Actual source code: dt.c
petsc-3.7.1 2016-05-15
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      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>            /*I "petscdt.h" I*/
     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";

     31: /*@
     32:   PetscQuadratureCreate - Create a PetscQuadrature object

     34:   Collective on MPI_Comm

     36:   Input Parameter:
     37: . comm - The communicator for the PetscQuadrature object

     39:   Output Parameter:
     40: . q  - The PetscQuadrature object

     42:   Level: beginner

     44: .keywords: PetscQuadrature, quadrature, create
     45: .seealso: PetscQuadratureDestroy(), PetscQuadratureGetData()
     46: @*/
     47: PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
     48: {

     53:   DMInitializePackage();
     54:   PetscHeaderCreate(*q,PETSC_OBJECT_CLASSID,"PetscQuadrature","Quadrature","DT",comm,PetscQuadratureDestroy,PetscQuadratureView);
     55:   (*q)->dim       = -1;
     56:   (*q)->order     = -1;
     57:   (*q)->numPoints = 0;
     58:   (*q)->points    = NULL;
     59:   (*q)->weights   = NULL;
     60:   return(0);
     61: }

     65: /*@
     66:   PetscQuadratureDuplicate - Create a deep copy of the PetscQuadrature object

     68:   Collective on PetscQuadrature

     70:   Input Parameter:
     71: . q  - The PetscQuadrature object

     73:   Output Parameter:
     74: . r  - The new PetscQuadrature object

     76:   Level: beginner

     78: .keywords: PetscQuadrature, quadrature, clone
     79: .seealso: PetscQuadratureCreate(), PetscQuadratureDestroy(), PetscQuadratureGetData()
     80: @*/
     81: PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
     82: {
     83:   PetscInt         order, dim, Nq;
     84:   const PetscReal *points, *weights;
     85:   PetscReal       *p, *w;
     86:   PetscErrorCode   ierr;

     90:   PetscQuadratureCreate(PetscObjectComm((PetscObject) q), r);
     91:   PetscQuadratureGetOrder(q, &order);
     92:   PetscQuadratureSetOrder(*r, order);
     93:   PetscQuadratureGetData(q, &dim, &Nq, &points, &weights);
     94:   PetscMalloc1(Nq*dim, &p);
     95:   PetscMalloc1(Nq, &w);
     96:   PetscMemcpy(p, points, Nq*dim * sizeof(PetscReal));
     97:   PetscMemcpy(w, weights, Nq * sizeof(PetscReal));
     98:   PetscQuadratureSetData(*r, dim, Nq, p, w);
     99:   return(0);
    100: }

    104: /*@
    105:   PetscQuadratureDestroy - Destroys a PetscQuadrature object

    107:   Collective on PetscQuadrature

    109:   Input Parameter:
    110: . q  - The PetscQuadrature object

    112:   Level: beginner

    114: .keywords: PetscQuadrature, quadrature, destroy
    115: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
    116: @*/
    117: PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
    118: {

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

    136: /*@
    137:   PetscQuadratureGetOrder - Return the quadrature information

    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:   Output Parameter:

    149:   Level: intermediate

    151: .seealso: PetscQuadratureSetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
    152: @*/
    153: PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature q, PetscInt *order)
    154: {
    158:   *order = q->order;
    159:   return(0);
    160: }

    164: /*@
    165:   PetscQuadratureSetOrder - Return the quadrature information

    167:   Not collective

    169:   Input Parameters:
    170: + q - The PetscQuadrature object
    171: - order - The order of the quadrature, i.e. the highest degree polynomial that is exactly integrated

    173:   Level: intermediate

    175: .seealso: PetscQuadratureGetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
    176: @*/
    177: PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature q, PetscInt order)
    178: {
    181:   q->order = order;
    182:   return(0);
    183: }

    187: /*@C
    188:   PetscQuadratureGetData - Returns the data defining the quadrature

    190:   Not collective

    192:   Input Parameter:
    193: . q  - The PetscQuadrature object

    195:   Output Parameters:
    196: + dim - The spatial dimension
    197: . npoints - The number of quadrature points
    198: . points - The coordinates of each quadrature point
    199: - weights - The weight of each quadrature point

    201:   Level: intermediate

    203: .keywords: PetscQuadrature, quadrature
    204: .seealso: PetscQuadratureCreate(), PetscQuadratureSetData()
    205: @*/
    206: PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
    207: {
    210:   if (dim) {
    212:     *dim = q->dim;
    213:   }
    214:   if (npoints) {
    216:     *npoints = q->numPoints;
    217:   }
    218:   if (points) {
    220:     *points = q->points;
    221:   }
    222:   if (weights) {
    224:     *weights = q->weights;
    225:   }
    226:   return(0);
    227: }

    231: /*@C
    232:   PetscQuadratureSetData - Sets the data defining the quadrature

    234:   Not collective

    236:   Input Parameters:
    237: + q  - The PetscQuadrature object
    238: . dim - The spatial dimension
    239: . npoints - The number of quadrature points
    240: . points - The coordinates of each quadrature point
    241: - weights - The weight of each quadrature point

    243:   Level: intermediate

    245: .keywords: PetscQuadrature, quadrature
    246: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
    247: @*/
    248: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
    249: {
    252:   if (dim >= 0)     q->dim       = dim;
    253:   if (npoints >= 0) q->numPoints = npoints;
    254:   if (points) {
    256:     q->points = points;
    257:   }
    258:   if (weights) {
    260:     q->weights = weights;
    261:   }
    262:   return(0);
    263: }

    267: /*@C
    268:   PetscQuadratureView - Views a PetscQuadrature object

    270:   Collective on PetscQuadrature

    272:   Input Parameters:
    273: + q  - The PetscQuadrature object
    274: - viewer - The PetscViewer object

    276:   Level: beginner

    278: .keywords: PetscQuadrature, quadrature, view
    279: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
    280: @*/
    281: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
    282: {
    283:   PetscInt       q, d;

    287:   PetscObjectPrintClassNamePrefixType((PetscObject)quad,viewer);
    288:   PetscViewerASCIIPrintf(viewer, "Quadrature on %d points\n  (", quad->numPoints);
    289:   for (q = 0; q < quad->numPoints; ++q) {
    290:     for (d = 0; d < quad->dim; ++d) {
    291:       if (d) PetscViewerASCIIPrintf(viewer, ", ");
    292:       PetscViewerASCIIPrintf(viewer, "%g\n", (double)quad->points[q*quad->dim+d]);
    293:     }
    294:     PetscViewerASCIIPrintf(viewer, ") %g\n", (double)quad->weights[q]);
    295:   }
    296:   return(0);
    297: }

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

    304:   Not collective

    306:   Input Parameter:
    307: + q - The original PetscQuadrature
    308: . numSubelements - The number of subelements the original element is divided into
    309: . v0 - An array of the initial points for each subelement
    310: - jac - An array of the Jacobian mappings from the reference to each subelement

    312:   Output Parameters:
    313: . dim - The dimension

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

    317:   Level: intermediate

    319: .seealso: PetscFECreate(), PetscSpaceGetDimension(), PetscDualSpaceGetDimension()
    320: @*/
    321: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
    322: {
    323:   const PetscReal *points,    *weights;
    324:   PetscReal       *pointsRef, *weightsRef;
    325:   PetscInt         dim, order, npoints, npointsRef, c, p, d, e;
    326:   PetscErrorCode   ierr;

    333:   PetscQuadratureCreate(PETSC_COMM_SELF, qref);
    334:   PetscQuadratureGetOrder(q, &order);
    335:   PetscQuadratureGetData(q, &dim, &npoints, &points, &weights);
    336:   npointsRef = npoints*numSubelements;
    337:   PetscMalloc1(npointsRef*dim,&pointsRef);
    338:   PetscMalloc1(npointsRef,&weightsRef);
    339:   for (c = 0; c < numSubelements; ++c) {
    340:     for (p = 0; p < npoints; ++p) {
    341:       for (d = 0; d < dim; ++d) {
    342:         pointsRef[(c*npoints + p)*dim+d] = v0[c*dim+d];
    343:         for (e = 0; e < dim; ++e) {
    344:           pointsRef[(c*npoints + p)*dim+d] += jac[(c*dim + d)*dim+e]*(points[p*dim+e] + 1.0);
    345:         }
    346:       }
    347:       /* Could also use detJ here */
    348:       weightsRef[c*npoints+p] = weights[p]/numSubelements;
    349:     }
    350:   }
    351:   PetscQuadratureSetOrder(*qref, order);
    352:   PetscQuadratureSetData(*qref, dim, npointsRef, pointsRef, weightsRef);
    353:   return(0);
    354: }

    358: /*@
    359:    PetscDTLegendreEval - evaluate Legendre polynomial at points

    361:    Not Collective

    363:    Input Arguments:
    364: +  npoints - number of spatial points to evaluate at
    365: .  points - array of locations to evaluate at
    366: .  ndegree - number of basis degrees to evaluate
    367: -  degrees - sorted array of degrees to evaluate

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

    374:    Level: intermediate

    376: .seealso: PetscDTGaussQuadrature()
    377: @*/
    378: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
    379: {
    380:   PetscInt i,maxdegree;

    383:   if (!npoints || !ndegree) return(0);
    384:   maxdegree = degrees[ndegree-1];
    385:   for (i=0; i<npoints; i++) {
    386:     PetscReal pm1,pm2,pd1,pd2,pdd1,pdd2,x;
    387:     PetscInt  j,k;
    388:     x    = points[i];
    389:     pm2  = 0;
    390:     pm1  = 1;
    391:     pd2  = 0;
    392:     pd1  = 0;
    393:     pdd2 = 0;
    394:     pdd1 = 0;
    395:     k    = 0;
    396:     if (degrees[k] == 0) {
    397:       if (B) B[i*ndegree+k] = pm1;
    398:       if (D) D[i*ndegree+k] = pd1;
    399:       if (D2) D2[i*ndegree+k] = pdd1;
    400:       k++;
    401:     }
    402:     for (j=1; j<=maxdegree; j++,k++) {
    403:       PetscReal p,d,dd;
    404:       p    = ((2*j-1)*x*pm1 - (j-1)*pm2)/j;
    405:       d    = pd2 + (2*j-1)*pm1;
    406:       dd   = pdd2 + (2*j-1)*pd1;
    407:       pm2  = pm1;
    408:       pm1  = p;
    409:       pd2  = pd1;
    410:       pd1  = d;
    411:       pdd2 = pdd1;
    412:       pdd1 = dd;
    413:       if (degrees[k] == j) {
    414:         if (B) B[i*ndegree+k] = p;
    415:         if (D) D[i*ndegree+k] = d;
    416:         if (D2) D2[i*ndegree+k] = dd;
    417:       }
    418:     }
    419:   }
    420:   return(0);
    421: }

    425: /*@
    426:    PetscDTGaussQuadrature - create Gauss quadrature

    428:    Not Collective

    430:    Input Arguments:
    431: +  npoints - number of points
    432: .  a - left end of interval (often-1)
    433: -  b - right end of interval (often +1)

    435:    Output Arguments:
    436: +  x - quadrature points
    437: -  w - quadrature weights

    439:    Level: intermediate

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

    444: .seealso: PetscDTLegendreEval()
    445: @*/
    446: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
    447: {
    449:   PetscInt       i;
    450:   PetscReal      *work;
    451:   PetscScalar    *Z;
    452:   PetscBLASInt   N,LDZ,info;

    455:   PetscCitationsRegister(GaussCitation, &GaussCite);
    456:   /* Set up the Golub-Welsch system */
    457:   for (i=0; i<npoints; i++) {
    458:     x[i] = 0;                   /* diagonal is 0 */
    459:     if (i) w[i-1] = 0.5 / PetscSqrtReal(1 - 1./PetscSqr(2*i));
    460:   }
    461:   PetscMalloc2(npoints*npoints,&Z,PetscMax(1,2*npoints-2),&work);
    462:   PetscBLASIntCast(npoints,&N);
    463:   LDZ  = N;
    464:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
    465:   PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&N,x,w,Z,&LDZ,work,&info));
    466:   PetscFPTrapPop();
    467:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");

    469:   for (i=0; i<(npoints+1)/2; i++) {
    470:     PetscReal y = 0.5 * (-x[i] + x[npoints-i-1]); /* enforces symmetry */
    471:     x[i]           = (a+b)/2 - y*(b-a)/2;
    472:     if (x[i] == -0.0) x[i] = 0.0;
    473:     x[npoints-i-1] = (a+b)/2 + y*(b-a)/2;

    475:     w[i] = w[npoints-1-i] = 0.5*(b-a)*(PetscSqr(PetscAbsScalar(Z[i*npoints])) + PetscSqr(PetscAbsScalar(Z[(npoints-i-1)*npoints])));
    476:   }
    477:   PetscFree2(Z,work);
    478:   return(0);
    479: }

    483: /*@
    484:   PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

    486:   Not Collective

    488:   Input Arguments:
    489: + dim     - The spatial dimension
    490: . npoints - number of points in one dimension
    491: . a       - left end of interval (often-1)
    492: - b       - right end of interval (often +1)

    494:   Output Argument:
    495: . q - A PetscQuadrature object

    497:   Level: intermediate

    499: .seealso: PetscDTGaussQuadrature(), PetscDTLegendreEval()
    500: @*/
    501: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
    502: {
    503:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k;
    504:   PetscReal     *x, *w, *xw, *ww;

    508:   PetscMalloc1(totpoints*dim,&x);
    509:   PetscMalloc1(totpoints,&w);
    510:   /* Set up the Golub-Welsch system */
    511:   switch (dim) {
    512:   case 0:
    513:     PetscFree(x);
    514:     PetscFree(w);
    515:     PetscMalloc1(1, &x);
    516:     PetscMalloc1(1, &w);
    517:     x[0] = 0.0;
    518:     w[0] = 1.0;
    519:     break;
    520:   case 1:
    521:     PetscDTGaussQuadrature(npoints, a, b, x, w);
    522:     break;
    523:   case 2:
    524:     PetscMalloc2(npoints,&xw,npoints,&ww);
    525:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
    526:     for (i = 0; i < npoints; ++i) {
    527:       for (j = 0; j < npoints; ++j) {
    528:         x[(i*npoints+j)*dim+0] = xw[i];
    529:         x[(i*npoints+j)*dim+1] = xw[j];
    530:         w[i*npoints+j]         = ww[i] * ww[j];
    531:       }
    532:     }
    533:     PetscFree2(xw,ww);
    534:     break;
    535:   case 3:
    536:     PetscMalloc2(npoints,&xw,npoints,&ww);
    537:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
    538:     for (i = 0; i < npoints; ++i) {
    539:       for (j = 0; j < npoints; ++j) {
    540:         for (k = 0; k < npoints; ++k) {
    541:           x[((i*npoints+j)*npoints+k)*dim+0] = xw[i];
    542:           x[((i*npoints+j)*npoints+k)*dim+1] = xw[j];
    543:           x[((i*npoints+j)*npoints+k)*dim+2] = xw[k];
    544:           w[(i*npoints+j)*npoints+k]         = ww[i] * ww[j] * ww[k];
    545:         }
    546:       }
    547:     }
    548:     PetscFree2(xw,ww);
    549:     break;
    550:   default:
    551:     SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
    552:   }
    553:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    554:   PetscQuadratureSetOrder(*q, npoints);
    555:   PetscQuadratureSetData(*q, dim, totpoints, x, w);
    556:   return(0);
    557: }

    561: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
    562:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
    563: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorial_Internal(PetscInt n, PetscReal *factorial)
    564: {
    565:   PetscReal f = 1.0;
    566:   PetscInt  i;

    569:   for (i = 1; i < n+1; ++i) f *= i;
    570:   *factorial = f;
    571:   return(0);
    572: }

    576: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
    577:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
    578: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
    579: {
    580:   PetscReal apb, pn1, pn2;
    581:   PetscInt  k;

    584:   if (!n) {*P = 1.0; return(0);}
    585:   if (n == 1) {*P = 0.5 * (a - b + (a + b + 2.0) * x); return(0);}
    586:   apb = a + b;
    587:   pn2 = 1.0;
    588:   pn1 = 0.5 * (a - b + (apb + 2.0) * x);
    589:   *P  = 0.0;
    590:   for (k = 2; k < n+1; ++k) {
    591:     PetscReal a1 = 2.0 * k * (k + apb) * (2.0*k + apb - 2.0);
    592:     PetscReal a2 = (2.0 * k + apb - 1.0) * (a*a - b*b);
    593:     PetscReal a3 = (2.0 * k + apb - 2.0) * (2.0 * k + apb - 1.0) * (2.0 * k + apb);
    594:     PetscReal a4 = 2.0 * (k + a - 1.0) * (k + b - 1.0) * (2.0 * k + apb);

    596:     a2  = a2 / a1;
    597:     a3  = a3 / a1;
    598:     a4  = a4 / a1;
    599:     *P  = (a2 + a3 * x) * pn1 - a4 * pn2;
    600:     pn2 = pn1;
    601:     pn1 = *P;
    602:   }
    603:   return(0);
    604: }

    608: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
    609: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
    610: {
    611:   PetscReal      nP;

    615:   if (!n) {*P = 0.0; return(0);}
    616:   PetscDTComputeJacobi(a+1, b+1, n-1, x, &nP);
    617:   *P   = 0.5 * (a + b + n + 1) * nP;
    618:   return(0);
    619: }

    623: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
    624: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapSquareToTriangle_Internal(PetscReal x, PetscReal y, PetscReal *xi, PetscReal *eta)
    625: {
    627:   *xi  = 0.5 * (1.0 + x) * (1.0 - y) - 1.0;
    628:   *eta = y;
    629:   return(0);
    630: }

    634: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
    635: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapCubeToTetrahedron_Internal(PetscReal x, PetscReal y, PetscReal z, PetscReal *xi, PetscReal *eta, PetscReal *zeta)
    636: {
    638:   *xi   = 0.25 * (1.0 + x) * (1.0 - y) * (1.0 - z) - 1.0;
    639:   *eta  = 0.5  * (1.0 + y) * (1.0 - z) - 1.0;
    640:   *zeta = z;
    641:   return(0);
    642: }

    646: static PetscErrorCode PetscDTGaussJacobiQuadrature1D_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
    647: {
    648:   PetscInt       maxIter = 100;
    649:   PetscReal      eps     = 1.0e-8;
    650:   PetscReal      a1, a2, a3, a4, a5, a6;
    651:   PetscInt       k;

    656:   a1      = PetscPowReal(2.0, a+b+1);
    657: #if defined(PETSC_HAVE_TGAMMA)
    658:   a2      = PetscTGamma(a + npoints + 1);
    659:   a3      = PetscTGamma(b + npoints + 1);
    660:   a4      = PetscTGamma(a + b + npoints + 1);
    661: #else
    662:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
    663: #endif

    665:   PetscDTFactorial_Internal(npoints, &a5);
    666:   a6   = a1 * a2 * a3 / a4 / a5;
    667:   /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
    668:    Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
    669:   for (k = 0; k < npoints; ++k) {
    670:     PetscReal r = -PetscCosReal((2.0*k + 1.0) * PETSC_PI / (2.0 * npoints)), dP;
    671:     PetscInt  j;

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

    678:       for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
    679:       PetscDTComputeJacobi(a, b, npoints, r, &f);
    680:       PetscDTComputeJacobiDerivative(a, b, npoints, r, &fp);
    681:       delta = f / (fp - f * s);
    682:       r     = r - delta;
    683:       if (PetscAbsReal(delta) < eps) break;
    684:     }
    685:     x[k] = r;
    686:     PetscDTComputeJacobiDerivative(a, b, npoints, x[k], &dP);
    687:     w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
    688:   }
    689:   return(0);
    690: }

    694: /*@C
    695:   PetscDTGaussJacobiQuadrature - create Gauss-Jacobi quadrature for a simplex

    697:   Not Collective

    699:   Input Arguments:
    700: + dim   - The simplex dimension
    701: . order - The number of points in one dimension
    702: . a     - left end of interval (often-1)
    703: - b     - right end of interval (often +1)

    705:   Output Argument:
    706: . q - A PetscQuadrature object

    708:   Level: intermediate

    710:   References:
    711: .  1. - Karniadakis and Sherwin.  FIAT

    713: .seealso: PetscDTGaussTensorQuadrature(), PetscDTGaussQuadrature()
    714: @*/
    715: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt dim, PetscInt order, PetscReal a, PetscReal b, PetscQuadrature *q)
    716: {
    717:   PetscInt       npoints = dim > 1 ? dim > 2 ? order*PetscSqr(order) : PetscSqr(order) : order;
    718:   PetscReal     *px, *wx, *py, *wy, *pz, *wz, *x, *w;
    719:   PetscInt       i, j, k;

    723:   if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
    724:   PetscMalloc1(npoints*dim, &x);
    725:   PetscMalloc1(npoints, &w);
    726:   switch (dim) {
    727:   case 0:
    728:     PetscFree(x);
    729:     PetscFree(w);
    730:     PetscMalloc1(1, &x);
    731:     PetscMalloc1(1, &w);
    732:     x[0] = 0.0;
    733:     w[0] = 1.0;
    734:     break;
    735:   case 1:
    736:     PetscDTGaussJacobiQuadrature1D_Internal(order, 0.0, 0.0, x, w);
    737:     break;
    738:   case 2:
    739:     PetscMalloc4(order,&px,order,&wx,order,&py,order,&wy);
    740:     PetscDTGaussJacobiQuadrature1D_Internal(order, 0.0, 0.0, px, wx);
    741:     PetscDTGaussJacobiQuadrature1D_Internal(order, 1.0, 0.0, py, wy);
    742:     for (i = 0; i < order; ++i) {
    743:       for (j = 0; j < order; ++j) {
    744:         PetscDTMapSquareToTriangle_Internal(px[i], py[j], &x[(i*order+j)*2+0], &x[(i*order+j)*2+1]);
    745:         w[i*order+j] = 0.5 * wx[i] * wy[j];
    746:       }
    747:     }
    748:     PetscFree4(px,wx,py,wy);
    749:     break;
    750:   case 3:
    751:     PetscMalloc6(order,&px,order,&wx,order,&py,order,&wy,order,&pz,order,&wz);
    752:     PetscDTGaussJacobiQuadrature1D_Internal(order, 0.0, 0.0, px, wx);
    753:     PetscDTGaussJacobiQuadrature1D_Internal(order, 1.0, 0.0, py, wy);
    754:     PetscDTGaussJacobiQuadrature1D_Internal(order, 2.0, 0.0, pz, wz);
    755:     for (i = 0; i < order; ++i) {
    756:       for (j = 0; j < order; ++j) {
    757:         for (k = 0; k < order; ++k) {
    758:           PetscDTMapCubeToTetrahedron_Internal(px[i], py[j], pz[k], &x[((i*order+j)*order+k)*3+0], &x[((i*order+j)*order+k)*3+1], &x[((i*order+j)*order+k)*3+2]);
    759:           w[(i*order+j)*order+k] = 0.125 * wx[i] * wy[j] * wz[k];
    760:         }
    761:       }
    762:     }
    763:     PetscFree6(px,wx,py,wy,pz,wz);
    764:     break;
    765:   default:
    766:     SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
    767:   }
    768:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    769:   PetscQuadratureSetOrder(*q, order);
    770:   PetscQuadratureSetData(*q, dim, npoints, x, w);
    771:   return(0);
    772: }

    776: /*@C
    777:   PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell

    779:   Not Collective

    781:   Input Arguments:
    782: + dim   - The cell dimension
    783: . level - The number of points in one dimension, 2^l
    784: . a     - left end of interval (often-1)
    785: - b     - right end of interval (often +1)

    787:   Output Argument:
    788: . q - A PetscQuadrature object

    790:   Level: intermediate

    792: .seealso: PetscDTGaussTensorQuadrature()
    793: @*/
    794: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
    795: {
    796:   const PetscInt  p     = 16;                        /* Digits of precision in the evaluation */
    797:   const PetscReal alpha = (b-a)/2.;                  /* Half-width of the integration interval */
    798:   const PetscReal beta  = (b+a)/2.;                  /* Center of the integration interval */
    799:   const PetscReal h     = PetscPowReal(2.0, -level); /* Step size, length between x_k */
    800:   PetscReal       xk;                                /* Quadrature point x_k on reference domain [-1, 1] */
    801:   PetscReal       wk    = 0.5*PETSC_PI;              /* Quadrature weight at x_k */
    802:   PetscReal      *x, *w;
    803:   PetscInt        K, k, npoints;
    804:   PetscErrorCode  ierr;

    807:   if (dim > 1) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %d not yet implemented", dim);
    808:   if (!level) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
    809:   /* Find K such that the weights are < 32 digits of precision */
    810:   for (K = 1; PetscAbsReal(PetscLog10Real(wk)) < 2*p; ++K) {
    811:     wk = 0.5*h*PETSC_PI*PetscCoshReal(K*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(K*h)));
    812:   }
    813:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    814:   PetscQuadratureSetOrder(*q, 2*K+1);
    815:   npoints = 2*K-1;
    816:   PetscMalloc1(npoints*dim, &x);
    817:   PetscMalloc1(npoints, &w);
    818:   /* Center term */
    819:   x[0] = beta;
    820:   w[0] = 0.5*alpha*PETSC_PI;
    821:   for (k = 1; k < K; ++k) {
    822:     wk = 0.5*alpha*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    823:     xk = tanh(0.5*PETSC_PI*PetscSinhReal(k*h));
    824:     x[2*k-1] = -alpha*xk+beta;
    825:     w[2*k-1] = wk;
    826:     x[2*k+0] =  alpha*xk+beta;
    827:     w[2*k+0] = wk;
    828:   }
    829:   PetscQuadratureSetData(*q, dim, npoints, x, w);
    830:   return(0);
    831: }

    835: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    836: {
    837:   const PetscInt  p     = 16;        /* Digits of precision in the evaluation */
    838:   const PetscReal alpha = (b-a)/2.;  /* Half-width of the integration interval */
    839:   const PetscReal beta  = (b+a)/2.;  /* Center of the integration interval */
    840:   PetscReal       h     = 1.0;       /* Step size, length between x_k */
    841:   PetscInt        l     = 0;         /* Level of refinement, h = 2^{-l} */
    842:   PetscReal       osum  = 0.0;       /* Integral on last level */
    843:   PetscReal       psum  = 0.0;       /* Integral on the level before the last level */
    844:   PetscReal       sum;               /* Integral on current level */
    845:   PetscReal       yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
    846:   PetscReal       lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
    847:   PetscReal       wk;                /* Quadrature weight at x_k */
    848:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
    849:   PetscInt        d;                 /* Digits of precision in the integral */

    852:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
    853:   /* Center term */
    854:   func(beta, &lval);
    855:   sum = 0.5*alpha*PETSC_PI*lval;
    856:   /* */
    857:   do {
    858:     PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
    859:     PetscInt  k = 1;

    861:     ++l;
    862:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
    863:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
    864:     psum = osum;
    865:     osum = sum;
    866:     h   *= 0.5;
    867:     sum *= 0.5;
    868:     do {
    869:       wk = 0.5*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    870:       yk = 1.0/(PetscExpReal(0.5*PETSC_PI*PetscSinhReal(k*h)) * PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    871:       lx = -alpha*(1.0 - yk)+beta;
    872:       rx =  alpha*(1.0 - yk)+beta;
    873:       func(lx, &lval);
    874:       func(rx, &rval);
    875:       lterm   = alpha*wk*lval;
    876:       maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
    877:       sum    += lterm;
    878:       rterm   = alpha*wk*rval;
    879:       maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
    880:       sum    += rterm;
    881:       /* if (l == 1) printf("k is %d and sum is %15.15f and wk is %15.15f\n", k, sum, wk); */
    882:       ++k;
    883:       /* Only need to evaluate every other point on refined levels */
    884:       if (l != 1) ++k;
    885:     } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */

    887:     d1 = PetscLog10Real(PetscAbsReal(sum - osum));
    888:     d2 = PetscLog10Real(PetscAbsReal(sum - psum));
    889:     d3 = PetscLog10Real(maxTerm) - p;
    890:     if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
    891:     else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
    892:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
    893:   } while (d < digits && l < 12);
    894:   *sol = sum;

    896:   return(0);
    897: }

    899: #ifdef PETSC_HAVE_MPFR
    902: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    903: {
    904:   const PetscInt  safetyFactor = 2;  /* Calculate abcissa until 2*p digits */
    905:   PetscInt        l            = 0;  /* Level of refinement, h = 2^{-l} */
    906:   mpfr_t          alpha;             /* Half-width of the integration interval */
    907:   mpfr_t          beta;              /* Center of the integration interval */
    908:   mpfr_t          h;                 /* Step size, length between x_k */
    909:   mpfr_t          osum;              /* Integral on last level */
    910:   mpfr_t          psum;              /* Integral on the level before the last level */
    911:   mpfr_t          sum;               /* Integral on current level */
    912:   mpfr_t          yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
    913:   mpfr_t          lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
    914:   mpfr_t          wk;                /* Quadrature weight at x_k */
    915:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
    916:   PetscInt        d;                 /* Digits of precision in the integral */
    917:   mpfr_t          pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;

    920:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
    921:   /* Create high precision storage */
    922:   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);
    923:   /* Initialization */
    924:   mpfr_set_d(alpha, 0.5*(b-a), MPFR_RNDN);
    925:   mpfr_set_d(beta,  0.5*(b+a), MPFR_RNDN);
    926:   mpfr_set_d(osum,  0.0,       MPFR_RNDN);
    927:   mpfr_set_d(psum,  0.0,       MPFR_RNDN);
    928:   mpfr_set_d(h,     1.0,       MPFR_RNDN);
    929:   mpfr_const_pi(pi2, MPFR_RNDN);
    930:   mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
    931:   /* Center term */
    932:   func(0.5*(b+a), &lval);
    933:   mpfr_set(sum, pi2, MPFR_RNDN);
    934:   mpfr_mul(sum, sum, alpha, MPFR_RNDN);
    935:   mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
    936:   /* */
    937:   do {
    938:     PetscReal d1, d2, d3, d4;
    939:     PetscInt  k = 1;

    941:     ++l;
    942:     mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
    943:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
    944:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
    945:     mpfr_set(psum, osum, MPFR_RNDN);
    946:     mpfr_set(osum,  sum, MPFR_RNDN);
    947:     mpfr_mul_d(h,   h,   0.5, MPFR_RNDN);
    948:     mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
    949:     do {
    950:       mpfr_set_si(kh, k, MPFR_RNDN);
    951:       mpfr_mul(kh, kh, h, MPFR_RNDN);
    952:       /* Weight */
    953:       mpfr_set(wk, h, MPFR_RNDN);
    954:       mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
    955:       mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
    956:       mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
    957:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
    958:       mpfr_sqr(tmp, tmp, MPFR_RNDN);
    959:       mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
    960:       mpfr_div(wk, wk, tmp, MPFR_RNDN);
    961:       /* Abscissa */
    962:       mpfr_set_d(yk, 1.0, MPFR_RNDZ);
    963:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
    964:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
    965:       mpfr_exp(tmp, msinh, MPFR_RNDN);
    966:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
    967:       /* Quadrature points */
    968:       mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
    969:       mpfr_mul(lx, lx, alpha, MPFR_RNDU);
    970:       mpfr_add(lx, lx, beta, MPFR_RNDU);
    971:       mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
    972:       mpfr_mul(rx, rx, alpha, MPFR_RNDD);
    973:       mpfr_add(rx, rx, beta, MPFR_RNDD);
    974:       /* Evaluation */
    975:       func(mpfr_get_d(lx, MPFR_RNDU), &lval);
    976:       func(mpfr_get_d(rx, MPFR_RNDD), &rval);
    977:       /* Update */
    978:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
    979:       mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
    980:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
    981:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    982:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
    983:       mpfr_set(curTerm, tmp, MPFR_RNDN);
    984:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
    985:       mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
    986:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
    987:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    988:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
    989:       mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
    990:       /* if (l == 1) printf("k is %d and sum is %15.15f and wk is %15.15f\n", k, sum, wk); */
    991:       ++k;
    992:       /* Only need to evaluate every other point on refined levels */
    993:       if (l != 1) ++k;
    994:       mpfr_log10(tmp, wk, MPFR_RNDN);
    995:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    996:     } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor*digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
    997:     mpfr_sub(tmp, sum, osum, MPFR_RNDN);
    998:     mpfr_abs(tmp, tmp, MPFR_RNDN);
    999:     mpfr_log10(tmp, tmp, MPFR_RNDN);
    1000:     d1 = mpfr_get_d(tmp, MPFR_RNDN);
    1001:     mpfr_sub(tmp, sum, psum, MPFR_RNDN);
    1002:     mpfr_abs(tmp, tmp, MPFR_RNDN);
    1003:     mpfr_log10(tmp, tmp, MPFR_RNDN);
    1004:     d2 = mpfr_get_d(tmp, MPFR_RNDN);
    1005:     mpfr_log10(tmp, maxTerm, MPFR_RNDN);
    1006:     d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
    1007:     mpfr_log10(tmp, curTerm, MPFR_RNDN);
    1008:     d4 = mpfr_get_d(tmp, MPFR_RNDN);
    1009:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
    1010:   } while (d < digits && l < 8);
    1011:   *sol = mpfr_get_d(sum, MPFR_RNDN);
    1012:   /* Cleanup */
    1013:   mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
    1014:   return(0);
    1015: }
    1016: #else
    1019: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    1020: {
    1021:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
    1022: }
    1023: #endif

    1027: /* Overwrites A. Can only handle full-rank problems with m>=n
    1028:  * A in column-major format
    1029:  * Ainv in row-major format
    1030:  * tau has length m
    1031:  * worksize must be >= max(1,n)
    1032:  */
    1033: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
    1034: {
    1036:   PetscBLASInt   M,N,K,lda,ldb,ldwork,info;
    1037:   PetscScalar    *A,*Ainv,*R,*Q,Alpha;

    1040: #if defined(PETSC_USE_COMPLEX)
    1041:   {
    1042:     PetscInt i,j;
    1043:     PetscMalloc2(m*n,&A,m*n,&Ainv);
    1044:     for (j=0; j<n; j++) {
    1045:       for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
    1046:     }
    1047:     mstride = m;
    1048:   }
    1049: #else
    1050:   A = A_in;
    1051:   Ainv = Ainv_out;
    1052: #endif

    1054:   PetscBLASIntCast(m,&M);
    1055:   PetscBLASIntCast(n,&N);
    1056:   PetscBLASIntCast(mstride,&lda);
    1057:   PetscBLASIntCast(worksize,&ldwork);
    1058:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
    1059:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info));
    1060:   PetscFPTrapPop();
    1061:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
    1062:   R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

    1064:   /* Extract an explicit representation of Q */
    1065:   Q = Ainv;
    1066:   PetscMemcpy(Q,A,mstride*n*sizeof(PetscScalar));
    1067:   K = N;                        /* full rank */
    1068:   PetscStackCallBLAS("LAPACKungqr",LAPACKungqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info));
    1069:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");

    1071:   /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
    1072:   Alpha = 1.0;
    1073:   ldb = lda;
    1074:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb));
    1075:   /* Ainv is Q, overwritten with inverse */

    1077: #if defined(PETSC_USE_COMPLEX)
    1078:   {
    1079:     PetscInt i;
    1080:     for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
    1081:     PetscFree2(A,Ainv);
    1082:   }
    1083: #endif
    1084:   return(0);
    1085: }

    1089: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
    1090: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
    1091: {
    1093:   PetscReal      *Bv;
    1094:   PetscInt       i,j;

    1097:   PetscMalloc1((ninterval+1)*ndegree,&Bv);
    1098:   /* Point evaluation of L_p on all the source vertices */
    1099:   PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
    1100:   /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
    1101:   for (i=0; i<ninterval; i++) {
    1102:     for (j=0; j<ndegree; j++) {
    1103:       if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
    1104:       else           B[i*ndegree+j]   = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
    1105:     }
    1106:   }
    1107:   PetscFree(Bv);
    1108:   return(0);
    1109: }

    1113: /*@
    1114:    PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

    1116:    Not Collective

    1118:    Input Arguments:
    1119: +  degree - degree of reconstruction polynomial
    1120: .  nsource - number of source intervals
    1121: .  sourcex - sorted coordinates of source cell boundaries (length nsource+1)
    1122: .  ntarget - number of target intervals
    1123: -  targetx - sorted coordinates of target cell boundaries (length ntarget+1)

    1125:    Output Arguments:
    1126: .  R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]

    1128:    Level: advanced

    1130: .seealso: PetscDTLegendreEval()
    1131: @*/
    1132: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
    1133: {
    1135:   PetscInt       i,j,k,*bdegrees,worksize;
    1136:   PetscReal      xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
    1137:   PetscScalar    *tau,*work;

    1143:   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);
    1144: #if defined(PETSC_USE_DEBUG)
    1145:   for (i=0; i<nsource; i++) {
    1146:     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]);
    1147:   }
    1148:   for (i=0; i<ntarget; i++) {
    1149:     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]);
    1150:   }
    1151: #endif
    1152:   xmin = PetscMin(sourcex[0],targetx[0]);
    1153:   xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
    1154:   center = (xmin + xmax)/2;
    1155:   hscale = (xmax - xmin)/2;
    1156:   worksize = nsource;
    1157:   PetscMalloc4(degree+1,&bdegrees,nsource+1,&sourcey,nsource*(degree+1),&Bsource,worksize,&work);
    1158:   PetscMalloc4(nsource,&tau,nsource*(degree+1),&Bsinv,ntarget+1,&targety,ntarget*(degree+1),&Btarget);
    1159:   for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
    1160:   for (i=0; i<=degree; i++) bdegrees[i] = i+1;
    1161:   PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
    1162:   PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
    1163:   for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
    1164:   PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
    1165:   for (i=0; i<ntarget; i++) {
    1166:     PetscReal rowsum = 0;
    1167:     for (j=0; j<nsource; j++) {
    1168:       PetscReal sum = 0;
    1169:       for (k=0; k<degree+1; k++) {
    1170:         sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
    1171:       }
    1172:       R[i*nsource+j] = sum;
    1173:       rowsum += sum;
    1174:     }
    1175:     for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
    1176:   }
    1177:   PetscFree4(bdegrees,sourcey,Bsource,work);
    1178:   PetscFree4(tau,Bsinv,targety,Btarget);
    1179:   return(0);
    1180: }