (s = 0, subtotal = 1, nexttotal = 1; c > 0;) {
3013: /* subtotal is the number of ways to have a tuple of length c with sum s */
3014: /* nexttotal is the number of ways to have a tuple of length c-1 with sum s */
3015: if ((index + subtotal) >= total) {
3016: coord[--c] = sum - s;
3017: index -= (total - subtotal);
3018: sum = s;
3019: total = nexttotal;
3020: subtotal = 1;
3021: nexttotal = 1;
3022: s = 0;
3023: } else {
3024: subtotal = (subtotal * (c + s)) / (s + 1);
3025: nexttotal = (nexttotal * (c - 1 + s)) / (s + 1);
3026: s++;
3027: }
3028: }
3029: PetscFunctionReturn(PETSC_SUCCESS);
3030: }
3032: /*@
3033: PetscDTBaryToIndex - convert a barycentric coordinate to an index
3035: Input Parameters:
3036: + 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)
3037: . sum - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
3038: - coord - a barycentric coordinate with the given length and sum
3040: Output Parameter:
3041: . index - the unique index for the coordinate, >= 0 and < Binomial(len - 1 + sum, sum)
3043: Level: beginner
3045: Note:
3046: The indices map to barycentric coordinates in lexicographic order, where the first index is the
3047: least significant and the last index is the most significant.
3049: .seealso: `PetscDTIndexToBary`
3050: @*/
3051: PetscErrorCode PetscDTBaryToIndex(PetscInt len, PetscInt sum, const PetscInt coord[], PetscInt *index)
3052: {
3053: PetscInt c;
3054: PetscInt i;
3055: PetscInt total;
3057: PetscFunctionBeginHot;
3058: PetscCheck(len >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
3059: if (!len) {
3060: if (!sum) {
3061: *index = 0;
3062: PetscFunctionReturn(PETSC_SUCCESS);
3063: }
3064: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
3065: }
3066: for (c = 1, total = 1; c < len; c++) total = (total * (sum + c)) / c;
3067: i = total - 1;
3068: c = len - 1;
3069: sum -= coord[c];
3070: while (sum > 0) {
3071: PetscInt subtotal;
3072: PetscInt s;
3074: for (s = 1, subtotal = 1; s < sum; s++) subtotal = (subtotal * (c + s)) / s;
3075: i -= subtotal;
3076: sum -= coord[--c];
3077: }
3078: *index = i;
3079: PetscFunctionReturn(PETSC_SUCCESS);
3080: }