Actual source code: dt.c
petsc-3.9.0 2018-04-07
1: /* Discretization tools */
3: #include <petscconf.h>
4: #if defined(PETSC_HAVE_MATHIMF_H)
5: #include <mathimf.h> /* this needs to be included before math.h */
6: #endif
7: #ifdef PETSC_HAVE_MPFR
8: #include <mpfr.h>
9: #endif
11: #include <petscdt.h>
12: #include <petscblaslapack.h>
13: #include <petsc/private/petscimpl.h>
14: #include <petsc/private/dtimpl.h>
15: #include <petscviewer.h>
16: #include <petscdmplex.h>
17: #include <petscdmshell.h>
19: static PetscBool GaussCite = PETSC_FALSE;
20: const char GaussCitation[] = "@article{GolubWelsch1969,\n"
21: " author = {Golub and Welsch},\n"
22: " title = {Calculation of Quadrature Rules},\n"
23: " journal = {Math. Comp.},\n"
24: " volume = {23},\n"
25: " number = {106},\n"
26: " pages = {221--230},\n"
27: " year = {1969}\n}\n";
29: /*@
30: PetscQuadratureCreate - Create a PetscQuadrature object
32: Collective on MPI_Comm
34: Input Parameter:
35: . comm - The communicator for the PetscQuadrature object
37: Output Parameter:
38: . q - The PetscQuadrature object
40: Level: beginner
42: .keywords: PetscQuadrature, quadrature, create
43: .seealso: PetscQuadratureDestroy(), PetscQuadratureGetData()
44: @*/
45: PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
46: {
51: PetscSysInitializePackage();
52: PetscHeaderCreate(*q,PETSC_OBJECT_CLASSID,"PetscQuadrature","Quadrature","DT",comm,PetscQuadratureDestroy,PetscQuadratureView);
53: (*q)->dim = -1;
54: (*q)->Nc = 1;
55: (*q)->order = -1;
56: (*q)->numPoints = 0;
57: (*q)->points = NULL;
58: (*q)->weights = NULL;
59: return(0);
60: }
62: /*@
63: PetscQuadratureDuplicate - Create a deep copy of the PetscQuadrature object
65: Collective on PetscQuadrature
67: Input Parameter:
68: . q - The PetscQuadrature object
70: Output Parameter:
71: . r - The new PetscQuadrature object
73: Level: beginner
75: .keywords: PetscQuadrature, quadrature, clone
76: .seealso: PetscQuadratureCreate(), PetscQuadratureDestroy(), PetscQuadratureGetData()
77: @*/
78: PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
79: {
80: PetscInt order, dim, Nc, Nq;
81: const PetscReal *points, *weights;
82: PetscReal *p, *w;
83: PetscErrorCode ierr;
87: PetscQuadratureCreate(PetscObjectComm((PetscObject) q), r);
88: PetscQuadratureGetOrder(q, &order);
89: PetscQuadratureSetOrder(*r, order);
90: PetscQuadratureGetData(q, &dim, &Nc, &Nq, &points, &weights);
91: PetscMalloc1(Nq*dim, &p);
92: PetscMalloc1(Nq*Nc, &w);
93: PetscMemcpy(p, points, Nq*dim * sizeof(PetscReal));
94: PetscMemcpy(w, weights, Nc * Nq * sizeof(PetscReal));
95: PetscQuadratureSetData(*r, dim, Nc, Nq, p, w);
96: return(0);
97: }
99: /*@
100: PetscQuadratureDestroy - Destroys a PetscQuadrature object
102: Collective on PetscQuadrature
104: Input Parameter:
105: . q - The PetscQuadrature object
107: Level: beginner
109: .keywords: PetscQuadrature, quadrature, destroy
110: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
111: @*/
112: PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
113: {
117: if (!*q) return(0);
119: if (--((PetscObject)(*q))->refct > 0) {
120: *q = NULL;
121: return(0);
122: }
123: PetscFree((*q)->points);
124: PetscFree((*q)->weights);
125: PetscHeaderDestroy(q);
126: return(0);
127: }
129: /*@
130: PetscQuadratureGetOrder - Return the order of the method
132: Not collective
134: Input Parameter:
135: . q - The PetscQuadrature object
137: Output Parameter:
138: . order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated
140: Level: intermediate
142: .seealso: PetscQuadratureSetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
143: @*/
144: PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature q, PetscInt *order)
145: {
149: *order = q->order;
150: return(0);
151: }
153: /*@
154: PetscQuadratureSetOrder - Return the order of the method
156: Not collective
158: Input Parameters:
159: + q - The PetscQuadrature object
160: - order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated
162: Level: intermediate
164: .seealso: PetscQuadratureGetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
165: @*/
166: PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature q, PetscInt order)
167: {
170: q->order = order;
171: return(0);
172: }
174: /*@
175: PetscQuadratureGetNumComponents - Return the number of components for functions to be integrated
177: Not collective
179: Input Parameter:
180: . q - The PetscQuadrature object
182: Output Parameter:
183: . Nc - The number of components
185: Note: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.
187: Level: intermediate
189: .seealso: PetscQuadratureSetNumComponents(), PetscQuadratureGetData(), PetscQuadratureSetData()
190: @*/
191: PetscErrorCode PetscQuadratureGetNumComponents(PetscQuadrature q, PetscInt *Nc)
192: {
196: *Nc = q->Nc;
197: return(0);
198: }
200: /*@
201: PetscQuadratureSetNumComponents - Return the number of components for functions to be integrated
203: Not collective
205: Input Parameters:
206: + q - The PetscQuadrature object
207: - Nc - The number of components
209: Note: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.
211: Level: intermediate
213: .seealso: PetscQuadratureGetNumComponents(), PetscQuadratureGetData(), PetscQuadratureSetData()
214: @*/
215: PetscErrorCode PetscQuadratureSetNumComponents(PetscQuadrature q, PetscInt Nc)
216: {
219: q->Nc = Nc;
220: return(0);
221: }
223: /*@C
224: PetscQuadratureGetData - Returns the data defining the quadrature
226: Not collective
228: Input Parameter:
229: . q - The PetscQuadrature object
231: Output Parameters:
232: + dim - The spatial dimension
233: , Nc - The number of components
234: . npoints - The number of quadrature points
235: . points - The coordinates of each quadrature point
236: - weights - The weight of each quadrature point
238: Level: intermediate
240: Fortran Notes: From Fortran you must call PetscQuadratureRestoreData() when you are done with the data
242: .keywords: PetscQuadrature, quadrature
243: .seealso: PetscQuadratureCreate(), PetscQuadratureSetData()
244: @*/
245: PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *Nc, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
246: {
249: if (dim) {
251: *dim = q->dim;
252: }
253: if (Nc) {
255: *Nc = q->Nc;
256: }
257: if (npoints) {
259: *npoints = q->numPoints;
260: }
261: if (points) {
263: *points = q->points;
264: }
265: if (weights) {
267: *weights = q->weights;
268: }
269: return(0);
270: }
272: /*@C
273: PetscQuadratureSetData - Sets the data defining the quadrature
275: Not collective
277: Input Parameters:
278: + q - The PetscQuadrature object
279: . dim - The spatial dimension
280: , Nc - The number of components
281: . npoints - The number of quadrature points
282: . points - The coordinates of each quadrature point
283: - weights - The weight of each quadrature point
285: Note: This routine owns the references to points and weights, so they msut be allocated using PetscMalloc() and the user should not free them.
287: Level: intermediate
289: .keywords: PetscQuadrature, quadrature
290: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
291: @*/
292: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt Nc, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
293: {
296: if (dim >= 0) q->dim = dim;
297: if (Nc >= 0) q->Nc = Nc;
298: if (npoints >= 0) q->numPoints = npoints;
299: if (points) {
301: q->points = points;
302: }
303: if (weights) {
305: q->weights = weights;
306: }
307: return(0);
308: }
310: /*@C
311: PetscQuadratureView - Views a PetscQuadrature object
313: Collective on PetscQuadrature
315: Input Parameters:
316: + q - The PetscQuadrature object
317: - viewer - The PetscViewer object
319: Level: beginner
321: .keywords: PetscQuadrature, quadrature, view
322: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
323: @*/
324: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
325: {
326: PetscInt q, d, c;
330: PetscObjectPrintClassNamePrefixType((PetscObject)quad,viewer);
331: if (quad->Nc > 1) {PetscViewerASCIIPrintf(viewer, "Quadrature on %D points with %D components\n (", quad->numPoints, quad->Nc);}
332: else {PetscViewerASCIIPrintf(viewer, "Quadrature on %D points\n (", quad->numPoints);}
333: for (q = 0; q < quad->numPoints; ++q) {
334: for (d = 0; d < quad->dim; ++d) {
335: if (d) PetscViewerASCIIPrintf(viewer, ", ");
336: PetscViewerASCIIPrintf(viewer, "%g\n", (double)quad->points[q*quad->dim+d]);
337: }
338: if (quad->Nc > 1) {
339: PetscViewerASCIIPrintf(viewer, ") (");
340: for (c = 0; c < quad->Nc; ++c) {
341: if (c) PetscViewerASCIIPrintf(viewer, ", ");
342: PetscViewerASCIIPrintf(viewer, "%g", (double)quad->weights[q*quad->Nc+c]);
343: }
344: PetscViewerASCIIPrintf(viewer, ")\n");
345: } else {
346: PetscViewerASCIIPrintf(viewer, ") %g\n", (double)quad->weights[q]);
347: }
348: }
349: return(0);
350: }
352: /*@C
353: PetscQuadratureExpandComposite - Return a quadrature over the composite element, which has the original quadrature in each subelement
355: Not collective
357: Input Parameter:
358: + q - The original PetscQuadrature
359: . numSubelements - The number of subelements the original element is divided into
360: . v0 - An array of the initial points for each subelement
361: - jac - An array of the Jacobian mappings from the reference to each subelement
363: Output Parameters:
364: . dim - The dimension
366: Note: Together v0 and jac define an affine mapping from the original reference element to each subelement
368: Not available from Fortran
370: Level: intermediate
372: .seealso: PetscFECreate(), PetscSpaceGetDimension(), PetscDualSpaceGetDimension()
373: @*/
374: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
375: {
376: const PetscReal *points, *weights;
377: PetscReal *pointsRef, *weightsRef;
378: PetscInt dim, Nc, order, npoints, npointsRef, c, p, cp, d, e;
379: PetscErrorCode ierr;
386: PetscQuadratureCreate(PETSC_COMM_SELF, qref);
387: PetscQuadratureGetOrder(q, &order);
388: PetscQuadratureGetData(q, &dim, &Nc, &npoints, &points, &weights);
389: npointsRef = npoints*numSubelements;
390: PetscMalloc1(npointsRef*dim,&pointsRef);
391: PetscMalloc1(npointsRef*Nc, &weightsRef);
392: for (c = 0; c < numSubelements; ++c) {
393: for (p = 0; p < npoints; ++p) {
394: for (d = 0; d < dim; ++d) {
395: pointsRef[(c*npoints + p)*dim+d] = v0[c*dim+d];
396: for (e = 0; e < dim; ++e) {
397: pointsRef[(c*npoints + p)*dim+d] += jac[(c*dim + d)*dim+e]*(points[p*dim+e] + 1.0);
398: }
399: }
400: /* Could also use detJ here */
401: for (cp = 0; cp < Nc; ++cp) weightsRef[(c*npoints+p)*Nc+cp] = weights[p*Nc+cp]/numSubelements;
402: }
403: }
404: PetscQuadratureSetOrder(*qref, order);
405: PetscQuadratureSetData(*qref, dim, Nc, npointsRef, pointsRef, weightsRef);
406: return(0);
407: }
409: /*@
410: PetscDTLegendreEval - evaluate Legendre polynomial at points
412: Not Collective
414: Input Arguments:
415: + npoints - number of spatial points to evaluate at
416: . points - array of locations to evaluate at
417: . ndegree - number of basis degrees to evaluate
418: - degrees - sorted array of degrees to evaluate
420: Output Arguments:
421: + B - row-oriented basis evaluation matrix B[point*ndegree + degree] (dimension npoints*ndegrees, allocated by caller) (or NULL)
422: . D - row-oriented derivative evaluation matrix (or NULL)
423: - D2 - row-oriented second derivative evaluation matrix (or NULL)
425: Level: intermediate
427: .seealso: PetscDTGaussQuadrature()
428: @*/
429: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
430: {
431: PetscInt i,maxdegree;
434: if (!npoints || !ndegree) return(0);
435: maxdegree = degrees[ndegree-1];
436: for (i=0; i<npoints; i++) {
437: PetscReal pm1,pm2,pd1,pd2,pdd1,pdd2,x;
438: PetscInt j,k;
439: x = points[i];
440: pm2 = 0;
441: pm1 = 1;
442: pd2 = 0;
443: pd1 = 0;
444: pdd2 = 0;
445: pdd1 = 0;
446: k = 0;
447: if (degrees[k] == 0) {
448: if (B) B[i*ndegree+k] = pm1;
449: if (D) D[i*ndegree+k] = pd1;
450: if (D2) D2[i*ndegree+k] = pdd1;
451: k++;
452: }
453: for (j=1; j<=maxdegree; j++,k++) {
454: PetscReal p,d,dd;
455: p = ((2*j-1)*x*pm1 - (j-1)*pm2)/j;
456: d = pd2 + (2*j-1)*pm1;
457: dd = pdd2 + (2*j-1)*pd1;
458: pm2 = pm1;
459: pm1 = p;
460: pd2 = pd1;
461: pd1 = d;
462: pdd2 = pdd1;
463: pdd1 = dd;
464: if (degrees[k] == j) {
465: if (B) B[i*ndegree+k] = p;
466: if (D) D[i*ndegree+k] = d;
467: if (D2) D2[i*ndegree+k] = dd;
468: }
469: }
470: }
471: return(0);
472: }
474: /*@
475: PetscDTGaussQuadrature - create Gauss quadrature
477: Not Collective
479: Input Arguments:
480: + npoints - number of points
481: . a - left end of interval (often-1)
482: - b - right end of interval (often +1)
484: Output Arguments:
485: + x - quadrature points
486: - w - quadrature weights
488: Level: intermediate
490: References:
491: . 1. - Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 1969.
493: .seealso: PetscDTLegendreEval()
494: @*/
495: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
496: {
498: PetscInt i;
499: PetscReal *work;
500: PetscScalar *Z;
501: PetscBLASInt N,LDZ,info;
504: PetscCitationsRegister(GaussCitation, &GaussCite);
505: /* Set up the Golub-Welsch system */
506: for (i=0; i<npoints; i++) {
507: x[i] = 0; /* diagonal is 0 */
508: if (i) w[i-1] = 0.5 / PetscSqrtReal(1 - 1./PetscSqr(2*i));
509: }
510: PetscMalloc2(npoints*npoints,&Z,PetscMax(1,2*npoints-2),&work);
511: PetscBLASIntCast(npoints,&N);
512: LDZ = N;
513: PetscFPTrapPush(PETSC_FP_TRAP_OFF);
514: PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&N,x,w,Z,&LDZ,work,&info));
515: PetscFPTrapPop();
516: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");
518: for (i=0; i<(npoints+1)/2; i++) {
519: PetscReal y = 0.5 * (-x[i] + x[npoints-i-1]); /* enforces symmetry */
520: x[i] = (a+b)/2 - y*(b-a)/2;
521: if (x[i] == -0.0) x[i] = 0.0;
522: x[npoints-i-1] = (a+b)/2 + y*(b-a)/2;
524: w[i] = w[npoints-1-i] = 0.5*(b-a)*(PetscSqr(PetscAbsScalar(Z[i*npoints])) + PetscSqr(PetscAbsScalar(Z[(npoints-i-1)*npoints])));
525: }
526: PetscFree2(Z,work);
527: return(0);
528: }
530: /*@
531: PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature
533: Not Collective
535: Input Arguments:
536: + dim - The spatial dimension
537: . Nc - The number of components
538: . npoints - number of points in one dimension
539: . a - left end of interval (often-1)
540: - b - right end of interval (often +1)
542: Output Argument:
543: . q - A PetscQuadrature object
545: Level: intermediate
547: .seealso: PetscDTGaussQuadrature(), PetscDTLegendreEval()
548: @*/
549: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
550: {
551: PetscInt totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k, c;
552: PetscReal *x, *w, *xw, *ww;
556: PetscMalloc1(totpoints*dim,&x);
557: PetscMalloc1(totpoints*Nc,&w);
558: /* Set up the Golub-Welsch system */
559: switch (dim) {
560: case 0:
561: PetscFree(x);
562: PetscFree(w);
563: PetscMalloc1(1, &x);
564: PetscMalloc1(Nc, &w);
565: x[0] = 0.0;
566: for (c = 0; c < Nc; ++c) w[c] = 1.0;
567: break;
568: case 1:
569: PetscMalloc1(npoints,&ww);
570: PetscDTGaussQuadrature(npoints, a, b, x, ww);
571: for (i = 0; i < npoints; ++i) for (c = 0; c < Nc; ++c) w[i*Nc+c] = ww[i];
572: PetscFree(ww);
573: break;
574: case 2:
575: PetscMalloc2(npoints,&xw,npoints,&ww);
576: PetscDTGaussQuadrature(npoints, a, b, xw, ww);
577: for (i = 0; i < npoints; ++i) {
578: for (j = 0; j < npoints; ++j) {
579: x[(i*npoints+j)*dim+0] = xw[i];
580: x[(i*npoints+j)*dim+1] = xw[j];
581: for (c = 0; c < Nc; ++c) w[(i*npoints+j)*Nc+c] = ww[i] * ww[j];
582: }
583: }
584: PetscFree2(xw,ww);
585: break;
586: case 3:
587: PetscMalloc2(npoints,&xw,npoints,&ww);
588: PetscDTGaussQuadrature(npoints, a, b, xw, ww);
589: for (i = 0; i < npoints; ++i) {
590: for (j = 0; j < npoints; ++j) {
591: for (k = 0; k < npoints; ++k) {
592: x[((i*npoints+j)*npoints+k)*dim+0] = xw[i];
593: x[((i*npoints+j)*npoints+k)*dim+1] = xw[j];
594: x[((i*npoints+j)*npoints+k)*dim+2] = xw[k];
595: for (c = 0; c < Nc; ++c) w[((i*npoints+j)*npoints+k)*Nc+c] = ww[i] * ww[j] * ww[k];
596: }
597: }
598: }
599: PetscFree2(xw,ww);
600: break;
601: default:
602: SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
603: }
604: PetscQuadratureCreate(PETSC_COMM_SELF, q);
605: PetscQuadratureSetOrder(*q, npoints-1);
606: PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);
607: return(0);
608: }
610: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
611: Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
612: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorial_Internal(PetscInt n, PetscReal *factorial)
613: {
614: PetscReal f = 1.0;
615: PetscInt i;
618: for (i = 1; i < n+1; ++i) f *= i;
619: *factorial = f;
620: return(0);
621: }
623: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
624: Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
625: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
626: {
627: PetscReal apb, pn1, pn2;
628: PetscInt k;
631: if (!n) {*P = 1.0; return(0);}
632: if (n == 1) {*P = 0.5 * (a - b + (a + b + 2.0) * x); return(0);}
633: apb = a + b;
634: pn2 = 1.0;
635: pn1 = 0.5 * (a - b + (apb + 2.0) * x);
636: *P = 0.0;
637: for (k = 2; k < n+1; ++k) {
638: PetscReal a1 = 2.0 * k * (k + apb) * (2.0*k + apb - 2.0);
639: PetscReal a2 = (2.0 * k + apb - 1.0) * (a*a - b*b);
640: PetscReal a3 = (2.0 * k + apb - 2.0) * (2.0 * k + apb - 1.0) * (2.0 * k + apb);
641: PetscReal a4 = 2.0 * (k + a - 1.0) * (k + b - 1.0) * (2.0 * k + apb);
643: a2 = a2 / a1;
644: a3 = a3 / a1;
645: a4 = a4 / a1;
646: *P = (a2 + a3 * x) * pn1 - a4 * pn2;
647: pn2 = pn1;
648: pn1 = *P;
649: }
650: return(0);
651: }
653: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
654: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
655: {
656: PetscReal nP;
660: if (!n) {*P = 0.0; return(0);}
661: PetscDTComputeJacobi(a+1, b+1, n-1, x, &nP);
662: *P = 0.5 * (a + b + n + 1) * nP;
663: return(0);
664: }
666: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
667: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapSquareToTriangle_Internal(PetscReal x, PetscReal y, PetscReal *xi, PetscReal *eta)
668: {
670: *xi = 0.5 * (1.0 + x) * (1.0 - y) - 1.0;
671: *eta = y;
672: return(0);
673: }
675: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
676: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapCubeToTetrahedron_Internal(PetscReal x, PetscReal y, PetscReal z, PetscReal *xi, PetscReal *eta, PetscReal *zeta)
677: {
679: *xi = 0.25 * (1.0 + x) * (1.0 - y) * (1.0 - z) - 1.0;
680: *eta = 0.5 * (1.0 + y) * (1.0 - z) - 1.0;
681: *zeta = z;
682: return(0);
683: }
685: static PetscErrorCode PetscDTGaussJacobiQuadrature1D_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
686: {
687: PetscInt maxIter = 100;
688: PetscReal eps = 1.0e-8;
689: PetscReal a1, a2, a3, a4, a5, a6;
690: PetscInt k;
695: a1 = PetscPowReal(2.0, a+b+1);
696: #if defined(PETSC_HAVE_TGAMMA)
697: a2 = PetscTGamma(a + npoints + 1);
698: a3 = PetscTGamma(b + npoints + 1);
699: a4 = PetscTGamma(a + b + npoints + 1);
700: #else
701: {
702: PetscInt ia, ib;
704: ia = (PetscInt) a;
705: ib = (PetscInt) b;
706: if (ia == a && ib == b && ia + npoints + 1 > 0 && ib + npoints + 1 > 0 && ia + ib + npoints + 1 > 0) { /* All gamma(x) terms are (x-1)! terms */
707: PetscDTFactorial_Internal(ia + npoints, &a2);
708: PetscDTFactorial_Internal(ib + npoints, &a3);
709: PetscDTFactorial_Internal(ia + ib + npoints, &a4);
710: } else {
711: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
712: }
713: }
714: #endif
716: PetscDTFactorial_Internal(npoints, &a5);
717: a6 = a1 * a2 * a3 / a4 / a5;
718: /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
719: Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
720: for (k = 0; k < npoints; ++k) {
721: PetscReal r = -PetscCosReal((2.0*k + 1.0) * PETSC_PI / (2.0 * npoints)), dP;
722: PetscInt j;
724: if (k > 0) r = 0.5 * (r + x[k-1]);
725: for (j = 0; j < maxIter; ++j) {
726: PetscReal s = 0.0, delta, f, fp;
727: PetscInt i;
729: for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
730: PetscDTComputeJacobi(a, b, npoints, r, &f);
731: PetscDTComputeJacobiDerivative(a, b, npoints, r, &fp);
732: delta = f / (fp - f * s);
733: r = r - delta;
734: if (PetscAbsReal(delta) < eps) break;
735: }
736: x[k] = r;
737: PetscDTComputeJacobiDerivative(a, b, npoints, x[k], &dP);
738: w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
739: }
740: return(0);
741: }
743: /*@
744: PetscDTGaussJacobiQuadrature - create Gauss-Jacobi quadrature for a simplex
746: Not Collective
748: Input Arguments:
749: + dim - The simplex dimension
750: . Nc - The number of components
751: . npoints - The number of points in one dimension
752: . a - left end of interval (often-1)
753: - b - right end of interval (often +1)
755: Output Argument:
756: . q - A PetscQuadrature object
758: Level: intermediate
760: References:
761: . 1. - Karniadakis and Sherwin. FIAT
763: .seealso: PetscDTGaussTensorQuadrature(), PetscDTGaussQuadrature()
764: @*/
765: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
766: {
767: PetscInt totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints;
768: PetscReal *px, *wx, *py, *wy, *pz, *wz, *x, *w;
769: PetscInt i, j, k, c;
773: if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
774: PetscMalloc1(totpoints*dim, &x);
775: PetscMalloc1(totpoints*Nc, &w);
776: switch (dim) {
777: case 0:
778: PetscFree(x);
779: PetscFree(w);
780: PetscMalloc1(1, &x);
781: PetscMalloc1(Nc, &w);
782: x[0] = 0.0;
783: for (c = 0; c < Nc; ++c) w[c] = 1.0;
784: break;
785: case 1:
786: PetscMalloc1(npoints,&wx);
787: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, x, wx);
788: for (i = 0; i < npoints; ++i) for (c = 0; c < Nc; ++c) w[i*Nc+c] = wx[i];
789: PetscFree(wx);
790: break;
791: case 2:
792: PetscMalloc4(npoints,&px,npoints,&wx,npoints,&py,npoints,&wy);
793: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, px, wx);
794: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 1.0, 0.0, py, wy);
795: for (i = 0; i < npoints; ++i) {
796: for (j = 0; j < npoints; ++j) {
797: PetscDTMapSquareToTriangle_Internal(px[i], py[j], &x[(i*npoints+j)*2+0], &x[(i*npoints+j)*2+1]);
798: for (c = 0; c < Nc; ++c) w[(i*npoints+j)*Nc+c] = 0.5 * wx[i] * wy[j];
799: }
800: }
801: PetscFree4(px,wx,py,wy);
802: break;
803: case 3:
804: PetscMalloc6(npoints,&px,npoints,&wx,npoints,&py,npoints,&wy,npoints,&pz,npoints,&wz);
805: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 0.0, 0.0, px, wx);
806: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 1.0, 0.0, py, wy);
807: PetscDTGaussJacobiQuadrature1D_Internal(npoints, 2.0, 0.0, pz, wz);
808: for (i = 0; i < npoints; ++i) {
809: for (j = 0; j < npoints; ++j) {
810: for (k = 0; k < npoints; ++k) {
811: PetscDTMapCubeToTetrahedron_Internal(px[i], py[j], pz[k], &x[((i*npoints+j)*npoints+k)*3+0], &x[((i*npoints+j)*npoints+k)*3+1], &x[((i*npoints+j)*npoints+k)*3+2]);
812: for (c = 0; c < Nc; ++c) w[((i*npoints+j)*npoints+k)*Nc+c] = 0.125 * wx[i] * wy[j] * wz[k];
813: }
814: }
815: }
816: PetscFree6(px,wx,py,wy,pz,wz);
817: break;
818: default:
819: SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
820: }
821: PetscQuadratureCreate(PETSC_COMM_SELF, q);
822: PetscQuadratureSetOrder(*q, npoints-1);
823: PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);
824: return(0);
825: }
827: /*@
828: PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell
830: Not Collective
832: Input Arguments:
833: + dim - The cell dimension
834: . level - The number of points in one dimension, 2^l
835: . a - left end of interval (often-1)
836: - b - right end of interval (often +1)
838: Output Argument:
839: . q - A PetscQuadrature object
841: Level: intermediate
843: .seealso: PetscDTGaussTensorQuadrature()
844: @*/
845: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
846: {
847: const PetscInt p = 16; /* Digits of precision in the evaluation */
848: const PetscReal alpha = (b-a)/2.; /* Half-width of the integration interval */
849: const PetscReal beta = (b+a)/2.; /* Center of the integration interval */
850: const PetscReal h = PetscPowReal(2.0, -level); /* Step size, length between x_k */
851: PetscReal xk; /* Quadrature point x_k on reference domain [-1, 1] */
852: PetscReal wk = 0.5*PETSC_PI; /* Quadrature weight at x_k */
853: PetscReal *x, *w;
854: PetscInt K, k, npoints;
855: PetscErrorCode ierr;
858: if (dim > 1) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %d not yet implemented", dim);
859: if (!level) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
860: /* Find K such that the weights are < 32 digits of precision */
861: for (K = 1; PetscAbsReal(PetscLog10Real(wk)) < 2*p; ++K) {
862: wk = 0.5*h*PETSC_PI*PetscCoshReal(K*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(K*h)));
863: }
864: PetscQuadratureCreate(PETSC_COMM_SELF, q);
865: PetscQuadratureSetOrder(*q, 2*K+1);
866: npoints = 2*K-1;
867: PetscMalloc1(npoints*dim, &x);
868: PetscMalloc1(npoints, &w);
869: /* Center term */
870: x[0] = beta;
871: w[0] = 0.5*alpha*PETSC_PI;
872: for (k = 1; k < K; ++k) {
873: wk = 0.5*alpha*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
874: xk = PetscTanhReal(0.5*PETSC_PI*PetscSinhReal(k*h));
875: x[2*k-1] = -alpha*xk+beta;
876: w[2*k-1] = wk;
877: x[2*k+0] = alpha*xk+beta;
878: w[2*k+0] = wk;
879: }
880: PetscQuadratureSetData(*q, dim, 1, npoints, x, w);
881: return(0);
882: }
884: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
885: {
886: const PetscInt p = 16; /* Digits of precision in the evaluation */
887: const PetscReal alpha = (b-a)/2.; /* Half-width of the integration interval */
888: const PetscReal beta = (b+a)/2.; /* Center of the integration interval */
889: PetscReal h = 1.0; /* Step size, length between x_k */
890: PetscInt l = 0; /* Level of refinement, h = 2^{-l} */
891: PetscReal osum = 0.0; /* Integral on last level */
892: PetscReal psum = 0.0; /* Integral on the level before the last level */
893: PetscReal sum; /* Integral on current level */
894: PetscReal yk; /* Quadrature point 1 - x_k on reference domain [-1, 1] */
895: PetscReal lx, rx; /* Quadrature points to the left and right of 0 on the real domain [a, b] */
896: PetscReal wk; /* Quadrature weight at x_k */
897: PetscReal lval, rval; /* Terms in the quadature sum to the left and right of 0 */
898: PetscInt d; /* Digits of precision in the integral */
901: if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
902: /* Center term */
903: func(beta, &lval);
904: sum = 0.5*alpha*PETSC_PI*lval;
905: /* */
906: do {
907: PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
908: PetscInt k = 1;
910: ++l;
911: /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
912: /* At each level of refinement, h --> h/2 and sum --> sum/2 */
913: psum = osum;
914: osum = sum;
915: h *= 0.5;
916: sum *= 0.5;
917: do {
918: wk = 0.5*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
919: yk = 1.0/(PetscExpReal(0.5*PETSC_PI*PetscSinhReal(k*h)) * PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
920: lx = -alpha*(1.0 - yk)+beta;
921: rx = alpha*(1.0 - yk)+beta;
922: func(lx, &lval);
923: func(rx, &rval);
924: lterm = alpha*wk*lval;
925: maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
926: sum += lterm;
927: rterm = alpha*wk*rval;
928: maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
929: sum += rterm;
930: ++k;
931: /* Only need to evaluate every other point on refined levels */
932: if (l != 1) ++k;
933: } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */
935: d1 = PetscLog10Real(PetscAbsReal(sum - osum));
936: d2 = PetscLog10Real(PetscAbsReal(sum - psum));
937: d3 = PetscLog10Real(maxTerm) - p;
938: if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
939: else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
940: d = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
941: } while (d < digits && l < 12);
942: *sol = sum;
944: return(0);
945: }
947: #ifdef PETSC_HAVE_MPFR
948: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
949: {
950: const PetscInt safetyFactor = 2; /* Calculate abcissa until 2*p digits */
951: PetscInt l = 0; /* Level of refinement, h = 2^{-l} */
952: mpfr_t alpha; /* Half-width of the integration interval */
953: mpfr_t beta; /* Center of the integration interval */
954: mpfr_t h; /* Step size, length between x_k */
955: mpfr_t osum; /* Integral on last level */
956: mpfr_t psum; /* Integral on the level before the last level */
957: mpfr_t sum; /* Integral on current level */
958: mpfr_t yk; /* Quadrature point 1 - x_k on reference domain [-1, 1] */
959: mpfr_t lx, rx; /* Quadrature points to the left and right of 0 on the real domain [a, b] */
960: mpfr_t wk; /* Quadrature weight at x_k */
961: PetscReal lval, rval; /* Terms in the quadature sum to the left and right of 0 */
962: PetscInt d; /* Digits of precision in the integral */
963: mpfr_t pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;
966: if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
967: /* Create high precision storage */
968: 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);
969: /* Initialization */
970: mpfr_set_d(alpha, 0.5*(b-a), MPFR_RNDN);
971: mpfr_set_d(beta, 0.5*(b+a), MPFR_RNDN);
972: mpfr_set_d(osum, 0.0, MPFR_RNDN);
973: mpfr_set_d(psum, 0.0, MPFR_RNDN);
974: mpfr_set_d(h, 1.0, MPFR_RNDN);
975: mpfr_const_pi(pi2, MPFR_RNDN);
976: mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
977: /* Center term */
978: func(0.5*(b+a), &lval);
979: mpfr_set(sum, pi2, MPFR_RNDN);
980: mpfr_mul(sum, sum, alpha, MPFR_RNDN);
981: mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
982: /* */
983: do {
984: PetscReal d1, d2, d3, d4;
985: PetscInt k = 1;
987: ++l;
988: mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
989: /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
990: /* At each level of refinement, h --> h/2 and sum --> sum/2 */
991: mpfr_set(psum, osum, MPFR_RNDN);
992: mpfr_set(osum, sum, MPFR_RNDN);
993: mpfr_mul_d(h, h, 0.5, MPFR_RNDN);
994: mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
995: do {
996: mpfr_set_si(kh, k, MPFR_RNDN);
997: mpfr_mul(kh, kh, h, MPFR_RNDN);
998: /* Weight */
999: mpfr_set(wk, h, MPFR_RNDN);
1000: mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
1001: mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
1002: mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
1003: mpfr_cosh(tmp, msinh, MPFR_RNDN);
1004: mpfr_sqr(tmp, tmp, MPFR_RNDN);
1005: mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
1006: mpfr_div(wk, wk, tmp, MPFR_RNDN);
1007: /* Abscissa */
1008: mpfr_set_d(yk, 1.0, MPFR_RNDZ);
1009: mpfr_cosh(tmp, msinh, MPFR_RNDN);
1010: mpfr_div(yk, yk, tmp, MPFR_RNDZ);
1011: mpfr_exp(tmp, msinh, MPFR_RNDN);
1012: mpfr_div(yk, yk, tmp, MPFR_RNDZ);
1013: /* Quadrature points */
1014: mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
1015: mpfr_mul(lx, lx, alpha, MPFR_RNDU);
1016: mpfr_add(lx, lx, beta, MPFR_RNDU);
1017: mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
1018: mpfr_mul(rx, rx, alpha, MPFR_RNDD);
1019: mpfr_add(rx, rx, beta, MPFR_RNDD);
1020: /* Evaluation */
1021: func(mpfr_get_d(lx, MPFR_RNDU), &lval);
1022: func(mpfr_get_d(rx, MPFR_RNDD), &rval);
1023: /* Update */
1024: mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
1025: mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
1026: mpfr_add(sum, sum, tmp, MPFR_RNDN);
1027: mpfr_abs(tmp, tmp, MPFR_RNDN);
1028: mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
1029: mpfr_set(curTerm, tmp, MPFR_RNDN);
1030: mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
1031: mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
1032: mpfr_add(sum, sum, tmp, MPFR_RNDN);
1033: mpfr_abs(tmp, tmp, MPFR_RNDN);
1034: mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
1035: mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
1036: ++k;
1037: /* Only need to evaluate every other point on refined levels */
1038: if (l != 1) ++k;
1039: mpfr_log10(tmp, wk, MPFR_RNDN);
1040: mpfr_abs(tmp, tmp, MPFR_RNDN);
1041: } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor*digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
1042: mpfr_sub(tmp, sum, osum, MPFR_RNDN);
1043: mpfr_abs(tmp, tmp, MPFR_RNDN);
1044: mpfr_log10(tmp, tmp, MPFR_RNDN);
1045: d1 = mpfr_get_d(tmp, MPFR_RNDN);
1046: mpfr_sub(tmp, sum, psum, MPFR_RNDN);
1047: mpfr_abs(tmp, tmp, MPFR_RNDN);
1048: mpfr_log10(tmp, tmp, MPFR_RNDN);
1049: d2 = mpfr_get_d(tmp, MPFR_RNDN);
1050: mpfr_log10(tmp, maxTerm, MPFR_RNDN);
1051: d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
1052: mpfr_log10(tmp, curTerm, MPFR_RNDN);
1053: d4 = mpfr_get_d(tmp, MPFR_RNDN);
1054: d = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
1055: } while (d < digits && l < 8);
1056: *sol = mpfr_get_d(sum, MPFR_RNDN);
1057: /* Cleanup */
1058: mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
1059: return(0);
1060: }
1061: #else
1063: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
1064: {
1065: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
1066: }
1067: #endif
1069: /* Overwrites A. Can only handle full-rank problems with m>=n
1070: * A in column-major format
1071: * Ainv in row-major format
1072: * tau has length m
1073: * worksize must be >= max(1,n)
1074: */
1075: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
1076: {
1078: PetscBLASInt M,N,K,lda,ldb,ldwork,info;
1079: PetscScalar *A,*Ainv,*R,*Q,Alpha;
1082: #if defined(PETSC_USE_COMPLEX)
1083: {
1084: PetscInt i,j;
1085: PetscMalloc2(m*n,&A,m*n,&Ainv);
1086: for (j=0; j<n; j++) {
1087: for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
1088: }
1089: mstride = m;
1090: }
1091: #else
1092: A = A_in;
1093: Ainv = Ainv_out;
1094: #endif
1096: PetscBLASIntCast(m,&M);
1097: PetscBLASIntCast(n,&N);
1098: PetscBLASIntCast(mstride,&lda);
1099: PetscBLASIntCast(worksize,&ldwork);
1100: PetscFPTrapPush(PETSC_FP_TRAP_OFF);
1101: PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info));
1102: PetscFPTrapPop();
1103: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
1104: R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */
1106: /* Extract an explicit representation of Q */
1107: Q = Ainv;
1108: PetscMemcpy(Q,A,mstride*n*sizeof(PetscScalar));
1109: K = N; /* full rank */
1110: PetscStackCallBLAS("LAPACKorgqr",LAPACKorgqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info));
1111: if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");
1113: /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
1114: Alpha = 1.0;
1115: ldb = lda;
1116: PetscStackCallBLAS("BLAStrsm",BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb));
1117: /* Ainv is Q, overwritten with inverse */
1119: #if defined(PETSC_USE_COMPLEX)
1120: {
1121: PetscInt i;
1122: for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
1123: PetscFree2(A,Ainv);
1124: }
1125: #endif
1126: return(0);
1127: }
1129: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
1130: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
1131: {
1133: PetscReal *Bv;
1134: PetscInt i,j;
1137: PetscMalloc1((ninterval+1)*ndegree,&Bv);
1138: /* Point evaluation of L_p on all the source vertices */
1139: PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
1140: /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
1141: for (i=0; i<ninterval; i++) {
1142: for (j=0; j<ndegree; j++) {
1143: if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
1144: else B[i*ndegree+j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
1145: }
1146: }
1147: PetscFree(Bv);
1148: return(0);
1149: }
1151: /*@
1152: PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals
1154: Not Collective
1156: Input Arguments:
1157: + degree - degree of reconstruction polynomial
1158: . nsource - number of source intervals
1159: . sourcex - sorted coordinates of source cell boundaries (length nsource+1)
1160: . ntarget - number of target intervals
1161: - targetx - sorted coordinates of target cell boundaries (length ntarget+1)
1163: Output Arguments:
1164: . R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]
1166: Level: advanced
1168: .seealso: PetscDTLegendreEval()
1169: @*/
1170: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
1171: {
1173: PetscInt i,j,k,*bdegrees,worksize;
1174: PetscReal xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
1175: PetscScalar *tau,*work;
1181: if (degree >= nsource) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP,"Reconstruction degree %D must be less than number of source intervals %D",degree,nsource);
1182: #if defined(PETSC_USE_DEBUG)
1183: for (i=0; i<nsource; i++) {
1184: if (sourcex[i] >= sourcex[i+1]) SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_ARG_CORRUPT,"Source interval %D has negative orientation (%g,%g)",i,(double)sourcex[i],(double)sourcex[i+1]);
1185: }
1186: for (i=0; i<ntarget; i++) {
1187: if (targetx[i] >= targetx[i+1]) SETERRQ3(PETSC_COMM_SELF,PETSC_ERR_ARG_CORRUPT,"Target interval %D has negative orientation (%g,%g)",i,(double)targetx[i],(double)targetx[i+1]);
1188: }
1189: #endif
1190: xmin = PetscMin(sourcex[0],targetx[0]);
1191: xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
1192: center = (xmin + xmax)/2;
1193: hscale = (xmax - xmin)/2;
1194: worksize = nsource;
1195: PetscMalloc4(degree+1,&bdegrees,nsource+1,&sourcey,nsource*(degree+1),&Bsource,worksize,&work);
1196: PetscMalloc4(nsource,&tau,nsource*(degree+1),&Bsinv,ntarget+1,&targety,ntarget*(degree+1),&Btarget);
1197: for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
1198: for (i=0; i<=degree; i++) bdegrees[i] = i+1;
1199: PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
1200: PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
1201: for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
1202: PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
1203: for (i=0; i<ntarget; i++) {
1204: PetscReal rowsum = 0;
1205: for (j=0; j<nsource; j++) {
1206: PetscReal sum = 0;
1207: for (k=0; k<degree+1; k++) {
1208: sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
1209: }
1210: R[i*nsource+j] = sum;
1211: rowsum += sum;
1212: }
1213: for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
1214: }
1215: PetscFree4(bdegrees,sourcey,Bsource,work);
1216: PetscFree4(tau,Bsinv,targety,Btarget);
1217: return(0);
1218: }