Date: 2021-03-03T22:27:04+00:00

Actual source code: dt.c
petsc-3.14.5 2021-03-03
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      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[] = {"gaussjacobi", "equispaced", "tanhsinh", "PETSCDTNODES_", NULL};

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

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

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

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

     38: PetscClassId PETSCQUADRATURE_CLASSID = 0;

     40: /*@
     41:   PetscQuadratureCreate - Create a PetscQuadrature object

     43:   Collective

     45:   Input Parameter:
     46: . comm - The communicator for the PetscQuadrature object

     48:   Output Parameter:
     49: . q  - The PetscQuadrature object

     51:   Level: beginner

     53: .seealso: PetscQuadratureDestroy(), PetscQuadratureGetData()
     54: @*/
     55: PetscErrorCode PetscQuadratureCreate(MPI_Comm comm, PetscQuadrature *q)
     56: {

     61:   DMInitializePackage();
     62:   PetscHeaderCreate(*q,PETSCQUADRATURE_CLASSID,"PetscQuadrature","Quadrature","DT",comm,PetscQuadratureDestroy,PetscQuadratureView);
     63:   (*q)->dim       = -1;
     64:   (*q)->Nc        =  1;
     65:   (*q)->order     = -1;
     66:   (*q)->numPoints = 0;
     67:   (*q)->points    = NULL;
     68:   (*q)->weights   = NULL;
     69:   return(0);
     70: }

     72: /*@
     73:   PetscQuadratureDuplicate - Create a deep copy of the PetscQuadrature object

     75:   Collective on q

     77:   Input Parameter:
     78: . q  - The PetscQuadrature object

     80:   Output Parameter:
     81: . r  - The new PetscQuadrature object

     83:   Level: beginner

     85: .seealso: PetscQuadratureCreate(), PetscQuadratureDestroy(), PetscQuadratureGetData()
     86: @*/
     87: PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature q, PetscQuadrature *r)
     88: {
     89:   PetscInt         order, dim, Nc, Nq;
     90:   const PetscReal *points, *weights;
     91:   PetscReal       *p, *w;
     92:   PetscErrorCode   ierr;

     96:   PetscQuadratureCreate(PetscObjectComm((PetscObject) q), r);
     97:   PetscQuadratureGetOrder(q, &order);
     98:   PetscQuadratureSetOrder(*r, order);
     99:   PetscQuadratureGetData(q, &dim, &Nc, &Nq, &points, &weights);
    100:   PetscMalloc1(Nq*dim, &p);
    101:   PetscMalloc1(Nq*Nc, &w);
    102:   PetscArraycpy(p, points, Nq*dim);
    103:   PetscArraycpy(w, weights, Nc * Nq);
    104:   PetscQuadratureSetData(*r, dim, Nc, Nq, p, w);
    105:   return(0);
    106: }

    108: /*@
    109:   PetscQuadratureDestroy - Destroys a PetscQuadrature object

    111:   Collective on q

    113:   Input Parameter:
    114: . q  - The PetscQuadrature object

    116:   Level: beginner

    118: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
    119: @*/
    120: PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *q)
    121: {

    125:   if (!*q) return(0);
    127:   if (--((PetscObject)(*q))->refct > 0) {
    128:     *q = NULL;
    129:     return(0);
    130:   }
    131:   PetscFree((*q)->points);
    132:   PetscFree((*q)->weights);
    133:   PetscHeaderDestroy(q);
    134:   return(0);
    135: }

    137: /*@
    138:   PetscQuadratureGetOrder - Return the order of the method

    140:   Not collective

    142:   Input Parameter:
    143: . q - The PetscQuadrature object

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

    148:   Level: intermediate

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

    161: /*@
    162:   PetscQuadratureSetOrder - Return the order of the method

    164:   Not collective

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

    170:   Level: intermediate

    172: .seealso: PetscQuadratureGetOrder(), PetscQuadratureGetData(), PetscQuadratureSetData()
    173: @*/
    174: PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature q, PetscInt order)
    175: {
    178:   q->order = order;
    179:   return(0);
    180: }

    182: /*@
    183:   PetscQuadratureGetNumComponents - Return the number of components for functions to be integrated

    185:   Not collective

    187:   Input Parameter:
    188: . q - The PetscQuadrature object

    190:   Output Parameter:
    191: . Nc - The number of components

    193:   Note: 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: PetscQuadratureSetNumComponents(), PetscQuadratureGetData(), PetscQuadratureSetData()
    198: @*/
    199: PetscErrorCode PetscQuadratureGetNumComponents(PetscQuadrature q, PetscInt *Nc)
    200: {
    204:   *Nc = q->Nc;
    205:   return(0);
    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: We are performing an integral int f(x) . w(x) dx, where both f and w (the weight) have Nc components.

    219:   Level: intermediate

    221: .seealso: PetscQuadratureGetNumComponents(), PetscQuadratureGetData(), PetscQuadratureSetData()
    222: @*/
    223: PetscErrorCode PetscQuadratureSetNumComponents(PetscQuadrature q, PetscInt Nc)
    224: {
    227:   q->Nc = Nc;
    228:   return(0);
    229: }

    231: /*@C
    232:   PetscQuadratureGetData - Returns the data defining the quadrature

    234:   Not collective

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

    239:   Output Parameters:
    240: + dim - The spatial dimension
    241: . Nc - The number of components
    242: . npoints - The number of quadrature points
    243: . points - The coordinates of each quadrature point
    244: - weights - The weight of each quadrature point

    246:   Level: intermediate

    248:   Fortran Notes:
    249:     From Fortran you must call PetscQuadratureRestoreData() when you are done with the data

    251: .seealso: PetscQuadratureCreate(), PetscQuadratureSetData()
    252: @*/
    253: PetscErrorCode PetscQuadratureGetData(PetscQuadrature q, PetscInt *dim, PetscInt *Nc, PetscInt *npoints, const PetscReal *points[], const PetscReal *weights[])
    254: {
    257:   if (dim) {
    259:     *dim = q->dim;
    260:   }
    261:   if (Nc) {
    263:     *Nc = q->Nc;
    264:   }
    265:   if (npoints) {
    267:     *npoints = q->numPoints;
    268:   }
    269:   if (points) {
    271:     *points = q->points;
    272:   }
    273:   if (weights) {
    275:     *weights = q->weights;
    276:   }
    277:   return(0);
    278: }

    280: static PetscErrorCode PetscDTJacobianInverse_Internal(PetscInt m, PetscInt n, const PetscReal J[], PetscReal Jinv[])
    281: {
    282:   PetscScalar    *Js, *Jinvs;
    283:   PetscInt       i, j, k;
    284:   PetscBLASInt   bm, bn, info;

    288:   if (!m || !n) return(0);
    289:   PetscBLASIntCast(m, &bm);
    290:   PetscBLASIntCast(n, &bn);
    291: #if defined(PETSC_USE_COMPLEX)
    292:   PetscMalloc2(m*n, &Js, m*n, &Jinvs);
    293:   for (i = 0; i < m*n; i++) Js[i] = J[i];
    294: #else
    295:   Js = (PetscReal *) J;
    296:   Jinvs = Jinv;
    297: #endif
    298:   if (m == n) {
    299:     PetscBLASInt *pivots;
    300:     PetscScalar *W;

    302:     PetscMalloc2(m, &pivots, m, &W);

    304:     PetscArraycpy(Jinvs, Js, m * m);
    305:     PetscStackCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bm, &bm, Jinvs, &bm, pivots, &info));
    306:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetrf %D",(PetscInt)info);
    307:     PetscStackCallBLAS("LAPACKgetri", LAPACKgetri_(&bm, Jinvs, &bm, pivots, W, &bm, &info));
    308:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetri %D",(PetscInt)info);
    309:     PetscFree2(pivots, W);
    310:   } else if (m < n) {
    311:     PetscScalar *JJT;
    312:     PetscBLASInt *pivots;
    313:     PetscScalar *W;

    315:     PetscMalloc1(m*m, &JJT);
    316:     PetscMalloc2(m, &pivots, m, &W);
    317:     for (i = 0; i < m; i++) {
    318:       for (j = 0; j < m; j++) {
    319:         PetscScalar val = 0.;

    321:         for (k = 0; k < n; k++) val += Js[i * n + k] * Js[j * n + k];
    322:         JJT[i * m + j] = val;
    323:       }
    324:     }

    326:     PetscStackCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bm, &bm, JJT, &bm, pivots, &info));
    327:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetrf %D",(PetscInt)info);
    328:     PetscStackCallBLAS("LAPACKgetri", LAPACKgetri_(&bm, JJT, &bm, pivots, W, &bm, &info));
    329:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetri %D",(PetscInt)info);
    330:     for (i = 0; i < n; i++) {
    331:       for (j = 0; j < m; j++) {
    332:         PetscScalar val = 0.;

    334:         for (k = 0; k < m; k++) val += Js[k * n + i] * JJT[k * m + j];
    335:         Jinvs[i * m + j] = val;
    336:       }
    337:     }
    338:     PetscFree2(pivots, W);
    339:     PetscFree(JJT);
    340:   } else {
    341:     PetscScalar *JTJ;
    342:     PetscBLASInt *pivots;
    343:     PetscScalar *W;

    345:     PetscMalloc1(n*n, &JTJ);
    346:     PetscMalloc2(n, &pivots, n, &W);
    347:     for (i = 0; i < n; i++) {
    348:       for (j = 0; j < n; j++) {
    349:         PetscScalar val = 0.;

    351:         for (k = 0; k < m; k++) val += Js[k * n + i] * Js[k * n + j];
    352:         JTJ[i * n + j] = val;
    353:       }
    354:     }

    356:     PetscStackCallBLAS("LAPACKgetrf", LAPACKgetrf_(&bn, &bn, JTJ, &bn, pivots, &info));
    357:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetrf %D",(PetscInt)info);
    358:     PetscStackCallBLAS("LAPACKgetri", LAPACKgetri_(&bn, JTJ, &bn, pivots, W, &bn, &info));
    359:     if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error returned from LAPACKgetri %D",(PetscInt)info);
    360:     for (i = 0; i < n; i++) {
    361:       for (j = 0; j < m; j++) {
    362:         PetscScalar val = 0.;

    364:         for (k = 0; k < n; k++) val += JTJ[i * n + k] * Js[j * n + k];
    365:         Jinvs[i * m + j] = val;
    366:       }
    367:     }
    368:     PetscFree2(pivots, W);
    369:     PetscFree(JTJ);
    370:   }
    371: #if defined(PETSC_USE_COMPLEX)
    372:   for (i = 0; i < m*n; i++) Jinv[i] = PetscRealPart(Jinvs[i]);
    373:   PetscFree2(Js, Jinvs);
    374: #endif
    375:   return(0);
    376: }

    378: /*@
    379:    PetscQuadraturePushForward - Push forward a quadrature functional under an affine transformation.

    381:    Collecive on PetscQuadrature

    383:    Input Arguments:
    384: +  q - the quadrature functional
    385: .  imageDim - the dimension of the image of the transformation
    386: .  origin - a point in the original space
    387: .  originImage - the image of the origin under the transformation
    388: .  J - the Jacobian of the image: an [imageDim x dim] matrix in row major order
    389: -  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]

    391:    Output Arguments:
    392: .  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.

    394:    Note: 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.

    396:    Level: intermediate

    398: .seealso: PetscDTAltVPullback(), PetscDTAltVPullbackMatrix()
    399: @*/
    400: PetscErrorCode PetscQuadraturePushForward(PetscQuadrature q, PetscInt imageDim, const PetscReal origin[], const PetscReal originImage[], const PetscReal J[], PetscInt formDegree, PetscQuadrature *Jinvstarq)
    401: {
    402:   PetscInt         dim, Nc, imageNc, formSize, Ncopies, imageFormSize, Npoints, pt, i, j, c;
    403:   const PetscReal *points;
    404:   const PetscReal *weights;
    405:   PetscReal       *imagePoints, *imageWeights;
    406:   PetscReal       *Jinv;
    407:   PetscReal       *Jinvstar;
    408:   PetscErrorCode   ierr;

    412:   if (imageDim < PetscAbsInt(formDegree)) SETERRQ2(PetscObjectComm((PetscObject)q), PETSC_ERR_ARG_INCOMP, "Cannot represent a %D-form in %D dimensions", PetscAbsInt(formDegree), imageDim);
    413:   PetscQuadratureGetData(q, &dim, &Nc, &Npoints, &points, &weights);
    414:   PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &formSize);
    415:   if (Nc % formSize) SETERRQ2(PetscObjectComm((PetscObject)q), PETSC_ERR_ARG_INCOMP, "Number of components %D is not a multiple of formSize %D\n", Nc, formSize);
    416:   Ncopies = Nc / formSize;
    417:   PetscDTBinomialInt(imageDim, PetscAbsInt(formDegree), &imageFormSize);
    418:   imageNc = Ncopies * imageFormSize;
    419:   PetscMalloc1(Npoints * imageDim, &imagePoints);
    420:   PetscMalloc1(Npoints * imageNc, &imageWeights);
    421:   PetscMalloc2(imageDim * dim, &Jinv, formSize * imageFormSize, &Jinvstar);
    422:   PetscDTJacobianInverse_Internal(imageDim, dim, J, Jinv);
    423:   PetscDTAltVPullbackMatrix(imageDim, dim, Jinv, formDegree, Jinvstar);
    424:   for (pt = 0; pt < Npoints; pt++) {
    425:     const PetscReal *point = &points[pt * dim];
    426:     PetscReal       *imagePoint = &imagePoints[pt * imageDim];

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

    431:       for (j = 0; j < dim; j++) val += J[i * dim + j] * (point[j] - origin[j]);
    432:       imagePoint[i] = val;
    433:     }
    434:     for (c = 0; c < Ncopies; c++) {
    435:       const PetscReal *form = &weights[pt * Nc + c * formSize];
    436:       PetscReal       *imageForm = &imageWeights[pt * imageNc + c * imageFormSize];

    438:       for (i = 0; i < imageFormSize; i++) {
    439:         PetscReal val = 0.;

    441:         for (j = 0; j < formSize; j++) val += Jinvstar[i * formSize + j] * form[j];
    442:         imageForm[i] = val;
    443:       }
    444:     }
    445:   }
    446:   PetscQuadratureCreate(PetscObjectComm((PetscObject)q), Jinvstarq);
    447:   PetscQuadratureSetData(*Jinvstarq, imageDim, imageNc, Npoints, imagePoints, imageWeights);
    448:   PetscFree2(Jinv, Jinvstar);
    449:   return(0);
    450: }

    452: /*@C
    453:   PetscQuadratureSetData - Sets the data defining the quadrature

    455:   Not collective

    457:   Input Parameters:
    458: + q  - The PetscQuadrature object
    459: . dim - The spatial dimension
    460: . Nc - The number of components
    461: . npoints - The number of quadrature points
    462: . points - The coordinates of each quadrature point
    463: - weights - The weight of each quadrature point

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

    467:   Level: intermediate

    469: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
    470: @*/
    471: PetscErrorCode PetscQuadratureSetData(PetscQuadrature q, PetscInt dim, PetscInt Nc, PetscInt npoints, const PetscReal points[], const PetscReal weights[])
    472: {
    475:   if (dim >= 0)     q->dim       = dim;
    476:   if (Nc >= 0)      q->Nc        = Nc;
    477:   if (npoints >= 0) q->numPoints = npoints;
    478:   if (points) {
    480:     q->points = points;
    481:   }
    482:   if (weights) {
    484:     q->weights = weights;
    485:   }
    486:   return(0);
    487: }

    489: static PetscErrorCode PetscQuadratureView_Ascii(PetscQuadrature quad, PetscViewer v)
    490: {
    491:   PetscInt          q, d, c;
    492:   PetscViewerFormat format;
    493:   PetscErrorCode    ierr;

    496:   if (quad->Nc > 1) {PetscViewerASCIIPrintf(v, "Quadrature of order %D on %D points (dim %D) with %D components\n", quad->order, quad->numPoints, quad->dim, quad->Nc);}
    497:   else              {PetscViewerASCIIPrintf(v, "Quadrature of order %D on %D points (dim %D)\n", quad->order, quad->numPoints, quad->dim);}
    498:   PetscViewerGetFormat(v, &format);
    499:   if (format != PETSC_VIEWER_ASCII_INFO_DETAIL) return(0);
    500:   for (q = 0; q < quad->numPoints; ++q) {
    501:     PetscViewerASCIIPrintf(v, "p%D (", q);
    502:     PetscViewerASCIIUseTabs(v, PETSC_FALSE);
    503:     for (d = 0; d < quad->dim; ++d) {
    504:       if (d) {PetscViewerASCIIPrintf(v, ", ");}
    505:       PetscViewerASCIIPrintf(v, "%+g", (double)quad->points[q*quad->dim+d]);
    506:     }
    507:     PetscViewerASCIIPrintf(v, ") ");
    508:     if (quad->Nc > 1) {PetscViewerASCIIPrintf(v, "w%D (", q);}
    509:     for (c = 0; c < quad->Nc; ++c) {
    510:       if (c) {PetscViewerASCIIPrintf(v, ", ");}
    511:       PetscViewerASCIIPrintf(v, "%+g", (double)quad->weights[q*quad->Nc+c]);
    512:     }
    513:     if (quad->Nc > 1) {PetscViewerASCIIPrintf(v, ")");}
    514:     PetscViewerASCIIPrintf(v, "\n");
    515:     PetscViewerASCIIUseTabs(v, PETSC_TRUE);
    516:   }
    517:   return(0);
    518: }

    520: /*@C
    521:   PetscQuadratureView - Views a PetscQuadrature object

    523:   Collective on quad

    525:   Input Parameters:
    526: + quad  - The PetscQuadrature object
    527: - viewer - The PetscViewer object

    529:   Level: beginner

    531: .seealso: PetscQuadratureCreate(), PetscQuadratureGetData()
    532: @*/
    533: PetscErrorCode PetscQuadratureView(PetscQuadrature quad, PetscViewer viewer)
    534: {
    535:   PetscBool      iascii;

    541:   if (!viewer) {PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject) quad), &viewer);}
    542:   PetscObjectTypeCompare((PetscObject) viewer, PETSCVIEWERASCII, &iascii);
    543:   PetscViewerASCIIPushTab(viewer);
    544:   if (iascii) {PetscQuadratureView_Ascii(quad, viewer);}
    545:   PetscViewerASCIIPopTab(viewer);
    546:   return(0);
    547: }

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

    552:   Not collective

    554:   Input Parameter:
    555: + q - The original PetscQuadrature
    556: . numSubelements - The number of subelements the original element is divided into
    557: . v0 - An array of the initial points for each subelement
    558: - jac - An array of the Jacobian mappings from the reference to each subelement

    560:   Output Parameters:
    561: . dim - The dimension

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

    565:  Not available from Fortran

    567:   Level: intermediate

    569: .seealso: PetscFECreate(), PetscSpaceGetDimension(), PetscDualSpaceGetDimension()
    570: @*/
    571: PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature q, PetscInt numSubelements, const PetscReal v0[], const PetscReal jac[], PetscQuadrature *qref)
    572: {
    573:   const PetscReal *points,    *weights;
    574:   PetscReal       *pointsRef, *weightsRef;
    575:   PetscInt         dim, Nc, order, npoints, npointsRef, c, p, cp, d, e;
    576:   PetscErrorCode   ierr;

    583:   PetscQuadratureCreate(PETSC_COMM_SELF, qref);
    584:   PetscQuadratureGetOrder(q, &order);
    585:   PetscQuadratureGetData(q, &dim, &Nc, &npoints, &points, &weights);
    586:   npointsRef = npoints*numSubelements;
    587:   PetscMalloc1(npointsRef*dim,&pointsRef);
    588:   PetscMalloc1(npointsRef*Nc, &weightsRef);
    589:   for (c = 0; c < numSubelements; ++c) {
    590:     for (p = 0; p < npoints; ++p) {
    591:       for (d = 0; d < dim; ++d) {
    592:         pointsRef[(c*npoints + p)*dim+d] = v0[c*dim+d];
    593:         for (e = 0; e < dim; ++e) {
    594:           pointsRef[(c*npoints + p)*dim+d] += jac[(c*dim + d)*dim+e]*(points[p*dim+e] + 1.0);
    595:         }
    596:       }
    597:       /* Could also use detJ here */
    598:       for (cp = 0; cp < Nc; ++cp) weightsRef[(c*npoints+p)*Nc+cp] = weights[p*Nc+cp]/numSubelements;
    599:     }
    600:   }
    601:   PetscQuadratureSetOrder(*qref, order);
    602:   PetscQuadratureSetData(*qref, dim, Nc, npointsRef, pointsRef, weightsRef);
    603:   return(0);
    604: }

    606: /* Compute the coefficients for the Jacobi polynomial recurrence,
    607:  *
    608:  * J^{a,b}_n(x) = (cnm1 + cnm1x * x) * J^{a,b}_{n-1}(x) - cnm2 * J^{a,b}_{n-2}(x).
    609:  */
    610: #define PetscDTJacobiRecurrence_Internal(n,a,b,cnm1,cnm1x,cnm2) \
    611: do {                                                            \
    612:   PetscReal _a = (a);                                           \
    613:   PetscReal _b = (b);                                           \
    614:   PetscReal _n = (n);                                           \
    615:   if (n == 1) {                                                 \
    616:     (cnm1) = (_a-_b) * 0.5;                                     \
    617:     (cnm1x) = (_a+_b+2.)*0.5;                                   \
    618:     (cnm2) = 0.;                                                \
    619:   } else {                                                      \
    620:     PetscReal _2n = _n+_n;                                      \
    621:     PetscReal _d = (_2n*(_n+_a+_b)*(_2n+_a+_b-2));              \
    622:     PetscReal _n1 = (_2n+_a+_b-1.)*(_a*_a-_b*_b);               \
    623:     PetscReal _n1x = (_2n+_a+_b-1.)*(_2n+_a+_b)*(_2n+_a+_b-2);  \
    624:     PetscReal _n2 = 2.*((_n+_a-1.)*(_n+_b-1.)*(_2n+_a+_b));     \
    625:     (cnm1) = _n1 / _d;                                          \
    626:     (cnm1x) = _n1x / _d;                                        \
    627:     (cnm2) = _n2 / _d;                                          \
    628:   }                                                             \
    629: } while (0)

    631: /*@
    632:   PetscDTJacobiNorm - Compute the weighted L2 norm of a Jacobi polynomial.

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

    636:   Input Arguments:
    637: - alpha - the left exponent > -1
    638: . beta - the right exponent > -1
    639: + n - the polynomial degree

    641:   Output Arguments:
    642: . norm - the weighted L2 norm

    644:   Level: beginner

    646: .seealso: PetscDTJacobiEval()
    647: @*/
    648: PetscErrorCode PetscDTJacobiNorm(PetscReal alpha, PetscReal beta, PetscInt n, PetscReal *norm)
    649: {
    650:   PetscReal twoab1;
    651:   PetscReal gr;

    654:   if (alpha <= -1.) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Exponent alpha %g <= -1. invalid\n", (double) alpha);
    655:   if (beta <= -1.) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Exponent beta %g <= -1. invalid\n", (double) beta);
    656:   if (n < 0) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "n %D < 0 invalid\n", n);
    657:   twoab1 = PetscPowReal(2., alpha + beta + 1.);
    658: #if defined(PETSC_HAVE_LGAMMA)
    659:   if (!n) {
    660:     gr = PetscExpReal(PetscLGamma(alpha+1.) + PetscLGamma(beta+1.) - PetscLGamma(alpha+beta+2.));
    661:   } else {
    662:     gr = PetscExpReal(PetscLGamma(n+alpha+1.) + PetscLGamma(n+beta+1.) - (PetscLGamma(n+1.) + PetscLGamma(n+alpha+beta+1.))) / (n+n+alpha+beta+1.);
    663:   }
    664: #else
    665:   {
    666:     PetscInt alphai = (PetscInt) alpha;
    667:     PetscInt betai = (PetscInt) beta;
    668:     PetscInt i;

    670:     gr = n ? (1. / (n+n+alpha+beta+1.)) : 1.;
    671:     if ((PetscReal) alphai == alpha) {
    672:       if (!n) {
    673:         for (i = 0; i < alphai; i++) gr *= (i+1.) / (beta+i+1.);
    674:         gr /= (alpha+beta+1.);
    675:       } else {
    676:         for (i = 0; i < alphai; i++) gr *= (n+i+1.) / (n+beta+i+1.);
    677:       }
    678:     } else if ((PetscReal) betai == beta) {
    679:       if (!n) {
    680:         for (i = 0; i < betai; i++) gr *= (i+1.) / (alpha+i+2.);
    681:         gr /= (alpha+beta+1.);
    682:       } else {
    683:         for (i = 0; i < betai; i++) gr *= (n+i+1.) / (n+alpha+i+1.);
    684:       }
    685:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"lgamma() - math routine is unavailable.");
    686:   }
    687: #endif
    688:   *norm = PetscSqrtReal(twoab1 * gr);
    689:   return(0);
    690: }

    692: static PetscErrorCode PetscDTJacobiEval_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscInt k, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *p)
    693: {
    694:   PetscReal ak, bk;
    695:   PetscReal abk1;
    696:   PetscInt i,l,maxdegree;

    699:   maxdegree = degrees[ndegree-1] - k;
    700:   ak = a + k;
    701:   bk = b + k;
    702:   abk1 = a + b + k + 1.;
    703:   if (maxdegree < 0) {
    704:     for (i = 0; i < npoints; i++) for (l = 0; l < ndegree; l++) p[i*ndegree+l] = 0.;
    705:     return(0);
    706:   }
    707:   for (i=0; i<npoints; i++) {
    708:     PetscReal pm1,pm2,x;
    709:     PetscReal cnm1, cnm1x, cnm2;
    710:     PetscInt  j,m;

    712:     x    = points[i];
    713:     pm2  = 1.;
    714:     PetscDTJacobiRecurrence_Internal(1,ak,bk,cnm1,cnm1x,cnm2);
    715:     pm1 = (cnm1 + cnm1x*x);
    716:     l    = 0;
    717:     while (l < ndegree && degrees[l] - k < 0) {
    718:       p[l++] = 0.;
    719:     }
    720:     while (l < ndegree && degrees[l] - k == 0) {
    721:       p[l] = pm2;
    722:       for (m = 0; m < k; m++) p[l] *= (abk1 + m) * 0.5;
    723:       l++;
    724:     }
    725:     while (l < ndegree && degrees[l] - k == 1) {
    726:       p[l] = pm1;
    727:       for (m = 0; m < k; m++) p[l] *= (abk1 + 1 + m) * 0.5;
    728:       l++;
    729:     }
    730:     for (j=2; j<=maxdegree; j++) {
    731:       PetscReal pp;

    733:       PetscDTJacobiRecurrence_Internal(j,ak,bk,cnm1,cnm1x,cnm2);
    734:       pp   = (cnm1 + cnm1x*x)*pm1 - cnm2*pm2;
    735:       pm2  = pm1;
    736:       pm1  = pp;
    737:       while (l < ndegree && degrees[l] - k == j) {
    738:         p[l] = pp;
    739:         for (m = 0; m < k; m++) p[l] *= (abk1 + j + m) * 0.5;
    740:         l++;
    741:       }
    742:     }
    743:     p += ndegree;
    744:   }
    745:   return(0);
    746: }

    748: /*@
    749:   PetscDTJacobiEvalJet - Evaluate the jet (function and derivatives) of the Jacobi polynomials basis up to a given degree.  The Jacobi polynomials with indices $\alpha$ and $\beta$ are orthogonal with respect to the weighted inner product $\langle f, g \rangle = \int_{-1}^1 (1+x)^{\alpha} (1-x)^{\beta) f(x) g(x) dx$.

    751:   Input Arguments:
    752: + alpha - the left exponent of the weight
    753: . beta - the right exponetn of the weight
    754: . npoints - the number of points to evaluate the polynomials at
    755: . points - [npoints] array of point coordinates
    756: . degree - the maximm degree polynomial space to evaluate, (degree + 1) will be evaluated total.
    757: - k - the maximum derivative to evaluate in the jet, (k + 1) will be evaluated total.

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

    765:   Level: advanced

    767: .seealso: PetscDTJacobiEval(), PetscDTPKDEvalJet()
    768: @*/
    769: PetscErrorCode PetscDTJacobiEvalJet(PetscReal alpha, PetscReal beta, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
    770: {
    771:   PetscInt        i, j, l;
    772:   PetscInt       *degrees;
    773:   PetscReal      *psingle;
    774:   PetscErrorCode  ierr;

    777:   if (degree == 0) {
    778:     PetscInt zero = 0;

    780:     for (i = 0; i <= k; i++) {
    781:       PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, 1, &zero, &p[i*npoints]);
    782:     }
    783:     return(0);
    784:   }
    785:   PetscMalloc1(degree + 1, &degrees);
    786:   PetscMalloc1((degree + 1) * npoints, &psingle);
    787:   for (i = 0; i <= degree; i++) degrees[i] = i;
    788:   for (i = 0; i <= k; i++) {
    789:     PetscDTJacobiEval_Internal(npoints, alpha, beta, i, points, degree + 1, degrees, psingle);
    790:     for (j = 0; j <= degree; j++) {
    791:       for (l = 0; l < npoints; l++) {
    792:         p[(j * (k + 1) + i) * npoints + l] = psingle[l * (degree + 1) + j];
    793:       }
    794:     }
    795:   }
    796:   PetscFree(psingle);
    797:   PetscFree(degrees);
    798:   return(0);
    799: }

    801: /*@
    802:    PetscDTJacobiEval - evaluate Jacobi polynomials for the weight function $(1.+x)^{\alpha} (1.-x)^{\beta}$
    803:                        at points

    805:    Not Collective

    807:    Input Arguments:
    808: +  npoints - number of spatial points to evaluate at
    809: .  alpha - the left exponent > -1
    810: .  beta - the right exponent > -1
    811: .  points - array of locations to evaluate at
    812: .  ndegree - number of basis degrees to evaluate
    813: -  degrees - sorted array of degrees to evaluate

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

    820:    Level: intermediate

    822: .seealso: PetscDTGaussQuadrature()
    823: @*/
    824: PetscErrorCode PetscDTJacobiEval(PetscInt npoints,PetscReal alpha, PetscReal beta, const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
    825: {

    829:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
    830:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");
    831:   if (!npoints || !ndegree) return(0);
    832:   if (B)  {PetscDTJacobiEval_Internal(npoints, alpha, beta, 0, points, ndegree, degrees, B);}
    833:   if (D)  {PetscDTJacobiEval_Internal(npoints, alpha, beta, 1, points, ndegree, degrees, D);}
    834:   if (D2) {PetscDTJacobiEval_Internal(npoints, alpha, beta, 2, points, ndegree, degrees, D2);}
    835:   return(0);
    836: }

    838: /*@
    839:    PetscDTLegendreEval - evaluate Legendre polynomials at points

    841:    Not Collective

    843:    Input Arguments:
    844: +  npoints - number of spatial points to evaluate at
    845: .  points - array of locations to evaluate at
    846: .  ndegree - number of basis degrees to evaluate
    847: -  degrees - sorted array of degrees to evaluate

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

    854:    Level: intermediate

    856: .seealso: PetscDTGaussQuadrature()
    857: @*/
    858: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
    859: {

    863:   PetscDTJacobiEval(npoints, 0., 0., points, ndegree, degrees, B, D, D2);
    864:   return(0);
    865: }

    867: /*@
    868:   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)

    870:   Input Parameters:
    871: + len - the desired length of the degree tuple
    872: - index - the index to convert: should be >= 0

    874:   Output Parameter:
    875: . degtup - will be filled with a tuple of degrees

    877:   Level: beginner

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

    883: .seealso: PetscDTGradedOrderToIndex()
    884: @*/
    885: PetscErrorCode PetscDTIndexToGradedOrder(PetscInt len, PetscInt index, PetscInt degtup[])
    886: {
    887:   PetscInt i, total;
    888:   PetscInt sum;

    891:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
    892:   if (index < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
    893:   total = 1;
    894:   sum = 0;
    895:   while (index >= total) {
    896:     index -= total;
    897:     total = (total * (len + sum)) / (sum + 1);
    898:     sum++;
    899:   }
    900:   for (i = 0; i < len; i++) {
    901:     PetscInt c;

    903:     degtup[i] = sum;
    904:     for (c = 0, total = 1; c < sum; c++) {
    905:       /* going into the loop, total is the number of way to have a tuple of sum exactly c with length len - 1 - i */
    906:       if (index < total) break;
    907:       index -= total;
    908:       total = (total * (len - 1 - i + c)) / (c + 1);
    909:       degtup[i]--;
    910:     }
    911:     sum -= degtup[i];
    912:   }
    913:   return(0);
    914: }

    916: /*@
    917:   PetscDTGradedOrderToIndex - convert a tuple into an index in a graded order, the inverse of PetscDTIndexToGradedOrder().

    919:   Input Parameters:
    920: + len - the length of the degree tuple
    921: - degtup - tuple with this length

    923:   Output Parameter:
    924: . index - index in graded order: >= 0

    926:   Level: Beginner

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

    932: .seealso: PetscDTIndexToGradedOrder()
    933: @*/
    934: PetscErrorCode PetscDTGradedOrderToIndex(PetscInt len, const PetscInt degtup[], PetscInt *index)
    935: {
    936:   PetscInt i, idx, sum, total;

    939:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
    940:   for (i = 0, sum = 0; i < len; i++) sum += degtup[i];
    941:   idx = 0;
    942:   total = 1;
    943:   for (i = 0; i < sum; i++) {
    944:     idx += total;
    945:     total = (total * (len + i)) / (i + 1);
    946:   }
    947:   for (i = 0; i < len - 1; i++) {
    948:     PetscInt c;

    950:     total = 1;
    951:     sum -= degtup[i];
    952:     for (c = 0; c < sum; c++) {
    953:       idx += total;
    954:       total = (total * (len - 1 - i + c)) / (c + 1);
    955:     }
    956:   }
    957:   *index = idx;
    958:   return(0);
    959: }

    961: static PetscBool PKDCite = PETSC_FALSE;
    962: const char       PKDCitation[] = "@article{Kirby2010,\n"
    963:                                  "  title={Singularity-free evaluation of collapsed-coordinate orthogonal polynomials},\n"
    964:                                  "  author={Kirby, Robert C},\n"
    965:                                  "  journal={ACM Transactions on Mathematical Software (TOMS)},\n"
    966:                                  "  volume={37},\n"
    967:                                  "  number={1},\n"
    968:                                  "  pages={1--16},\n"
    969:                                  "  year={2010},\n"
    970:                                  "  publisher={ACM New York, NY, USA}\n}\n";

    972: /*@
    973:   PetscDTPKDEvalJet - Evaluate the jet (function and derivatives) of the Prioriol-Koornwinder-Dubiner (PKD) basis for
    974:   the space of polynomials up to a given degree.  The PKD basis is L2-orthonormal on the biunit simplex (which is used
    975:   as the reference element for finite elements in PETSc), which makes it a stable basis to use for evaluating
    976:   polynomials in that domain.

    978:   Input Arguments:
    979: + dim - the number of variables in the multivariate polynomials
    980: . npoints - the number of points to evaluate the polynomials at
    981: . points - [npoints x dim] array of point coordinates
    982: . 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.
    983: - k - the maximum order partial derivative to evaluate in the jet.  There are (dim + k choose dim) partial derivatives
    984:   in the jet.  Choosing k = 0 means to evaluate just the function and no derivatives

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

    992:   Level: advanced

    994:   Note: The ordering of the basis functions, and the ordering of the derivatives in the jet, both follow the graded
    995:   ordering of PetscDTIndexToGradedOrder() and PetscDTGradedOrderToIndex().  For example, in 3D, the polynomial with
    996:   leading monomial x^3,y^1,z^2, which as degree tuple (2,0,1), which by PetscDTGradedOrderToIndex() has index 12 (it is the 13th basis function in the space);
    997:   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).

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

    1001: .seealso: PetscDTGradedOrderToIndex(), PetscDTIndexToGradedOrder(), PetscDTJacobiEvalJet()
    1002: @*/
    1003: PetscErrorCode PetscDTPKDEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
    1004: {
    1005:   PetscInt        degidx, kidx, d, pt;
    1006:   PetscInt        Nk, Ndeg;
    1007:   PetscInt       *ktup, *degtup;
    1008:   PetscReal      *scales, initscale, scaleexp;
    1009:   PetscErrorCode  ierr;

    1012:   PetscCitationsRegister(PKDCitation, &PKDCite);
    1013:   PetscDTBinomialInt(dim + k, k, &Nk);
    1014:   PetscDTBinomialInt(degree + dim, degree, &Ndeg);
    1015:   PetscMalloc2(dim, &degtup, dim, &ktup);
    1016:   PetscMalloc1(Ndeg, &scales);
    1017:   initscale = 1.;
    1018:   if (dim > 1) {
    1019:     PetscDTBinomial(dim,2,&scaleexp);
    1020:     initscale = PetscPowReal(2.,scaleexp*0.5);
    1021:   }
    1022:   for (degidx = 0; degidx < Ndeg; degidx++) {
    1023:     PetscInt e, i;
    1024:     PetscInt m1idx = -1, m2idx = -1;
    1025:     PetscInt n;
    1026:     PetscInt degsum;
    1027:     PetscReal alpha;
    1028:     PetscReal cnm1, cnm1x, cnm2;
    1029:     PetscReal norm;

    1031:     PetscDTIndexToGradedOrder(dim, degidx, degtup);
    1032:     for (d = dim - 1; d >= 0; d--) if (degtup[d]) break;
    1033:     if (d < 0) { /* constant is 1 everywhere, all derivatives are zero */
    1034:       scales[degidx] = initscale;
    1035:       for (e = 0; e < dim; e++) {
    1036:         PetscDTJacobiNorm(e,0.,0,&norm);
    1037:         scales[degidx] /= norm;
    1038:       }
    1039:       for (i = 0; i < npoints; i++) p[degidx * Nk * npoints + i] = 1.;
    1040:       for (i = 0; i < (Nk - 1) * npoints; i++) p[(degidx * Nk + 1) * npoints + i] = 0.;
    1041:       continue;
    1042:     }
    1043:     n = degtup[d];
    1044:     degtup[d]--;
    1045:     PetscDTGradedOrderToIndex(dim, degtup, &m1idx);
    1046:     if (degtup[d] > 0) {
    1047:       degtup[d]--;
    1048:       PetscDTGradedOrderToIndex(dim, degtup, &m2idx);
    1049:       degtup[d]++;
    1050:     }
    1051:     degtup[d]++;
    1052:     for (e = 0, degsum = 0; e < d; e++) degsum += degtup[e];
    1053:     alpha = 2 * degsum + d;
    1054:     PetscDTJacobiRecurrence_Internal(n,alpha,0.,cnm1,cnm1x,cnm2);

    1057:     scales[degidx] = initscale;
    1058:     for (e = 0, degsum = 0; e < dim; e++) {
    1059:       PetscInt  f;
    1060:       PetscReal ealpha;
    1061:       PetscReal enorm;

    1063:       ealpha = 2 * degsum + e;
    1064:       for (f = 0; f < degsum; f++) scales[degidx] *= 2.;
    1065:       PetscDTJacobiNorm(ealpha,0.,degtup[e],&enorm);
    1066:       scales[degidx] /= enorm;
    1067:       degsum += degtup[e];
    1068:     }

    1070:     for (pt = 0; pt < npoints; pt++) {
    1071:       /* compute the multipliers */
    1072:       PetscReal thetanm1, thetanm1x, thetanm2;

    1074:       thetanm1x = dim - (d+1) + 2.*points[pt * dim + d];
    1075:       for (e = d+1; e < dim; e++) thetanm1x += points[pt * dim + e];
    1076:       thetanm1x *= 0.5;
    1077:       thetanm1 = (2. - (dim-(d+1)));
    1078:       for (e = d+1; e < dim; e++) thetanm1 -= points[pt * dim + e];
    1079:       thetanm1 *= 0.5;
    1080:       thetanm2 = thetanm1 * thetanm1;

    1082:       for (kidx = 0; kidx < Nk; kidx++) {
    1083:         PetscInt f;

    1085:         PetscDTIndexToGradedOrder(dim, kidx, ktup);
    1086:         /* first sum in the same derivative terms */
    1087:         p[(degidx * Nk + kidx) * npoints + pt] = (cnm1 * thetanm1 + cnm1x * thetanm1x) * p[(m1idx * Nk + kidx) * npoints + pt];
    1088:         if (m2idx >= 0) {
    1089:           p[(degidx * Nk + kidx) * npoints + pt] -= cnm2 * thetanm2 * p[(m2idx * Nk + kidx) * npoints + pt];
    1090:         }

    1092:         for (f = d; f < dim; f++) {
    1093:           PetscInt km1idx, mplty = ktup[f];

    1095:           if (!mplty) continue;
    1096:           ktup[f]--;
    1097:           PetscDTGradedOrderToIndex(dim, ktup, &km1idx);

    1099:           /* the derivative of  cnm1x * thetanm1x  wrt x variable f is 0.5 * cnm1x if f > d otherwise it is cnm1x */
    1100:           /* the derivative of  cnm1  * thetanm1   wrt x variable f is 0 if f == d, otherwise it is -0.5 * cnm1 */
    1101:           /* the derivative of -cnm2  * thetanm2   wrt x variable f is 0 if f == d, otherwise it is cnm2 * thetanm1 */
    1102:           if (f > d) {
    1103:             PetscInt f2;

    1105:             p[(degidx * Nk + kidx) * npoints + pt] += mplty * 0.5 * (cnm1x - cnm1) * p[(m1idx * Nk + km1idx) * npoints + pt];
    1106:             if (m2idx >= 0) {
    1107:               p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm2 * thetanm1 * p[(m2idx * Nk + km1idx) * npoints + pt];
    1108:               /* second derivatives of -cnm2  * thetanm2   wrt x variable f,f2 is like - 0.5 * cnm2 */
    1109:               for (f2 = f; f2 < dim; f2++) {
    1110:                 PetscInt km2idx, mplty2 = ktup[f2];
    1111:                 PetscInt factor;

    1113:                 if (!mplty2) continue;
    1114:                 ktup[f2]--;
    1115:                 PetscDTGradedOrderToIndex(dim, ktup, &km2idx);

    1117:                 factor = mplty * mplty2;
    1118:                 if (f == f2) factor /= 2;
    1119:                 p[(degidx * Nk + kidx) * npoints + pt] -= 0.5 * factor * cnm2 * p[(m2idx * Nk + km2idx) * npoints + pt];
    1120:                 ktup[f2]++;
    1121:               }
    1122:             }
    1123:           } else {
    1124:             p[(degidx * Nk + kidx) * npoints + pt] += mplty * cnm1x * p[(m1idx * Nk + km1idx) * npoints + pt];
    1125:           }
    1126:           ktup[f]++;
    1127:         }
    1128:       }
    1129:     }
    1130:   }
    1131:   for (degidx = 0; degidx < Ndeg; degidx++) {
    1132:     PetscReal scale = scales[degidx];
    1133:     PetscInt i;

    1135:     for (i = 0; i < Nk * npoints; i++) p[degidx*Nk*npoints + i] *= scale;
    1136:   }
    1137:   PetscFree(scales);
    1138:   PetscFree2(degtup, ktup);
    1139:   return(0);
    1140: }

    1142: /* solve the symmetric tridiagonal eigenvalue system, writing the eigenvalues into eigs and the eigenvectors into V
    1143:  * with lds n; diag and subdiag are overwritten */
    1144: static PetscErrorCode PetscDTSymmetricTridiagonalEigensolve(PetscInt n, PetscReal diag[], PetscReal subdiag[],
    1145:                                                             PetscReal eigs[], PetscScalar V[])
    1146: {
    1147:   char jobz = 'V'; /* eigenvalues and eigenvectors */
    1148:   char range = 'A'; /* all eigenvalues will be found */
    1149:   PetscReal VL = 0.; /* ignored because range is 'A' */
    1150:   PetscReal VU = 0.; /* ignored because range is 'A' */
    1151:   PetscBLASInt IL = 0; /* ignored because range is 'A' */
    1152:   PetscBLASInt IU = 0; /* ignored because range is 'A' */
    1153:   PetscReal abstol = 0.; /* unused */
    1154:   PetscBLASInt bn, bm, ldz; /* bm will equal bn on exit */
    1155:   PetscBLASInt *isuppz;
    1156:   PetscBLASInt lwork, liwork;
    1157:   PetscReal workquery;
    1158:   PetscBLASInt  iworkquery;
    1159:   PetscBLASInt *iwork;
    1160:   PetscBLASInt info;
    1161:   PetscReal *work = NULL;

    1165: #if !defined(PETSCDTGAUSSIANQUADRATURE_EIG)
    1166:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
    1167: #endif
    1168:   PetscBLASIntCast(n, &bn);
    1169:   PetscBLASIntCast(n, &ldz);
    1170: #if !defined(PETSC_MISSING_LAPACK_STEGR)
    1171:   PetscMalloc1(2 * n, &isuppz);
    1172:   lwork = -1;
    1173:   liwork = -1;
    1174:   PetscStackCallBLAS("LAPACKstegr",LAPACKstegr_(&jobz,&range,&bn,diag,subdiag,&VL,&VU,&IL,&IU,&abstol,&bm,eigs,V,&ldz,isuppz,&workquery,&lwork,&iworkquery,&liwork,&info));
    1175:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEGR error");
    1176:   lwork = (PetscBLASInt) workquery;
    1177:   liwork = (PetscBLASInt) iworkquery;
    1178:   PetscMalloc2(lwork, &work, liwork, &iwork);
    1179:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
    1180:   PetscStackCallBLAS("LAPACKstegr",LAPACKstegr_(&jobz,&range,&bn,diag,subdiag,&VL,&VU,&IL,&IU,&abstol,&bm,eigs,V,&ldz,isuppz,work,&lwork,iwork,&liwork,&info));
    1181:   PetscFPTrapPop();
    1182:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEGR error");
    1183:   PetscFree2(work, iwork);
    1184:   PetscFree(isuppz);
    1185: #elif !defined(PETSC_MISSING_LAPACK_STEQR)
    1186:   jobz = 'I'; /* Compute eigenvalues and eigenvectors of the
    1187:                  tridiagonal matrix.  Z is initialized to the identity
    1188:                  matrix. */
    1189:   PetscMalloc1(PetscMax(1,2*n-2),&work);
    1190:   PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&bn,diag,subdiag,V,&ldz,work,&info));
    1191:   PetscFPTrapPop();
    1192:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");
    1193:   PetscFree(work);
    1194:   PetscArraycpy(eigs,diag,n);
    1195: #endif
    1196:   return(0);
    1197: }

    1199: /* Formula for the weights at the endpoints (-1 and 1) of Gauss-Lobatto-Jacobi
    1200:  * quadrature rules on the interval [-1, 1] */
    1201: static PetscErrorCode PetscDTGaussLobattoJacobiEndweights_Internal(PetscInt n, PetscReal alpha, PetscReal beta, PetscReal *leftw, PetscReal *rightw)
    1202: {
    1203:   PetscReal twoab1;
    1204:   PetscInt  m = n - 2;
    1205:   PetscReal a = alpha + 1.;
    1206:   PetscReal b = beta + 1.;
    1207:   PetscReal gra, grb;

    1210:   twoab1 = PetscPowReal(2., a + b - 1.);
    1211: #if defined(PETSC_HAVE_LGAMMA)
    1212:   grb = PetscExpReal(2. * PetscLGamma(b+1.) + PetscLGamma(m+1.) + PetscLGamma(m+a+1.) -
    1213:                      (PetscLGamma(m+b+1) + PetscLGamma(m+a+b+1.)));
    1214:   gra = PetscExpReal(2. * PetscLGamma(a+1.) + PetscLGamma(m+1.) + PetscLGamma(m+b+1.) -
    1215:                      (PetscLGamma(m+a+1) + PetscLGamma(m+a+b+1.)));
    1216: #else
    1217:   {
    1218:     PetscInt alphai = (PetscInt) alpha;
    1219:     PetscInt betai = (PetscInt) beta;

    1222:     if ((PetscReal) alphai == alpha && (PetscReal) betai == beta) {
    1223:       PetscReal binom1, binom2;

    1225:       PetscDTBinomial(m+b, b, &binom1);
    1226:       PetscDTBinomial(m+a+b, b, &binom2);
    1227:       grb = 1./ (binom1 * binom2);
    1228:       PetscDTBinomial(m+a, a, &binom1);
    1229:       PetscDTBinomial(m+a+b, a, &binom2);
    1230:       gra = 1./ (binom1 * binom2);
    1231:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"lgamma() - math routine is unavailable.");
    1232:   }
    1233: #endif
    1234:   *leftw = twoab1 * grb / b;
    1235:   *rightw = twoab1 * gra / a;
    1236:   return(0);
    1237: }

    1239: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
    1240:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
    1241: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
    1242: {
    1243:   PetscReal pn1, pn2;
    1244:   PetscReal cnm1, cnm1x, cnm2;
    1245:   PetscInt  k;

    1248:   if (!n) {*P = 1.0; return(0);}
    1249:   PetscDTJacobiRecurrence_Internal(1,a,b,cnm1,cnm1x,cnm2);
    1250:   pn2 = 1.;
    1251:   pn1 = cnm1 + cnm1x*x;
    1252:   if (n == 1) {*P = pn1; return(0);}
    1253:   *P  = 0.0;
    1254:   for (k = 2; k < n+1; ++k) {
    1255:     PetscDTJacobiRecurrence_Internal(k,a,b,cnm1,cnm1x,cnm2);

    1257:     *P  = (cnm1 + cnm1x*x)*pn1 - cnm2*pn2;
    1258:     pn2 = pn1;
    1259:     pn1 = *P;
    1260:   }
    1261:   return(0);
    1262: }

    1264: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
    1265: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscInt k, PetscReal *P)
    1266: {
    1267:   PetscReal      nP;
    1268:   PetscInt       i;

    1272:   *P = 0.0;
    1273:   if (k > n) return(0);
    1274:   PetscDTComputeJacobi(a+k, b+k, n-k, x, &nP);
    1275:   for (i = 0; i < k; i++) nP *= (a + b + n + 1. + i) * 0.5;
    1276:   *P = nP;
    1277:   return(0);
    1278: }

    1280: static PetscErrorCode PetscDTGaussJacobiQuadrature_Newton_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal x[], PetscReal w[])
    1281: {
    1282:   PetscInt       maxIter = 100;
    1283:   PetscReal      eps     = PetscExpReal(0.75 * PetscLogReal(PETSC_MACHINE_EPSILON));
    1284:   PetscReal      a1, a6, gf;
    1285:   PetscInt       k;

    1290:   a1 = PetscPowReal(2.0, a+b+1);
    1291: #if defined(PETSC_HAVE_LGAMMA)
    1292:   {
    1293:     PetscReal a2, a3, a4, a5;
    1294:     a2 = PetscLGamma(a + npoints + 1);
    1295:     a3 = PetscLGamma(b + npoints + 1);
    1296:     a4 = PetscLGamma(a + b + npoints + 1);
    1297:     a5 = PetscLGamma(npoints + 1);
    1298:     gf = PetscExpReal(a2 + a3 - (a4 + a5));
    1299:   }
    1300: #else
    1301:   {
    1302:     PetscInt ia, ib;

    1304:     ia = (PetscInt) a;
    1305:     ib = (PetscInt) b;
    1306:     gf = 1.;
    1307:     if (ia == a && ia >= 0) { /* compute ratio of rising factorals wrt a */
    1308:       for (k = 0; k < ia; k++) gf *= (npoints + 1. + k) / (npoints + b + 1. + k);
    1309:     } else if (b == b && ib >= 0) { /* compute ratio of rising factorials wrt b */
    1310:       for (k = 0; k < ib; k++) gf *= (npoints + 1. + k) / (npoints + a + 1. + k);
    1311:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"lgamma() - math routine is unavailable.");
    1312:   }
    1313: #endif

    1315:   a6   = a1 * gf;
    1316:   /* Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method with Chebyshev points as initial guesses.
    1317:    Algorithm implemented from the pseudocode given by Karniadakis and Sherwin and Python in FIAT */
    1318:   for (k = 0; k < npoints; ++k) {
    1319:     PetscReal r = PetscCosReal(PETSC_PI * (1. - (4.*k + 3. + 2.*b) / (4.*npoints + 2.*(a + b + 1.)))), dP;
    1320:     PetscInt  j;

    1322:     if (k > 0) r = 0.5 * (r + x[k-1]);
    1323:     for (j = 0; j < maxIter; ++j) {
    1324:       PetscReal s = 0.0, delta, f, fp;
    1325:       PetscInt  i;

    1327:       for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
    1328:       PetscDTComputeJacobi(a, b, npoints, r, &f);
    1329:       PetscDTComputeJacobiDerivative(a, b, npoints, r, 1, &fp);
    1330:       delta = f / (fp - f * s);
    1331:       r     = r - delta;
    1332:       if (PetscAbsReal(delta) < eps) break;
    1333:     }
    1334:     x[k] = r;
    1335:     PetscDTComputeJacobiDerivative(a, b, npoints, x[k], 1, &dP);
    1336:     w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
    1337:   }
    1338:   return(0);
    1339: }

    1341: /* Compute the diagonals of the Jacobi matrix used in Golub & Welsch algorithms for Gauss-Jacobi
    1342:  * quadrature weight calculations on [-1,1] for exponents (1. + x)^a (1.-x)^b */
    1343: static PetscErrorCode PetscDTJacobiMatrix_Internal(PetscInt nPoints, PetscReal a, PetscReal b, PetscReal *d, PetscReal *s)
    1344: {
    1345:   PetscInt       i;

    1348:   for (i = 0; i < nPoints; i++) {
    1349:     PetscReal A, B, C;

    1351:     PetscDTJacobiRecurrence_Internal(i+1,a,b,A,B,C);
    1352:     d[i] = -A / B;
    1353:     if (i) s[i-1] *= C / B;
    1354:     if (i < nPoints - 1) s[i] = 1. / B;
    1355:   }
    1356:   return(0);
    1357: }

    1359: static PetscErrorCode PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
    1360: {
    1361:   PetscReal mu0;
    1362:   PetscReal ga, gb, gab;
    1363:   PetscInt i;

    1367:   PetscCitationsRegister(GolubWelschCitation, &GolubWelschCite);

    1369: #if defined(PETSC_HAVE_TGAMMA)
    1370:   ga  = PetscTGamma(a + 1);
    1371:   gb  = PetscTGamma(b + 1);
    1372:   gab = PetscTGamma(a + b + 2);
    1373: #else
    1374:   {
    1375:     PetscInt ia, ib;

    1377:     ia = (PetscInt) a;
    1378:     ib = (PetscInt) b;
    1379:     if (ia == a && ib == b && ia + 1 > 0 && ib + 1 > 0 && ia + ib + 2 > 0) { /* All gamma(x) terms are (x-1)! terms */
    1380:       PetscDTFactorial(ia, &ga);
    1381:       PetscDTFactorial(ib, &gb);
    1382:       PetscDTFactorial(ia + ib + 1, &gb);
    1383:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
    1384:   }
    1385: #endif
    1386:   mu0 = PetscPowReal(2.,a + b + 1.) * ga * gb / gab;

    1388: #if defined(PETSCDTGAUSSIANQUADRATURE_EIG)
    1389:   {
    1390:     PetscReal *diag, *subdiag;
    1391:     PetscScalar *V;

    1393:     PetscMalloc2(npoints, &diag, npoints, &subdiag);
    1394:     PetscMalloc1(npoints*npoints, &V);
    1395:     PetscDTJacobiMatrix_Internal(npoints, a, b, diag, subdiag);
    1396:     for (i = 0; i < npoints - 1; i++) subdiag[i] = PetscSqrtReal(subdiag[i]);
    1397:     PetscDTSymmetricTridiagonalEigensolve(npoints, diag, subdiag, x, V);
    1398:     for (i = 0; i < npoints; i++) w[i] = PetscSqr(PetscRealPart(V[i * npoints])) * mu0;
    1399:     PetscFree(V);
    1400:     PetscFree2(diag, subdiag);
    1401:   }
    1402: #else
    1403:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
    1404: #endif
    1405:   { /* As of March 2, 2020, The Sun Performance Library breaks the LAPACK contract for xstegr and xsteqr: the
    1406:        eigenvalues are not guaranteed to be in ascending order.  So we heave a passive aggressive sigh and check that
    1407:        the eigenvalues are sorted */
    1408:     PetscBool sorted;

    1410:     PetscSortedReal(npoints, x, &sorted);
    1411:     if (!sorted) {
    1412:       PetscInt *order, i;
    1413:       PetscReal *tmp;

    1415:       PetscMalloc2(npoints, &order, npoints, &tmp);
    1416:       for (i = 0; i < npoints; i++) order[i] = i;
    1417:       PetscSortRealWithPermutation(npoints, x, order);
    1418:       PetscArraycpy(tmp, x, npoints);
    1419:       for (i = 0; i < npoints; i++) x[i] = tmp[order[i]];
    1420:       PetscArraycpy(tmp, w, npoints);
    1421:       for (i = 0; i < npoints; i++) w[i] = tmp[order[i]];
    1422:       PetscFree2(order, tmp);
    1423:     }
    1424:   }
    1425:   return(0);
    1426: }

    1428: static PetscErrorCode PetscDTGaussJacobiQuadrature_Internal(PetscInt npoints,PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
    1429: {

    1433:   if (npoints < 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of points must be positive");
    1434:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
    1435:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
    1436:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");

    1438:   if (newton) {
    1439:     PetscDTGaussJacobiQuadrature_Newton_Internal(npoints, alpha, beta, x, w);
    1440:   } else {
    1441:     PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(npoints, alpha, beta, x, w);
    1442:   }
    1443:   if (alpha == beta) { /* symmetrize */
    1444:     PetscInt i;
    1445:     for (i = 0; i < (npoints + 1) / 2; i++) {
    1446:       PetscInt  j  = npoints - 1 - i;
    1447:       PetscReal xi = x[i];
    1448:       PetscReal xj = x[j];
    1449:       PetscReal wi = w[i];
    1450:       PetscReal wj = w[j];

    1452:       x[i] = (xi - xj) / 2.;
    1453:       x[j] = (xj - xi) / 2.;
    1454:       w[i] = w[j] = (wi + wj) / 2.;
    1455:     }
    1456:   }
    1457:   return(0);
    1458: }

    1460: /*@
    1461:   PetscDTGaussJacobiQuadrature - quadrature for the interval [a, b] with the weight function
    1462:   $(x-a)^\alpha (x-b)^\beta$.

    1464:   Not collective

    1466:   Input Parameters:
    1467: + npoints - the number of points in the quadrature rule
    1468: . a - the left endpoint of the interval
    1469: . b - the right endpoint of the interval
    1470: . alpha - the left exponent
    1471: - beta - the right exponent

    1473:   Output Parameters:
    1474: + x - array of length npoints, the locations of the quadrature points
    1475: - w - array of length npoints, the weights of the quadrature points

    1477:   Level: intermediate

    1479:   Note: this quadrature rule is exact for polynomials up to degree 2*npoints - 1.
    1480: @*/
    1481: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt npoints,PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
    1482: {
    1483:   PetscInt       i;

    1487:   PetscDTGaussJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal);
    1488:   if (a != -1. || b != 1.) { /* shift */
    1489:     for (i = 0; i < npoints; i++) {
    1490:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
    1491:       w[i] *= (b - a) / 2.;
    1492:     }
    1493:   }
    1494:   return(0);
    1495: }

    1497: static PetscErrorCode PetscDTGaussLobattoJacobiQuadrature_Internal(PetscInt npoints,PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
    1498: {
    1499:   PetscInt       i;

    1503:   if (npoints < 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of points must be positive");
    1504:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
    1505:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
    1506:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");

    1508:   x[0] = -1.;
    1509:   x[npoints-1] = 1.;
    1510:   if (npoints > 2) {
    1511:     PetscDTGaussJacobiQuadrature_Internal(npoints-2, alpha+1., beta+1., &x[1], &w[1], newton);
    1512:   }
    1513:   for (i = 1; i < npoints - 1; i++) {
    1514:     w[i] /= (1. - x[i]*x[i]);
    1515:   }
    1516:   PetscDTGaussLobattoJacobiEndweights_Internal(npoints, alpha, beta, &w[0], &w[npoints-1]);
    1517:   return(0);
    1518: }

    1520: /*@
    1521:   PetscDTGaussLobattoJacobiQuadrature - quadrature for the interval [a, b] with the weight function
    1522:   $(x-a)^\alpha (x-b)^\beta$, with endpoints a and b included as quadrature points.

    1524:   Not collective

    1526:   Input Parameters:
    1527: + npoints - the number of points in the quadrature rule
    1528: . a - the left endpoint of the interval
    1529: . b - the right endpoint of the interval
    1530: . alpha - the left exponent
    1531: - beta - the right exponent

    1533:   Output Parameters:
    1534: + x - array of length npoints, the locations of the quadrature points
    1535: - w - array of length npoints, the weights of the quadrature points

    1537:   Level: intermediate

    1539:   Note: this quadrature rule is exact for polynomials up to degree 2*npoints - 3.
    1540: @*/
    1541: PetscErrorCode PetscDTGaussLobattoJacobiQuadrature(PetscInt npoints,PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
    1542: {
    1543:   PetscInt       i;

    1547:   PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal);
    1548:   if (a != -1. || b != 1.) { /* shift */
    1549:     for (i = 0; i < npoints; i++) {
    1550:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
    1551:       w[i] *= (b - a) / 2.;
    1552:     }
    1553:   }
    1554:   return(0);
    1555: }

    1557: /*@
    1558:    PetscDTGaussQuadrature - create Gauss-Legendre quadrature

    1560:    Not Collective

    1562:    Input Arguments:
    1563: +  npoints - number of points
    1564: .  a - left end of interval (often-1)
    1565: -  b - right end of interval (often +1)

    1567:    Output Arguments:
    1568: +  x - quadrature points
    1569: -  w - quadrature weights

    1571:    Level: intermediate

    1573:    References:
    1574: .   1. - Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 1969.

    1576: .seealso: PetscDTLegendreEval()
    1577: @*/
    1578: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
    1579: {
    1580:   PetscInt       i;

    1584:   PetscDTGaussJacobiQuadrature_Internal(npoints, 0., 0., x, w, PetscDTGaussQuadratureNewton_Internal);
    1585:   if (a != -1. || b != 1.) { /* shift */
    1586:     for (i = 0; i < npoints; i++) {
    1587:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
    1588:       w[i] *= (b - a) / 2.;
    1589:     }
    1590:   }
    1591:   return(0);
    1592: }

    1594: /*@C
    1595:    PetscDTGaussLobattoLegendreQuadrature - creates a set of the locations and weights of the Gauss-Lobatto-Legendre
    1596:                       nodes of a given size on the domain [-1,1]

    1598:    Not Collective

    1600:    Input Parameter:
    1601: +  n - number of grid nodes
    1602: -  type - PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA or PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON

    1604:    Output Arguments:
    1605: +  x - quadrature points
    1606: -  w - quadrature weights

    1608:    Notes:
    1609:     For n > 30  the Newton approach computes duplicate (incorrect) values for some nodes because the initial guess is apparently not
    1610:           close enough to the desired solution

    1612:    These are useful for implementing spectral methods based on Gauss-Lobatto-Legendre (GLL) nodes

    1614:    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

    1616:    Level: intermediate

    1618: .seealso: PetscDTGaussQuadrature()

    1620: @*/
    1621: PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt npoints,PetscGaussLobattoLegendreCreateType type,PetscReal *x,PetscReal *w)
    1622: {
    1623:   PetscBool      newton;

    1627:   if (npoints < 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Must provide at least 2 grid points per element");
    1628:   newton = (PetscBool) (type == PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON);
    1629:   PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, 0., 0., x, w, newton);
    1630:   return(0);
    1631: }

    1633: /*@
    1634:   PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

    1636:   Not Collective

    1638:   Input Arguments:
    1639: + dim     - The spatial dimension
    1640: . Nc      - The number of components
    1641: . npoints - number of points in one dimension
    1642: . a       - left end of interval (often-1)
    1643: - b       - right end of interval (often +1)

    1645:   Output Argument:
    1646: . q - A PetscQuadrature object

    1648:   Level: intermediate

    1650: .seealso: PetscDTGaussQuadrature(), PetscDTLegendreEval()
    1651: @*/
    1652: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
    1653: {
    1654:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k, c;
    1655:   PetscReal     *x, *w, *xw, *ww;

    1659:   PetscMalloc1(totpoints*dim,&x);
    1660:   PetscMalloc1(totpoints*Nc,&w);
    1661:   /* Set up the Golub-Welsch system */
    1662:   switch (dim) {
    1663:   case 0:
    1664:     PetscFree(x);
    1665:     PetscFree(w);
    1666:     PetscMalloc1(1, &x);
    1667:     PetscMalloc1(Nc, &w);
    1668:     x[0] = 0.0;
    1669:     for (c = 0; c < Nc; ++c) w[c] = 1.0;
    1670:     break;
    1671:   case 1:
    1672:     PetscMalloc1(npoints,&ww);
    1673:     PetscDTGaussQuadrature(npoints, a, b, x, ww);
    1674:     for (i = 0; i < npoints; ++i) for (c = 0; c < Nc; ++c) w[i*Nc+c] = ww[i];
    1675:     PetscFree(ww);
    1676:     break;
    1677:   case 2:
    1678:     PetscMalloc2(npoints,&xw,npoints,&ww);
    1679:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
    1680:     for (i = 0; i < npoints; ++i) {
    1681:       for (j = 0; j < npoints; ++j) {
    1682:         x[(i*npoints+j)*dim+0] = xw[i];
    1683:         x[(i*npoints+j)*dim+1] = xw[j];
    1684:         for (c = 0; c < Nc; ++c) w[(i*npoints+j)*Nc+c] = ww[i] * ww[j];
    1685:       }
    1686:     }
    1687:     PetscFree2(xw,ww);
    1688:     break;
    1689:   case 3:
    1690:     PetscMalloc2(npoints,&xw,npoints,&ww);
    1691:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
    1692:     for (i = 0; i < npoints; ++i) {
    1693:       for (j = 0; j < npoints; ++j) {
    1694:         for (k = 0; k < npoints; ++k) {
    1695:           x[((i*npoints+j)*npoints+k)*dim+0] = xw[i];
    1696:           x[((i*npoints+j)*npoints+k)*dim+1] = xw[j];
    1697:           x[((i*npoints+j)*npoints+k)*dim+2] = xw[k];
    1698:           for (c = 0; c < Nc; ++c) w[((i*npoints+j)*npoints+k)*Nc+c] = ww[i] * ww[j] * ww[k];
    1699:         }
    1700:       }
    1701:     }
    1702:     PetscFree2(xw,ww);
    1703:     break;
    1704:   default:
    1705:     SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
    1706:   }
    1707:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    1708:   PetscQuadratureSetOrder(*q, 2*npoints-1);
    1709:   PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);
    1710:   PetscObjectChangeTypeName((PetscObject)*q,"GaussTensor");
    1711:   return(0);
    1712: }

    1714: /*@
    1715:   PetscDTStroudConicalQuadrature - create Stroud conical quadrature for a simplex

    1717:   Not Collective

    1719:   Input Arguments:
    1720: + dim     - The simplex dimension
    1721: . Nc      - The number of components
    1722: . npoints - The number of points in one dimension
    1723: . a       - left end of interval (often-1)
    1724: - b       - right end of interval (often +1)

    1726:   Output Argument:
    1727: . q - A PetscQuadrature object

    1729:   Level: intermediate

    1731:   References:
    1732: .  1. - Karniadakis and Sherwin.  FIAT

    1734:   Note: For dim == 1, this is Gauss-Legendre quadrature

    1736: .seealso: PetscDTGaussTensorQuadrature(), PetscDTGaussQuadrature()
    1737: @*/
    1738: PetscErrorCode PetscDTStroudConicalQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
    1739: {
    1740:   PetscInt       totprev, totrem;
    1741:   PetscInt       totpoints;
    1742:   PetscReal     *p1, *w1;
    1743:   PetscReal     *x, *w;
    1744:   PetscInt       i, j, k, l, m, pt, c;

    1748:   if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
    1749:   totpoints = 1;
    1750:   for (i = 0, totpoints = 1; i < dim; i++) totpoints *= npoints;
    1751:   PetscMalloc1(totpoints*dim, &x);
    1752:   PetscMalloc1(totpoints*Nc, &w);
    1753:   PetscMalloc2(npoints, &p1, npoints, &w1);
    1754:   for (i = 0; i < totpoints*Nc; i++) w[i] = 1.;
    1755:   for (i = 0, totprev = 1, totrem = totpoints / npoints; i < dim; i++) {
    1756:     PetscReal mul;

    1758:     mul = PetscPowReal(2.,-i);
    1759:     PetscDTGaussJacobiQuadrature(npoints, -1., 1., i, 0.0, p1, w1);
    1760:     for (pt = 0, l = 0; l < totprev; l++) {
    1761:       for (j = 0; j < npoints; j++) {
    1762:         for (m = 0; m < totrem; m++, pt++) {
    1763:           for (k = 0; k < i; k++) x[pt*dim+k] = (x[pt*dim+k]+1.)*(1.-p1[j])*0.5 - 1.;
    1764:           x[pt * dim + i] = p1[j];
    1765:           for (c = 0; c < Nc; c++) w[pt*Nc + c] *= mul * w1[j];
    1766:         }
    1767:       }
    1768:     }
    1769:     totprev *= npoints;
    1770:     totrem /= npoints;
    1771:   }
    1772:   PetscFree2(p1, w1);
    1773:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    1774:   PetscQuadratureSetOrder(*q, 2*npoints-1);
    1775:   PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);
    1776:   PetscObjectChangeTypeName((PetscObject)*q,"StroudConical");
    1777:   return(0);
    1778: }

    1780: /*@
    1781:   PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell

    1783:   Not Collective

    1785:   Input Arguments:
    1786: + dim   - The cell dimension
    1787: . level - The number of points in one dimension, 2^l
    1788: . a     - left end of interval (often-1)
    1789: - b     - right end of interval (often +1)

    1791:   Output Argument:
    1792: . q - A PetscQuadrature object

    1794:   Level: intermediate

    1796: .seealso: PetscDTGaussTensorQuadrature()
    1797: @*/
    1798: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
    1799: {
    1800:   const PetscInt  p     = 16;                        /* Digits of precision in the evaluation */
    1801:   const PetscReal alpha = (b-a)/2.;                  /* Half-width of the integration interval */
    1802:   const PetscReal beta  = (b+a)/2.;                  /* Center of the integration interval */
    1803:   const PetscReal h     = PetscPowReal(2.0, -level); /* Step size, length between x_k */
    1804:   PetscReal       xk;                                /* Quadrature point x_k on reference domain [-1, 1] */
    1805:   PetscReal       wk    = 0.5*PETSC_PI;              /* Quadrature weight at x_k */
    1806:   PetscReal      *x, *w;
    1807:   PetscInt        K, k, npoints;
    1808:   PetscErrorCode  ierr;

    1811:   if (dim > 1) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %d not yet implemented", dim);
    1812:   if (!level) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
    1813:   /* Find K such that the weights are < 32 digits of precision */
    1814:   for (K = 1; PetscAbsReal(PetscLog10Real(wk)) < 2*p; ++K) {
    1815:     wk = 0.5*h*PETSC_PI*PetscCoshReal(K*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(K*h)));
    1816:   }
    1817:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    1818:   PetscQuadratureSetOrder(*q, 2*K+1);
    1819:   npoints = 2*K-1;
    1820:   PetscMalloc1(npoints*dim, &x);
    1821:   PetscMalloc1(npoints, &w);
    1822:   /* Center term */
    1823:   x[0] = beta;
    1824:   w[0] = 0.5*alpha*PETSC_PI;
    1825:   for (k = 1; k < K; ++k) {
    1826:     wk = 0.5*alpha*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    1827:     xk = PetscTanhReal(0.5*PETSC_PI*PetscSinhReal(k*h));
    1828:     x[2*k-1] = -alpha*xk+beta;
    1829:     w[2*k-1] = wk;
    1830:     x[2*k+0] =  alpha*xk+beta;
    1831:     w[2*k+0] = wk;
    1832:   }
    1833:   PetscQuadratureSetData(*q, dim, 1, npoints, x, w);
    1834:   return(0);
    1835: }

    1837: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    1838: {
    1839:   const PetscInt  p     = 16;        /* Digits of precision in the evaluation */
    1840:   const PetscReal alpha = (b-a)/2.;  /* Half-width of the integration interval */
    1841:   const PetscReal beta  = (b+a)/2.;  /* Center of the integration interval */
    1842:   PetscReal       h     = 1.0;       /* Step size, length between x_k */
    1843:   PetscInt        l     = 0;         /* Level of refinement, h = 2^{-l} */
    1844:   PetscReal       osum  = 0.0;       /* Integral on last level */
    1845:   PetscReal       psum  = 0.0;       /* Integral on the level before the last level */
    1846:   PetscReal       sum;               /* Integral on current level */
    1847:   PetscReal       yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
    1848:   PetscReal       lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
    1849:   PetscReal       wk;                /* Quadrature weight at x_k */
    1850:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
    1851:   PetscInt        d;                 /* Digits of precision in the integral */

    1854:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
    1855:   /* Center term */
    1856:   func(beta, &lval);
    1857:   sum = 0.5*alpha*PETSC_PI*lval;
    1858:   /* */
    1859:   do {
    1860:     PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
    1861:     PetscInt  k = 1;

    1863:     ++l;
    1864:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
    1865:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
    1866:     psum = osum;
    1867:     osum = sum;
    1868:     h   *= 0.5;
    1869:     sum *= 0.5;
    1870:     do {
    1871:       wk = 0.5*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    1872:       yk = 1.0/(PetscExpReal(0.5*PETSC_PI*PetscSinhReal(k*h)) * PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    1873:       lx = -alpha*(1.0 - yk)+beta;
    1874:       rx =  alpha*(1.0 - yk)+beta;
    1875:       func(lx, &lval);
    1876:       func(rx, &rval);
    1877:       lterm   = alpha*wk*lval;
    1878:       maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
    1879:       sum    += lterm;
    1880:       rterm   = alpha*wk*rval;
    1881:       maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
    1882:       sum    += rterm;
    1883:       ++k;
    1884:       /* Only need to evaluate every other point on refined levels */
    1885:       if (l != 1) ++k;
    1886:     } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */

    1888:     d1 = PetscLog10Real(PetscAbsReal(sum - osum));
    1889:     d2 = PetscLog10Real(PetscAbsReal(sum - psum));
    1890:     d3 = PetscLog10Real(maxTerm) - p;
    1891:     if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
    1892:     else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
    1893:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
    1894:   } while (d < digits && l < 12);
    1895:   *sol = sum;

    1897:   return(0);
    1898: }

    1900: #if defined(PETSC_HAVE_MPFR)
    1901: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    1902: {
    1903:   const PetscInt  safetyFactor = 2;  /* Calculate abcissa until 2*p digits */
    1904:   PetscInt        l            = 0;  /* Level of refinement, h = 2^{-l} */
    1905:   mpfr_t          alpha;             /* Half-width of the integration interval */
    1906:   mpfr_t          beta;              /* Center of the integration interval */
    1907:   mpfr_t          h;                 /* Step size, length between x_k */
    1908:   mpfr_t          osum;              /* Integral on last level */
    1909:   mpfr_t          psum;              /* Integral on the level before the last level */
    1910:   mpfr_t          sum;               /* Integral on current level */
    1911:   mpfr_t          yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
    1912:   mpfr_t          lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
    1913:   mpfr_t          wk;                /* Quadrature weight at x_k */
    1914:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
    1915:   PetscInt        d;                 /* Digits of precision in the integral */
    1916:   mpfr_t          pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;

    1919:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
    1920:   /* Create high precision storage */
    1921:   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);
    1922:   /* Initialization */
    1923:   mpfr_set_d(alpha, 0.5*(b-a), MPFR_RNDN);
    1924:   mpfr_set_d(beta,  0.5*(b+a), MPFR_RNDN);
    1925:   mpfr_set_d(osum,  0.0,       MPFR_RNDN);
    1926:   mpfr_set_d(psum,  0.0,       MPFR_RNDN);
    1927:   mpfr_set_d(h,     1.0,       MPFR_RNDN);
    1928:   mpfr_const_pi(pi2, MPFR_RNDN);
    1929:   mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
    1930:   /* Center term */
    1931:   func(0.5*(b+a), &lval);
    1932:   mpfr_set(sum, pi2, MPFR_RNDN);
    1933:   mpfr_mul(sum, sum, alpha, MPFR_RNDN);
    1934:   mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
    1935:   /* */
    1936:   do {
    1937:     PetscReal d1, d2, d3, d4;
    1938:     PetscInt  k = 1;

    1940:     ++l;
    1941:     mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
    1942:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
    1943:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
    1944:     mpfr_set(psum, osum, MPFR_RNDN);
    1945:     mpfr_set(osum,  sum, MPFR_RNDN);
    1946:     mpfr_mul_d(h,   h,   0.5, MPFR_RNDN);
    1947:     mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
    1948:     do {
    1949:       mpfr_set_si(kh, k, MPFR_RNDN);
    1950:       mpfr_mul(kh, kh, h, MPFR_RNDN);
    1951:       /* Weight */
    1952:       mpfr_set(wk, h, MPFR_RNDN);
    1953:       mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
    1954:       mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
    1955:       mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
    1956:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
    1957:       mpfr_sqr(tmp, tmp, MPFR_RNDN);
    1958:       mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
    1959:       mpfr_div(wk, wk, tmp, MPFR_RNDN);
    1960:       /* Abscissa */
    1961:       mpfr_set_d(yk, 1.0, MPFR_RNDZ);
    1962:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
    1963:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
    1964:       mpfr_exp(tmp, msinh, MPFR_RNDN);
    1965:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
    1966:       /* Quadrature points */
    1967:       mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
    1968:       mpfr_mul(lx, lx, alpha, MPFR_RNDU);
    1969:       mpfr_add(lx, lx, beta, MPFR_RNDU);
    1970:       mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
    1971:       mpfr_mul(rx, rx, alpha, MPFR_RNDD);
    1972:       mpfr_add(rx, rx, beta, MPFR_RNDD);
    1973:       /* Evaluation */
    1974:       func(mpfr_get_d(lx, MPFR_RNDU), &lval);
    1975:       func(mpfr_get_d(rx, MPFR_RNDD), &rval);
    1976:       /* Update */
    1977:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
    1978:       mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
    1979:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
    1980:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    1981:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
    1982:       mpfr_set(curTerm, tmp, MPFR_RNDN);
    1983:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
    1984:       mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
    1985:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
    1986:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    1987:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
    1988:       mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
    1989:       ++k;
    1990:       /* Only need to evaluate every other point on refined levels */
    1991:       if (l != 1) ++k;
    1992:       mpfr_log10(tmp, wk, MPFR_RNDN);
    1993:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    1994:     } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor*digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
    1995:     mpfr_sub(tmp, sum, osum, MPFR_RNDN);
    1996:     mpfr_abs(tmp, tmp, MPFR_RNDN);
    1997:     mpfr_log10(tmp, tmp, MPFR_RNDN);
    1998:     d1 = mpfr_get_d(tmp, MPFR_RNDN);
    1999:     mpfr_sub(tmp, sum, psum, MPFR_RNDN);
    2000:     mpfr_abs(tmp, tmp, MPFR_RNDN);
    2001:     mpfr_log10(tmp, tmp, MPFR_RNDN);
    2002:     d2 = mpfr_get_d(tmp, MPFR_RNDN);
    2003:     mpfr_log10(tmp, maxTerm, MPFR_RNDN);
    2004:     d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
    2005:     mpfr_log10(tmp, curTerm, MPFR_RNDN);
    2006:     d4 = mpfr_get_d(tmp, MPFR_RNDN);
    2007:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
    2008:   } while (d < digits && l < 8);
    2009:   *sol = mpfr_get_d(sum, MPFR_RNDN);
    2010:   /* Cleanup */
    2011:   mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
    2012:   return(0);
    2013: }
    2014: #else

    2016: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    2017: {
    2018:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
    2019: }
    2020: #endif

    2022: /* Overwrites A. Can only handle full-rank problems with m>=n
    2023:  * A in column-major format
    2024:  * Ainv in row-major format
    2025:  * tau has length m
    2026:  * worksize must be >= max(1,n)
    2027:  */
    2028: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
    2029: {
    2031:   PetscBLASInt   M,N,K,lda,ldb,ldwork,info;
    2032:   PetscScalar    *A,*Ainv,*R,*Q,Alpha;

    2035: #if defined(PETSC_USE_COMPLEX)
    2036:   {
    2037:     PetscInt i,j;
    2038:     PetscMalloc2(m*n,&A,m*n,&Ainv);
    2039:     for (j=0; j<n; j++) {
    2040:       for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
    2041:     }
    2042:     mstride = m;
    2043:   }
    2044: #else
    2045:   A = A_in;
    2046:   Ainv = Ainv_out;
    2047: #endif

    2049:   PetscBLASIntCast(m,&M);
    2050:   PetscBLASIntCast(n,&N);
    2051:   PetscBLASIntCast(mstride,&lda);
    2052:   PetscBLASIntCast(worksize,&ldwork);
    2053:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
    2054:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info));
    2055:   PetscFPTrapPop();
    2056:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
    2057:   R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

    2059:   /* Extract an explicit representation of Q */
    2060:   Q = Ainv;
    2061:   PetscArraycpy(Q,A,mstride*n);
    2062:   K = N;                        /* full rank */
    2063:   PetscStackCallBLAS("LAPACKorgqr",LAPACKorgqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info));
    2064:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");

    2066:   /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
    2067:   Alpha = 1.0;
    2068:   ldb = lda;
    2069:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb));
    2070:   /* Ainv is Q, overwritten with inverse */

    2072: #if defined(PETSC_USE_COMPLEX)
    2073:   {
    2074:     PetscInt i;
    2075:     for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
    2076:     PetscFree2(A,Ainv);
    2077:   }
    2078: #endif
    2079:   return(0);
    2080: }

    2082: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
    2083: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
    2084: {
    2086:   PetscReal      *Bv;
    2087:   PetscInt       i,j;

    2090:   PetscMalloc1((ninterval+1)*ndegree,&Bv);
    2091:   /* Point evaluation of L_p on all the source vertices */
    2092:   PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
    2093:   /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
    2094:   for (i=0; i<ninterval; i++) {
    2095:     for (j=0; j<ndegree; j++) {
    2096:       if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
    2097:       else           B[i*ndegree+j]   = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
    2098:     }
    2099:   }
    2100:   PetscFree(Bv);
    2101:   return(0);
    2102: }

    2104: /*@
    2105:    PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

    2107:    Not Collective

    2109:    Input Arguments:
    2110: +  degree - degree of reconstruction polynomial
    2111: .  nsource - number of source intervals
    2112: .  sourcex - sorted coordinates of source cell boundaries (length nsource+1)
    2113: .  ntarget - number of target intervals
    2114: -  targetx - sorted coordinates of target cell boundaries (length ntarget+1)

    2116:    Output Arguments:
    2117: .  R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]

    2119:    Level: advanced

    2121: .seealso: PetscDTLegendreEval()
    2122: @*/
    2123: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
    2124: {
    2126:   PetscInt       i,j,k,*bdegrees,worksize;
    2127:   PetscReal      xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
    2128:   PetscScalar    *tau,*work;

    2134:   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);
    2135:   if (PetscDefined(USE_DEBUG)) {
    2136:     for (i=0; i<nsource; i++) {
    2137:       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]);
    2138:     }
    2139:     for (i=0; i<ntarget; i++) {
    2140:       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]);
    2141:     }
    2142:   }
    2143:   xmin = PetscMin(sourcex[0],targetx[0]);
    2144:   xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
    2145:   center = (xmin + xmax)/2;
    2146:   hscale = (xmax - xmin)/2;
    2147:   worksize = nsource;
    2148:   PetscMalloc4(degree+1,&bdegrees,nsource+1,&sourcey,nsource*(degree+1),&Bsource,worksize,&work);
    2149:   PetscMalloc4(nsource,&tau,nsource*(degree+1),&Bsinv,ntarget+1,&targety,ntarget*(degree+1),&Btarget);
    2150:   for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
    2151:   for (i=0; i<=degree; i++) bdegrees[i] = i+1;
    2152:   PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
    2153:   PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
    2154:   for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
    2155:   PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
    2156:   for (i=0; i<ntarget; i++) {
    2157:     PetscReal rowsum = 0;
    2158:     for (j=0; j<nsource; j++) {
    2159:       PetscReal sum = 0;
    2160:       for (k=0; k<degree+1; k++) {
    2161:         sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
    2162:       }
    2163:       R[i*nsource+j] = sum;
    2164:       rowsum += sum;
    2165:     }
    2166:     for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
    2167:   }
    2168:   PetscFree4(bdegrees,sourcey,Bsource,work);
    2169:   PetscFree4(tau,Bsinv,targety,Btarget);
    2170:   return(0);
    2171: }

    2173: /*@C
    2174:    PetscGaussLobattoLegendreIntegrate - Compute the L2 integral of a function on the GLL points

    2176:    Not Collective

    2178:    Input Parameter:
    2179: +  n - the number of GLL nodes
    2180: .  nodes - the GLL nodes
    2181: .  weights - the GLL weights
    2182: -  f - the function values at the nodes

    2184:    Output Parameter:
    2185: .  in - the value of the integral

    2187:    Level: beginner

    2189: .seealso: PetscDTGaussLobattoLegendreQuadrature()

    2191: @*/
    2192: PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt n,PetscReal *nodes,PetscReal *weights,const PetscReal *f,PetscReal *in)
    2193: {
    2194:   PetscInt          i;

    2197:   *in = 0.;
    2198:   for (i=0; i<n; i++) {
    2199:     *in += f[i]*f[i]*weights[i];
    2200:   }
    2201:   return(0);
    2202: }

    2204: /*@C
    2205:    PetscGaussLobattoLegendreElementLaplacianCreate - computes the Laplacian for a single 1d GLL element

    2207:    Not Collective

    2209:    Input Parameter:
    2210: +  n - the number of GLL nodes
    2211: .  nodes - the GLL nodes
    2212: -  weights - the GLL weights

    2214:    Output Parameter:
    2215: .  A - the stiffness element

    2217:    Level: beginner

    2219:    Notes:
    2220:     Destroy this with PetscGaussLobattoLegendreElementLaplacianDestroy()

    2222:    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)

    2224: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianDestroy()

    2226: @*/
    2227: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2228: {
    2229:   PetscReal        **A;
    2230:   PetscErrorCode  ierr;
    2231:   const PetscReal  *gllnodes = nodes;
    2232:   const PetscInt   p = n-1;
    2233:   PetscReal        z0,z1,z2 = -1,x,Lpj,Lpr;
    2234:   PetscInt         i,j,nn,r;

    2237:   PetscMalloc1(n,&A);
    2238:   PetscMalloc1(n*n,&A[0]);
    2239:   for (i=1; i<n; i++) A[i] = A[i-1]+n;

    2241:   for (j=1; j<p; j++) {
    2242:     x  = gllnodes[j];
    2243:     z0 = 1.;
    2244:     z1 = x;
    2245:     for (nn=1; nn<p; nn++) {
    2246:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
    2247:       z0 = z1;
    2248:       z1 = z2;
    2249:     }
    2250:     Lpj=z2;
    2251:     for (r=1; r<p; r++) {
    2252:       if (r == j) {
    2253:         A[j][j]=2./(3.*(1.-gllnodes[j]*gllnodes[j])*Lpj*Lpj);
    2254:       } else {
    2255:         x  = gllnodes[r];
    2256:         z0 = 1.;
    2257:         z1 = x;
    2258:         for (nn=1; nn<p; nn++) {
    2259:           z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
    2260:           z0 = z1;
    2261:           z1 = z2;
    2262:         }
    2263:         Lpr     = z2;
    2264:         A[r][j] = 4./(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*Lpr*(gllnodes[j]-gllnodes[r])*(gllnodes[j]-gllnodes[r]));
    2265:       }
    2266:     }
    2267:   }
    2268:   for (j=1; j<p+1; j++) {
    2269:     x  = gllnodes[j];
    2270:     z0 = 1.;
    2271:     z1 = x;
    2272:     for (nn=1; nn<p; nn++) {
    2273:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
    2274:       z0 = z1;
    2275:       z1 = z2;
    2276:     }
    2277:     Lpj     = z2;
    2278:     A[j][0] = 4.*PetscPowRealInt(-1.,p)/(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*(1.+gllnodes[j])*(1.+gllnodes[j]));
    2279:     A[0][j] = A[j][0];
    2280:   }
    2281:   for (j=0; j<p; j++) {
    2282:     x  = gllnodes[j];
    2283:     z0 = 1.;
    2284:     z1 = x;
    2285:     for (nn=1; nn<p; nn++) {
    2286:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
    2287:       z0 = z1;
    2288:       z1 = z2;
    2289:     }
    2290:     Lpj=z2;

    2292:     A[p][j] = 4./(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*(1.-gllnodes[j])*(1.-gllnodes[j]));
    2293:     A[j][p] = A[p][j];
    2294:   }
    2295:   A[0][0]=0.5+(((PetscReal)p)*(((PetscReal)p)+1.)-2.)/6.;
    2296:   A[p][p]=A[0][0];
    2297:   *AA = A;
    2298:   return(0);
    2299: }

    2301: /*@C
    2302:    PetscGaussLobattoLegendreElementLaplacianDestroy - frees the Laplacian for a single 1d GLL element

    2304:    Not Collective

    2306:    Input Parameter:
    2307: +  n - the number of GLL nodes
    2308: .  nodes - the GLL nodes
    2309: .  weights - the GLL weightss
    2310: -  A - the stiffness element

    2312:    Level: beginner

    2314: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate()

    2316: @*/
    2317: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2318: {

    2322:   PetscFree((*AA)[0]);
    2323:   PetscFree(*AA);
    2324:   *AA  = NULL;
    2325:   return(0);
    2326: }

    2328: /*@C
    2329:    PetscGaussLobattoLegendreElementGradientCreate - computes the gradient for a single 1d GLL element

    2331:    Not Collective

    2333:    Input Parameter:
    2334: +  n - the number of GLL nodes
    2335: .  nodes - the GLL nodes
    2336: .  weights - the GLL weights

    2338:    Output Parameter:
    2339: .  AA - the stiffness element
    2340: -  AAT - the transpose of AA (pass in NULL if you do not need this array)

    2342:    Level: beginner

    2344:    Notes:
    2345:     Destroy this with PetscGaussLobattoLegendreElementGradientDestroy()

    2347:    You can access entries in these arrays with AA[i][j] but in memory it is stored in contiguous memory, row oriented

    2349: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianDestroy()

    2351: @*/
    2352: PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA,PetscReal ***AAT)
    2353: {
    2354:   PetscReal        **A, **AT = NULL;
    2355:   PetscErrorCode  ierr;
    2356:   const PetscReal  *gllnodes = nodes;
    2357:   const PetscInt   p = n-1;
    2358:   PetscReal        Li, Lj,d0;
    2359:   PetscInt         i,j;

    2362:   PetscMalloc1(n,&A);
    2363:   PetscMalloc1(n*n,&A[0]);
    2364:   for (i=1; i<n; i++) A[i] = A[i-1]+n;

    2366:   if (AAT) {
    2367:     PetscMalloc1(n,&AT);
    2368:     PetscMalloc1(n*n,&AT[0]);
    2369:     for (i=1; i<n; i++) AT[i] = AT[i-1]+n;
    2370:   }

    2372:   if (n==1) {A[0][0] = 0.;}
    2373:   d0 = (PetscReal)p*((PetscReal)p+1.)/4.;
    2374:   for  (i=0; i<n; i++) {
    2375:     for  (j=0; j<n; j++) {
    2376:       A[i][j] = 0.;
    2377:       PetscDTComputeJacobi(0., 0., p, gllnodes[i], &Li);
    2378:       PetscDTComputeJacobi(0., 0., p, gllnodes[j], &Lj);
    2379:       if (i!=j)             A[i][j] = Li/(Lj*(gllnodes[i]-gllnodes[j]));
    2380:       if ((j==i) && (i==0)) A[i][j] = -d0;
    2381:       if (j==i && i==p)     A[i][j] = d0;
    2382:       if (AT) AT[j][i] = A[i][j];
    2383:     }
    2384:   }
    2385:   if (AAT) *AAT = AT;
    2386:   *AA  = A;
    2387:   return(0);
    2388: }

    2390: /*@C
    2391:    PetscGaussLobattoLegendreElementGradientDestroy - frees the gradient for a single 1d GLL element obtained with PetscGaussLobattoLegendreElementGradientCreate()

    2393:    Not Collective

    2395:    Input Parameter:
    2396: +  n - the number of GLL nodes
    2397: .  nodes - the GLL nodes
    2398: .  weights - the GLL weights
    2399: .  AA - the stiffness element
    2400: -  AAT - the transpose of the element

    2402:    Level: beginner

    2404: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate(), PetscGaussLobattoLegendreElementAdvectionCreate()

    2406: @*/
    2407: PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA,PetscReal ***AAT)
    2408: {

    2412:   PetscFree((*AA)[0]);
    2413:   PetscFree(*AA);
    2414:   *AA  = NULL;
    2415:   if (*AAT) {
    2416:     PetscFree((*AAT)[0]);
    2417:     PetscFree(*AAT);
    2418:     *AAT  = NULL;
    2419:   }
    2420:   return(0);
    2421: }

    2423: /*@C
    2424:    PetscGaussLobattoLegendreElementAdvectionCreate - computes the advection operator for a single 1d GLL element

    2426:    Not Collective

    2428:    Input Parameter:
    2429: +  n - the number of GLL nodes
    2430: .  nodes - the GLL nodes
    2431: -  weights - the GLL weightss

    2433:    Output Parameter:
    2434: .  AA - the stiffness element

    2436:    Level: beginner

    2438:    Notes:
    2439:     Destroy this with PetscGaussLobattoLegendreElementAdvectionDestroy()

    2441:    This is the same as the Gradient operator multiplied by the diagonal mass matrix

    2443:    You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented

    2445: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate(), PetscGaussLobattoLegendreElementAdvectionDestroy()

    2447: @*/
    2448: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2449: {
    2450:   PetscReal       **D;
    2451:   PetscErrorCode  ierr;
    2452:   const PetscReal  *gllweights = weights;
    2453:   const PetscInt   glln = n;
    2454:   PetscInt         i,j;

    2457:   PetscGaussLobattoLegendreElementGradientCreate(n,nodes,weights,&D,NULL);
    2458:   for (i=0; i<glln; i++){
    2459:     for (j=0; j<glln; j++) {
    2460:       D[i][j] = gllweights[i]*D[i][j];
    2461:     }
    2462:   }
    2463:   *AA = D;
    2464:   return(0);
    2465: }

    2467: /*@C
    2468:    PetscGaussLobattoLegendreElementAdvectionDestroy - frees the advection stiffness for a single 1d GLL element

    2470:    Not Collective

    2472:    Input Parameter:
    2473: +  n - the number of GLL nodes
    2474: .  nodes - the GLL nodes
    2475: .  weights - the GLL weights
    2476: -  A - advection

    2478:    Level: beginner

    2480: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementAdvectionCreate()

    2482: @*/
    2483: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2484: {

    2488:   PetscFree((*AA)[0]);
    2489:   PetscFree(*AA);
    2490:   *AA  = NULL;
    2491:   return(0);
    2492: }

    2494: PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2495: {
    2496:   PetscReal        **A;
    2497:   PetscErrorCode  ierr;
    2498:   const PetscReal  *gllweights = weights;
    2499:   const PetscInt   glln = n;
    2500:   PetscInt         i,j;

    2503:   PetscMalloc1(glln,&A);
    2504:   PetscMalloc1(glln*glln,&A[0]);
    2505:   for (i=1; i<glln; i++) A[i] = A[i-1]+glln;
    2506:   if (glln==1) {A[0][0] = 0.;}
    2507:   for  (i=0; i<glln; i++) {
    2508:     for  (j=0; j<glln; j++) {
    2509:       A[i][j] = 0.;
    2510:       if (j==i)     A[i][j] = gllweights[i];
    2511:     }
    2512:   }
    2513:   *AA  = A;
    2514:   return(0);
    2515: }

    2517: PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2518: {

    2522:   PetscFree((*AA)[0]);
    2523:   PetscFree(*AA);
    2524:   *AA  = NULL;
    2525:   return(0);
    2526: }

    2528: /*@
    2529:   PetscDTIndexToBary - convert an index into a barycentric coordinate.

    2531:   Input Parameters:
    2532: + 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)
    2533: . sum - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
    2534: - index - the index to convert: should be >= 0 and < Binomial(len - 1 + sum, sum)

    2536:   Output Parameter:
    2537: . coord - will be filled with the barycentric coordinate

    2539:   Level: beginner

    2541:   Note: the indices map to barycentric coordinates in lexicographic order, where the first index is the
    2542:   least significant and the last index is the most significant.

    2544: .seealso: PetscDTBaryToIndex()
    2545: @*/
    2546: PetscErrorCode PetscDTIndexToBary(PetscInt len, PetscInt sum, PetscInt index, PetscInt coord[])
    2547: {
    2548:   PetscInt c, d, s, total, subtotal, nexttotal;

    2551:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
    2552:   if (index < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
    2553:   if (!len) {
    2554:     if (!sum && !index) return(0);
    2555:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
    2556:   }
    2557:   for (c = 1, total = 1; c <= len; c++) {
    2558:     /* total is the number of ways to have a tuple of length c with sum */
    2559:     if (index < total) break;
    2560:     total = (total * (sum + c)) / c;
    2561:   }
    2562:   if (c > len) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index out of range");
    2563:   for (d = c; d < len; d++) coord[d] = 0;
    2564:   for (s = 0, subtotal = 1, nexttotal = 1; c > 0;) {
    2565:     /* subtotal is the number of ways to have a tuple of length c with sum s */
    2566:     /* nexttotal is the number of ways to have a tuple of length c-1 with sum s */
    2567:     if ((index + subtotal) >= total) {
    2568:       coord[--c] = sum - s;
    2569:       index -= (total - subtotal);
    2570:       sum = s;
    2571:       total = nexttotal;
    2572:       subtotal = 1;
    2573:       nexttotal = 1;
    2574:       s = 0;
    2575:     } else {
    2576:       subtotal = (subtotal * (c + s)) / (s + 1);
    2577:       nexttotal = (nexttotal * (c - 1 + s)) / (s + 1);
    2578:       s++;
    2579:     }
    2580:   }
    2581:   return(0);
    2582: }

    2584: /*@
    2585:   PetscDTBaryToIndex - convert a barycentric coordinate to an index

    2587:   Input Parameters:
    2588: + 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)
    2589: . sum - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
    2590: - coord - a barycentric coordinate with the given length and sum

    2592:   Output Parameter:
    2593: . index - the unique index for the coordinate, >= 0 and < Binomial(len - 1 + sum, sum)

    2595:   Level: beginner

    2597:   Note: the indices map to barycentric coordinates in lexicographic order, where the first index is the
    2598:   least significant and the last index is the most significant.

    2600: .seealso: PetscDTIndexToBary
    2601: @*/
    2602: PetscErrorCode PetscDTBaryToIndex(PetscInt len, PetscInt sum, const PetscInt coord[], PetscInt *index)
    2603: {
    2604:   PetscInt c;
    2605:   PetscInt i;
    2606:   PetscInt total;

    2609:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
    2610:   if (!len) {
    2611:     if (!sum) {
    2612:       *index = 0;
    2613:       return(0);
    2614:     }
    2615:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
    2616:   }
    2617:   for (c = 1, total = 1; c < len; c++) total = (total * (sum + c)) / c;
    2618:   i = total - 1;
    2619:   c = len - 1;
    2620:   sum -= coord[c];
    2621:   while (sum > 0) {
    2622:     PetscInt subtotal;
    2623:     PetscInt s;

    2625:     for (s = 1, subtotal = 1; s < sum; s++) subtotal = (subtotal * (c + s)) / s;
    2626:     i   -= subtotal;
    2627:     sum -= coord[--c];
    2628:   }
    2629:   *index = i;
    2630:   return(0);
    2631: }