Actual source code: dt.c
petsc-3.13.6 2020-09-29
Report Typos and Errors

      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_", 0};

     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: static PetscErrorCode PetscDTJacobiEval_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscInt k, const PetscReal *points, PetscInt ndegree, const PetscInt *degrees, PetscReal *p)
    632: {
    633:   PetscReal ak, bk;
    634:   PetscReal abk1;
    635:   PetscInt i,l,maxdegree;

    638:   maxdegree = degrees[ndegree-1] - k;
    639:   ak = a + k;
    640:   bk = b + k;
    641:   abk1 = a + b + k + 1.;
    642:   if (maxdegree < 0) {
    643:     for (i = 0; i < npoints; i++) for (l = 0; l < ndegree; l++) p[i*ndegree+l] = 0.;
    644:     return(0);
    645:   }
    646:   for (i=0; i<npoints; i++) {
    647:     PetscReal pm1,pm2,x;
    648:     PetscReal cnm1, cnm1x, cnm2;
    649:     PetscInt  j,m;

    651:     x    = points[i];
    652:     pm2  = 1.;
    653:     PetscDTJacobiRecurrence_Internal(1,ak,bk,cnm1,cnm1x,cnm2);
    654:     pm1 = (cnm1 + cnm1x*x);
    655:     l    = 0;
    656:     while (l < ndegree && degrees[l] - k < 0) {
    657:       p[l++] = 0.;
    658:     }
    659:     while (l < ndegree && degrees[l] - k == 0) {
    660:       p[l] = pm2;
    661:       for (m = 0; m < k; m++) p[l] *= (abk1 + m) * 0.5;
    662:       l++;
    663:     }
    664:     while (l < ndegree && degrees[l] - k == 1) {
    665:       p[l] = pm1;
    666:       for (m = 0; m < k; m++) p[l] *= (abk1 + 1 + m) * 0.5;
    667:       l++;
    668:     }
    669:     for (j=2; j<=maxdegree; j++) {
    670:       PetscReal pp;

    672:       PetscDTJacobiRecurrence_Internal(j,ak,bk,cnm1,cnm1x,cnm2);
    673:       pp   = (cnm1 + cnm1x*x)*pm1 - cnm2*pm2;
    674:       pm2  = pm1;
    675:       pm1  = pp;
    676:       while (l < ndegree && degrees[l] - k == j) {
    677:         p[l] = pp;
    678:         for (m = 0; m < k; m++) p[l] *= (abk1 + j + m) * 0.5;
    679:         l++;
    680:       }
    681:     }
    682:     p += ndegree;
    683:   }
    684:   return(0);
    685: }

    687: /*@
    688:    PetscDTJacobiEval - evaluate Jacobi polynomials for the weight function $(1.+x)^{\alpha} (1.-x)^{\beta}$
    689:                        at points

    691:    Not Collective

    693:    Input Arguments:
    694: +  npoints - number of spatial points to evaluate at
    695: .  alpha - the left exponent > -1
    696: .  beta - the right exponent > -1
    697: .  points - array of locations to evaluate at
    698: .  ndegree - number of basis degrees to evaluate
    699: -  degrees - sorted array of degrees to evaluate

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

    706:    Level: intermediate

    708: .seealso: PetscDTGaussQuadrature()
    709: @*/
    710: PetscErrorCode PetscDTJacobiEval(PetscInt npoints,PetscReal alpha, PetscReal beta, const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
    711: {

    715:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
    716:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");
    717:   if (!npoints || !ndegree) return(0);
    718:   if (B)  {PetscDTJacobiEval_Internal(npoints, alpha, beta, 0, points, ndegree, degrees, B);}
    719:   if (D)  {PetscDTJacobiEval_Internal(npoints, alpha, beta, 1, points, ndegree, degrees, D);}
    720:   if (D2) {PetscDTJacobiEval_Internal(npoints, alpha, beta, 2, points, ndegree, degrees, D2);}
    721:   return(0);
    722: }

    724: /*@
    725:    PetscDTLegendreEval - evaluate Legendre polynomials at points

    727:    Not Collective

    729:    Input Arguments:
    730: +  npoints - number of spatial points to evaluate at
    731: .  points - array of locations to evaluate at
    732: .  ndegree - number of basis degrees to evaluate
    733: -  degrees - sorted array of degrees to evaluate

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

    740:    Level: intermediate

    742: .seealso: PetscDTGaussQuadrature()
    743: @*/
    744: PetscErrorCode PetscDTLegendreEval(PetscInt npoints,const PetscReal *points,PetscInt ndegree,const PetscInt *degrees,PetscReal *B,PetscReal *D,PetscReal *D2)
    745: {

    749:   PetscDTJacobiEval(npoints, 0., 0., points, ndegree, degrees, B, D, D2);
    750:   return(0);
    751: }

    753: /* solve the symmetric tridiagonal eigenvalue system, writing the eigenvalues into eigs and the eigenvectors into V
    754:  * with lds n; diag and subdiag are overwritten */
    755: static PetscErrorCode PetscDTSymmetricTridiagonalEigensolve(PetscInt n, PetscReal diag[], PetscReal subdiag[],
    756:                                                             PetscReal eigs[], PetscScalar V[])
    757: {
    758:   char jobz = 'V'; /* eigenvalues and eigenvectors */
    759:   char range = 'A'; /* all eigenvalues will be found */
    760:   PetscReal VL = 0.; /* ignored because range is 'A' */
    761:   PetscReal VU = 0.; /* ignored because range is 'A' */
    762:   PetscBLASInt IL = 0; /* ignored because range is 'A' */
    763:   PetscBLASInt IU = 0; /* ignored because range is 'A' */
    764:   PetscReal abstol = 0.; /* unused */
    765:   PetscBLASInt bn, bm, ldz; /* bm will equal bn on exit */
    766:   PetscBLASInt *isuppz;
    767:   PetscBLASInt lwork, liwork;
    768:   PetscReal workquery;
    769:   PetscBLASInt  iworkquery;
    770:   PetscBLASInt *iwork;
    771:   PetscBLASInt info;
    772:   PetscReal *work = NULL;

    776: #if !defined(PETSCDTGAUSSIANQUADRATURE_EIG)
    777:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
    778: #endif
    779:   PetscBLASIntCast(n, &bn);
    780:   PetscBLASIntCast(n, &ldz);
    781: #if !defined(PETSC_MISSING_LAPACK_STEGR)
    782:   PetscMalloc1(2 * n, &isuppz);
    783:   lwork = -1;
    784:   liwork = -1;
    785:   PetscStackCallBLAS("LAPACKstegr",LAPACKstegr_(&jobz,&range,&bn,diag,subdiag,&VL,&VU,&IL,&IU,&abstol,&bm,eigs,V,&ldz,isuppz,&workquery,&lwork,&iworkquery,&liwork,&info));
    786:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEGR error");
    787:   lwork = (PetscBLASInt) workquery;
    788:   liwork = (PetscBLASInt) iworkquery;
    789:   PetscMalloc2(lwork, &work, liwork, &iwork);
    790:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
    791:   PetscStackCallBLAS("LAPACKstegr",LAPACKstegr_(&jobz,&range,&bn,diag,subdiag,&VL,&VU,&IL,&IU,&abstol,&bm,eigs,V,&ldz,isuppz,work,&lwork,iwork,&liwork,&info));
    792:   PetscFPTrapPop();
    793:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEGR error");
    794:   PetscFree2(work, iwork);
    795:   PetscFree(isuppz);
    796: #elif !defined(PETSC_MISSING_LAPACK_STEQR)
    797:   jobz = 'I'; /* Compute eigenvalues and eigenvectors of the
    798:                  tridiagonal matrix.  Z is initialized to the identity
    799:                  matrix. */
    800:   PetscMalloc1(PetscMax(1,2*n-2),&work);
    801:   PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("I",&bn,diag,subdiag,V,&ldz,work,&info));
    802:   PetscFPTrapPop();
    803:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"xSTEQR error");
    804:   PetscFree(work);
    805:   PetscArraycpy(eigs,diag,n);
    806: #endif
    807:   return(0);
    808: }

    810: /* Formula for the weights at the endpoints (-1 and 1) of Gauss-Lobatto-Jacobi
    811:  * quadrature rules on the interval [-1, 1] */
    812: static PetscErrorCode PetscDTGaussLobattoJacobiEndweights_Internal(PetscInt n, PetscReal alpha, PetscReal beta, PetscReal *leftw, PetscReal *rightw)
    813: {
    814:   PetscReal twoab1;
    815:   PetscInt  m = n - 2;
    816:   PetscReal a = alpha + 1.;
    817:   PetscReal b = beta + 1.;
    818:   PetscReal gra, grb;

    821:   twoab1 = PetscPowReal(2., a + b - 1.);
    822: #if defined(PETSC_HAVE_LGAMMA)
    823:   grb = PetscExpReal(2. * PetscLGamma(b+1.) + PetscLGamma(m+1.) + PetscLGamma(m+a+1.) -
    824:                      (PetscLGamma(m+b+1) + PetscLGamma(m+a+b+1.)));
    825:   gra = PetscExpReal(2. * PetscLGamma(a+1.) + PetscLGamma(m+1.) + PetscLGamma(m+b+1.) -
    826:                      (PetscLGamma(m+a+1) + PetscLGamma(m+a+b+1.)));
    827: #else
    828:   {
    829:     PetscInt alphai = (PetscInt) alpha;
    830:     PetscInt betai = (PetscInt) beta;

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

    836:       PetscDTBinomial(m+b, b, &binom1);
    837:       PetscDTBinomial(m+a+b, b, &binom2);
    838:       grb = 1./ (binom1 * binom2);
    839:       PetscDTBinomial(m+a, a, &binom1);
    840:       PetscDTBinomial(m+a+b, a, &binom2);
    841:       gra = 1./ (binom1 * binom2);
    842:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"lgamma() - math routine is unavailable.");
    843:   }
    844: #endif
    845:   *leftw = twoab1 * grb / b;
    846:   *rightw = twoab1 * gra / a;
    847:   return(0);
    848: }

    850: /* Evaluates the nth jacobi polynomial with weight parameters a,b at a point x.
    851:    Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B */
    852: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobi(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscReal *P)
    853: {
    854:   PetscReal pn1, pn2;
    855:   PetscReal cnm1, cnm1x, cnm2;
    856:   PetscInt  k;

    859:   if (!n) {*P = 1.0; return(0);}
    860:   PetscDTJacobiRecurrence_Internal(1,a,b,cnm1,cnm1x,cnm2);
    861:   pn2 = 1.;
    862:   pn1 = cnm1 + cnm1x*x;
    863:   if (n == 1) {*P = pn1; return(0);}
    864:   *P  = 0.0;
    865:   for (k = 2; k < n+1; ++k) {
    866:     PetscDTJacobiRecurrence_Internal(k,a,b,cnm1,cnm1x,cnm2);

    868:     *P  = (cnm1 + cnm1x*x)*pn1 - cnm2*pn2;
    869:     pn2 = pn1;
    870:     pn1 = *P;
    871:   }
    872:   return(0);
    873: }

    875: /* Evaluates the first derivative of P_{n}^{a,b} at a point x. */
    876: PETSC_STATIC_INLINE PetscErrorCode PetscDTComputeJacobiDerivative(PetscReal a, PetscReal b, PetscInt n, PetscReal x, PetscInt k, PetscReal *P)
    877: {
    878:   PetscReal      nP;
    879:   PetscInt       i;

    883:   *P = 0.0;
    884:   if (k > n) return(0);
    885:   PetscDTComputeJacobi(a+k, b+k, n-k, x, &nP);
    886:   for (i = 0; i < k; i++) nP *= (a + b + n + 1. + i) * 0.5;
    887:   *P = nP;
    888:   return(0);
    889: }

    891: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
    892: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapSquareToTriangle_Internal(PetscReal x, PetscReal y, PetscReal *xi, PetscReal *eta)
    893: {
    895:   *xi  = 0.5 * (1.0 + x) * (1.0 - y) - 1.0;
    896:   *eta = y;
    897:   return(0);
    898: }

    900: static PetscErrorCode PetscDTGaussJacobiQuadrature_Newton_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal x[], PetscReal w[])
    901: {
    902:   PetscInt       maxIter = 100;
    903:   PetscReal      eps     = PetscExpReal(0.75 * PetscLogReal(PETSC_MACHINE_EPSILON));
    904:   PetscReal      a1, a6, gf;
    905:   PetscInt       k;

    910:   a1 = PetscPowReal(2.0, a+b+1);
    911: #if defined(PETSC_HAVE_LGAMMA)
    912:   {
    913:     PetscReal a2, a3, a4, a5;
    914:     a2 = PetscLGamma(a + npoints + 1);
    915:     a3 = PetscLGamma(b + npoints + 1);
    916:     a4 = PetscLGamma(a + b + npoints + 1);
    917:     a5 = PetscLGamma(npoints + 1);
    918:     gf = PetscExpReal(a2 + a3 - (a4 + a5));
    919:   }
    920: #else
    921:   {
    922:     PetscInt ia, ib;

    924:     ia = (PetscInt) a;
    925:     ib = (PetscInt) b;
    926:     gf = 1.;
    927:     if (ia == a && ia >= 0) { /* compute ratio of rising factorals wrt a */
    928:       for (k = 0; k < ia; k++) gf *= (npoints + 1. + k) / (npoints + b + 1. + k);
    929:     } else if (b == b && ib >= 0) { /* compute ratio of rising factorials wrt b */
    930:       for (k = 0; k < ib; k++) gf *= (npoints + 1. + k) / (npoints + a + 1. + k);
    931:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"lgamma() - math routine is unavailable.");
    932:   }
    933: #endif

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

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

    947:       for (i = 0; i < k; ++i) s = s + 1.0 / (r - x[i]);
    948:       PetscDTComputeJacobi(a, b, npoints, r, &f);
    949:       PetscDTComputeJacobiDerivative(a, b, npoints, r, 1, &fp);
    950:       delta = f / (fp - f * s);
    951:       r     = r - delta;
    952:       if (PetscAbsReal(delta) < eps) break;
    953:     }
    954:     x[k] = r;
    955:     PetscDTComputeJacobiDerivative(a, b, npoints, x[k], 1, &dP);
    956:     w[k] = a6 / (1.0 - PetscSqr(x[k])) / PetscSqr(dP);
    957:   }
    958:   return(0);
    959: }

    961: /* Compute the diagonals of the Jacobi matrix used in Golub & Welsch algorithms for Gauss-Jacobi
    962:  * quadrature weight calculations on [-1,1] for exponents (1. + x)^a (1.-x)^b */
    963: static PetscErrorCode PetscDTJacobiMatrix_Internal(PetscInt nPoints, PetscReal a, PetscReal b, PetscReal *d, PetscReal *s)
    964: {
    965:   PetscInt       i;

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

    971:     PetscDTJacobiRecurrence_Internal(i+1,a,b,A,B,C);
    972:     d[i] = -A / B;
    973:     if (i) s[i-1] *= C / B;
    974:     if (i < nPoints - 1) s[i] = 1. / B;
    975:   }
    976:   return(0);
    977: }

    979: static PetscErrorCode PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(PetscInt npoints, PetscReal a, PetscReal b, PetscReal *x, PetscReal *w)
    980: {
    981:   PetscReal mu0;
    982:   PetscReal ga, gb, gab;
    983:   PetscInt i;

    987:   PetscCitationsRegister(GolubWelschCitation, &GolubWelschCite);

    989: #if defined(PETSC_HAVE_TGAMMA)
    990:   ga  = PetscTGamma(a + 1);
    991:   gb  = PetscTGamma(b + 1);
    992:   gab = PetscTGamma(a + b + 2);
    993: #else
    994:   {
    995:     PetscInt ia, ib;

    997:     ia = (PetscInt) a;
    998:     ib = (PetscInt) b;
    999:     if (ia == a && ib == b && ia + 1 > 0 && ib + 1 > 0 && ia + ib + 2 > 0) { /* All gamma(x) terms are (x-1)! terms */
    1000:       PetscDTFactorial(ia, &ga);
    1001:       PetscDTFactorial(ib, &gb);
    1002:       PetscDTFactorial(ia + ib + 1, &gb);
    1003:     } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"tgamma() - math routine is unavailable.");
    1004:   }
    1005: #endif
    1006:   mu0 = PetscPowReal(2.,a + b + 1.) * ga * gb / gab;

    1008: #if defined(PETSCDTGAUSSIANQUADRATURE_EIG)
    1009:   {
    1010:     PetscReal *diag, *subdiag;
    1011:     PetscScalar *V;

    1013:     PetscMalloc2(npoints, &diag, npoints, &subdiag);
    1014:     PetscMalloc1(npoints*npoints, &V);
    1015:     PetscDTJacobiMatrix_Internal(npoints, a, b, diag, subdiag);
    1016:     for (i = 0; i < npoints - 1; i++) subdiag[i] = PetscSqrtReal(subdiag[i]);
    1017:     PetscDTSymmetricTridiagonalEigensolve(npoints, diag, subdiag, x, V);
    1018:     for (i = 0; i < npoints; i++) w[i] = PetscSqr(PetscRealPart(V[i * npoints])) * mu0;
    1019:     PetscFree(V);
    1020:     PetscFree2(diag, subdiag);
    1021:   }
    1022: #else
    1023:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP_SYS, "A LAPACK symmetric tridiagonal eigensolver could not be found");
    1024: #endif
    1025:   { /* As of March 2, 2020, The Sun Performance Library breaks the LAPACK contract for xstegr and xsteqr: the
    1026:        eigenvalues are not guaranteed to be in ascending order.  So we heave a passive aggressive sigh and check that
    1027:        the eigenvalues are sorted */
    1028:     PetscBool sorted;

    1030:     PetscSortedReal(npoints, x, &sorted);
    1031:     if (!sorted) {
    1032:       PetscInt *order, i;
    1033:       PetscReal *tmp;

    1035:       PetscMalloc2(npoints, &order, npoints, &tmp);
    1036:       for (i = 0; i < npoints; i++) order[i] = i;
    1037:       PetscSortRealWithPermutation(npoints, x, order);
    1038:       PetscArraycpy(tmp, x, npoints);
    1039:       for (i = 0; i < npoints; i++) x[i] = tmp[order[i]];
    1040:       PetscArraycpy(tmp, w, npoints);
    1041:       for (i = 0; i < npoints; i++) w[i] = tmp[order[i]];
    1042:       PetscFree2(order, tmp);
    1043:     }
    1044:   }
    1045:   return(0);
    1046: }

    1048: static PetscErrorCode PetscDTGaussJacobiQuadrature_Internal(PetscInt npoints,PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
    1049: {

    1053:   if (npoints < 1) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of points must be positive");
    1054:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
    1055:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
    1056:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");

    1058:   if (newton) {
    1059:     PetscDTGaussJacobiQuadrature_Newton_Internal(npoints, alpha, beta, x, w);
    1060:   } else {
    1061:     PetscDTGaussJacobiQuadrature_GolubWelsch_Internal(npoints, alpha, beta, x, w);
    1062:   }
    1063:   if (alpha == beta) { /* symmetrize */
    1064:     PetscInt i;
    1065:     for (i = 0; i < (npoints + 1) / 2; i++) {
    1066:       PetscInt  j  = npoints - 1 - i;
    1067:       PetscReal xi = x[i];
    1068:       PetscReal xj = x[j];
    1069:       PetscReal wi = w[i];
    1070:       PetscReal wj = w[j];

    1072:       x[i] = (xi - xj) / 2.;
    1073:       x[j] = (xj - xi) / 2.;
    1074:       w[i] = w[j] = (wi + wj) / 2.;
    1075:     }
    1076:   }
    1077:   return(0);
    1078: }

    1080: /*@
    1081:   PetscDTGaussJacobiQuadrature - quadrature for the interval [a, b] with the weight function
    1082:   $(x-a)^\alpha (x-b)^\beta$.

    1084:   Not collective

    1086:   Input Parameters:
    1087: + npoints - the number of points in the quadrature rule
    1088: . a - the left endpoint of the interval
    1089: . b - the right endpoint of the interval
    1090: . alpha - the left exponent
    1091: - beta - the right exponent

    1093:   Output Parameters:
    1094: + x - array of length npoints, the locations of the quadrature points
    1095: - w - array of length npoints, the weights of the quadrature points

    1097:   Level: intermediate

    1099:   Note: this quadrature rule is exact for polynomials up to degree 2*npoints - 1.
    1100: @*/
    1101: PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt npoints,PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
    1102: {
    1103:   PetscInt       i;

    1107:   PetscDTGaussJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal);
    1108:   if (a != -1. || b != 1.) { /* shift */
    1109:     for (i = 0; i < npoints; i++) {
    1110:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
    1111:       w[i] *= (b - a) / 2.;
    1112:     }
    1113:   }
    1114:   return(0);
    1115: }

    1117: static PetscErrorCode PetscDTGaussLobattoJacobiQuadrature_Internal(PetscInt npoints,PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[], PetscBool newton)
    1118: {
    1119:   PetscInt       i;

    1123:   if (npoints < 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of points must be positive");
    1124:   /* If asking for a 1D Lobatto point, just return the non-Lobatto 1D point */
    1125:   if (alpha <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"alpha must be > -1.");
    1126:   if (beta <= -1.) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"beta must be > -1.");

    1128:   x[0] = -1.;
    1129:   x[npoints-1] = 1.;
    1130:   if (npoints > 2) {
    1131:     PetscDTGaussJacobiQuadrature_Internal(npoints-2, alpha+1., beta+1., &x[1], &w[1], newton);
    1132:   }
    1133:   for (i = 1; i < npoints - 1; i++) {
    1134:     w[i] /= (1. - x[i]*x[i]);
    1135:   }
    1136:   PetscDTGaussLobattoJacobiEndweights_Internal(npoints, alpha, beta, &w[0], &w[npoints-1]);
    1137:   return(0);
    1138: }

    1140: /*@
    1141:   PetscDTGaussLobattoJacobiQuadrature - quadrature for the interval [a, b] with the weight function
    1142:   $(x-a)^\alpha (x-b)^\beta$, with endpoints a and b included as quadrature points.

    1144:   Not collective

    1146:   Input Parameters:
    1147: + npoints - the number of points in the quadrature rule
    1148: . a - the left endpoint of the interval
    1149: . b - the right endpoint of the interval
    1150: . alpha - the left exponent
    1151: - beta - the right exponent

    1153:   Output Parameters:
    1154: + x - array of length npoints, the locations of the quadrature points
    1155: - w - array of length npoints, the weights of the quadrature points

    1157:   Level: intermediate

    1159:   Note: this quadrature rule is exact for polynomials up to degree 2*npoints - 3.
    1160: @*/
    1161: PetscErrorCode PetscDTGaussLobattoJacobiQuadrature(PetscInt npoints,PetscReal a, PetscReal b, PetscReal alpha, PetscReal beta, PetscReal x[], PetscReal w[])
    1162: {
    1163:   PetscInt       i;

    1167:   PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, alpha, beta, x, w, PetscDTGaussQuadratureNewton_Internal);
    1168:   if (a != -1. || b != 1.) { /* shift */
    1169:     for (i = 0; i < npoints; i++) {
    1170:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
    1171:       w[i] *= (b - a) / 2.;
    1172:     }
    1173:   }
    1174:   return(0);
    1175: }

    1177: /*@
    1178:    PetscDTGaussQuadrature - create Gauss-Legendre quadrature

    1180:    Not Collective

    1182:    Input Arguments:
    1183: +  npoints - number of points
    1184: .  a - left end of interval (often-1)
    1185: -  b - right end of interval (often +1)

    1187:    Output Arguments:
    1188: +  x - quadrature points
    1189: -  w - quadrature weights

    1191:    Level: intermediate

    1193:    References:
    1194: .   1. - Golub and Welsch, Calculation of Quadrature Rules, Math. Comp. 23(106), 1969.

    1196: .seealso: PetscDTLegendreEval()
    1197: @*/
    1198: PetscErrorCode PetscDTGaussQuadrature(PetscInt npoints,PetscReal a,PetscReal b,PetscReal *x,PetscReal *w)
    1199: {
    1200:   PetscInt       i;

    1204:   PetscDTGaussJacobiQuadrature_Internal(npoints, 0., 0., x, w, PetscDTGaussQuadratureNewton_Internal);
    1205:   if (a != -1. || b != 1.) { /* shift */
    1206:     for (i = 0; i < npoints; i++) {
    1207:       x[i] = (x[i] + 1.) * ((b - a) / 2.) + a;
    1208:       w[i] *= (b - a) / 2.;
    1209:     }
    1210:   }
    1211:   return(0);
    1212: }

    1214: /*@C
    1215:    PetscDTGaussLobattoLegendreQuadrature - creates a set of the locations and weights of the Gauss-Lobatto-Legendre
    1216:                       nodes of a given size on the domain [-1,1]

    1218:    Not Collective

    1220:    Input Parameter:
    1221: +  n - number of grid nodes
    1222: -  type - PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA or PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON

    1224:    Output Arguments:
    1225: +  x - quadrature points
    1226: -  w - quadrature weights

    1228:    Notes:
    1229:     For n > 30  the Newton approach computes duplicate (incorrect) values for some nodes because the initial guess is apparently not
    1230:           close enough to the desired solution

    1232:    These are useful for implementing spectral methods based on Gauss-Lobatto-Legendre (GLL) nodes

    1234:    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

    1236:    Level: intermediate

    1238: .seealso: PetscDTGaussQuadrature()

    1240: @*/
    1241: PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt npoints,PetscGaussLobattoLegendreCreateType type,PetscReal *x,PetscReal *w)
    1242: {
    1243:   PetscBool      newton;

    1247:   if (npoints < 2) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Must provide at least 2 grid points per element");
    1248:   newton = (PetscBool) (type == PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON);
    1249:   PetscDTGaussLobattoJacobiQuadrature_Internal(npoints, 0., 0., x, w, newton);
    1250:   return(0);
    1251: }

    1253: /*@
    1254:   PetscDTGaussTensorQuadrature - creates a tensor-product Gauss quadrature

    1256:   Not Collective

    1258:   Input Arguments:
    1259: + dim     - The spatial dimension
    1260: . Nc      - The number of components
    1261: . npoints - number of points in one dimension
    1262: . a       - left end of interval (often-1)
    1263: - b       - right end of interval (often +1)

    1265:   Output Argument:
    1266: . q - A PetscQuadrature object

    1268:   Level: intermediate

    1270: .seealso: PetscDTGaussQuadrature(), PetscDTLegendreEval()
    1271: @*/
    1272: PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
    1273: {
    1274:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints, i, j, k, c;
    1275:   PetscReal     *x, *w, *xw, *ww;

    1279:   PetscMalloc1(totpoints*dim,&x);
    1280:   PetscMalloc1(totpoints*Nc,&w);
    1281:   /* Set up the Golub-Welsch system */
    1282:   switch (dim) {
    1283:   case 0:
    1284:     PetscFree(x);
    1285:     PetscFree(w);
    1286:     PetscMalloc1(1, &x);
    1287:     PetscMalloc1(Nc, &w);
    1288:     x[0] = 0.0;
    1289:     for (c = 0; c < Nc; ++c) w[c] = 1.0;
    1290:     break;
    1291:   case 1:
    1292:     PetscMalloc1(npoints,&ww);
    1293:     PetscDTGaussQuadrature(npoints, a, b, x, ww);
    1294:     for (i = 0; i < npoints; ++i) for (c = 0; c < Nc; ++c) w[i*Nc+c] = ww[i];
    1295:     PetscFree(ww);
    1296:     break;
    1297:   case 2:
    1298:     PetscMalloc2(npoints,&xw,npoints,&ww);
    1299:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
    1300:     for (i = 0; i < npoints; ++i) {
    1301:       for (j = 0; j < npoints; ++j) {
    1302:         x[(i*npoints+j)*dim+0] = xw[i];
    1303:         x[(i*npoints+j)*dim+1] = xw[j];
    1304:         for (c = 0; c < Nc; ++c) w[(i*npoints+j)*Nc+c] = ww[i] * ww[j];
    1305:       }
    1306:     }
    1307:     PetscFree2(xw,ww);
    1308:     break;
    1309:   case 3:
    1310:     PetscMalloc2(npoints,&xw,npoints,&ww);
    1311:     PetscDTGaussQuadrature(npoints, a, b, xw, ww);
    1312:     for (i = 0; i < npoints; ++i) {
    1313:       for (j = 0; j < npoints; ++j) {
    1314:         for (k = 0; k < npoints; ++k) {
    1315:           x[((i*npoints+j)*npoints+k)*dim+0] = xw[i];
    1316:           x[((i*npoints+j)*npoints+k)*dim+1] = xw[j];
    1317:           x[((i*npoints+j)*npoints+k)*dim+2] = xw[k];
    1318:           for (c = 0; c < Nc; ++c) w[((i*npoints+j)*npoints+k)*Nc+c] = ww[i] * ww[j] * ww[k];
    1319:         }
    1320:       }
    1321:     }
    1322:     PetscFree2(xw,ww);
    1323:     break;
    1324:   default:
    1325:     SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
    1326:   }
    1327:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    1328:   PetscQuadratureSetOrder(*q, 2*npoints-1);
    1329:   PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);
    1330:   PetscObjectChangeTypeName((PetscObject)*q,"GaussTensor");
    1331:   return(0);
    1332: }

    1334: /* Maps from [-1,1]^2 to the (-1,1) reference triangle */
    1335: PETSC_STATIC_INLINE PetscErrorCode PetscDTMapCubeToTetrahedron_Internal(PetscReal x, PetscReal y, PetscReal z, PetscReal *xi, PetscReal *eta, PetscReal *zeta)
    1336: {
    1338:   *xi   = 0.25 * (1.0 + x) * (1.0 - y) * (1.0 - z) - 1.0;
    1339:   *eta  = 0.5  * (1.0 + y) * (1.0 - z) - 1.0;
    1340:   *zeta = z;
    1341:   return(0);
    1342: }

    1345: /*@
    1346:   PetscDTStroudConicalQuadrature - create Stroud conical quadrature for a simplex

    1348:   Not Collective

    1350:   Input Arguments:
    1351: + dim     - The simplex dimension
    1352: . Nc      - The number of components
    1353: . npoints - The number of points in one dimension
    1354: . a       - left end of interval (often-1)
    1355: - b       - right end of interval (often +1)

    1357:   Output Argument:
    1358: . q - A PetscQuadrature object

    1360:   Level: intermediate

    1362:   References:
    1363: .  1. - Karniadakis and Sherwin.  FIAT

    1365:   Note: For dim == 1, this is Gauss-Legendre quadrature

    1367: .seealso: PetscDTGaussTensorQuadrature(), PetscDTGaussQuadrature()
    1368: @*/
    1369: PetscErrorCode PetscDTStroudConicalQuadrature(PetscInt dim, PetscInt Nc, PetscInt npoints, PetscReal a, PetscReal b, PetscQuadrature *q)
    1370: {
    1371:   PetscInt       totpoints = dim > 1 ? dim > 2 ? npoints*PetscSqr(npoints) : PetscSqr(npoints) : npoints;
    1372:   PetscReal     *px, *wx, *py, *wy, *pz, *wz, *x, *w;

    1376:   if ((a != -1.0) || (b != 1.0)) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must use default internal right now");
    1377:   PetscMalloc1(totpoints*dim, &x);
    1378:   PetscMalloc1(totpoints*Nc, &w);
    1379:   switch (dim) {
    1380:   case 0:
    1381:     PetscFree(x);
    1382:     PetscFree(w);
    1383:     PetscMalloc1(1, &x);
    1384:     PetscMalloc1(Nc, &w);
    1385:     x[0] = 0.0;
    1386:     for (c = 0; c < Nc; ++c) w[c] = 1.0;
    1387:     break;
    1388:   case 1:
    1389:     PetscMalloc1(npoints,&wx);
    1390:     PetscDTGaussJacobiQuadrature(npoints, -1., 1., 0.0, 0.0, x, wx);
    1391:     for (i = 0; i < npoints; ++i) for (c = 0; c < Nc; ++c) w[i*Nc+c] = wx[i];
    1392:     PetscFree(wx);
    1393:     break;
    1394:   case 2:
    1395:     PetscMalloc4(npoints,&px,npoints,&wx,npoints,&py,npoints,&wy);
    1396:     PetscDTGaussJacobiQuadrature(npoints, -1., 1., 0.0, 0.0, px, wx);
    1397:     PetscDTGaussJacobiQuadrature(npoints, -1., 1., 1.0, 0.0, py, wy);
    1398:     for (i = 0; i < npoints; ++i) {
    1399:       for (j = 0; j < npoints; ++j) {
    1400:         PetscDTMapSquareToTriangle_Internal(px[i], py[j], &x[(i*npoints+j)*2+0], &x[(i*npoints+j)*2+1]);
    1401:         for (c = 0; c < Nc; ++c) w[(i*npoints+j)*Nc+c] = 0.5 * wx[i] * wy[j];
    1402:       }
    1403:     }
    1404:     PetscFree4(px,wx,py,wy);
    1405:     break;
    1406:   case 3:
    1407:     PetscMalloc6(npoints,&px,npoints,&wx,npoints,&py,npoints,&wy,npoints,&pz,npoints,&wz);
    1408:     PetscDTGaussJacobiQuadrature(npoints, -1., 1., 0.0, 0.0, px, wx);
    1409:     PetscDTGaussJacobiQuadrature(npoints, -1., 1., 1.0, 0.0, py, wy);
    1410:     PetscDTGaussJacobiQuadrature(npoints, -1., 1., 2.0, 0.0, pz, wz);
    1411:     for (i = 0; i < npoints; ++i) {
    1412:       for (j = 0; j < npoints; ++j) {
    1413:         for (k = 0; k < npoints; ++k) {
    1414:           PetscDTMapCubeToTetrahedron_Internal(px[i], py[j], pz[k], &x[((i*npoints+j)*npoints+k)*3+0], &x[((i*npoints+j)*npoints+k)*3+1], &x[((i*npoints+j)*npoints+k)*3+2]);
    1415:           for (c = 0; c < Nc; ++c) w[((i*npoints+j)*npoints+k)*Nc+c] = 0.125 * wx[i] * wy[j] * wz[k];
    1416:         }
    1417:       }
    1418:     }
    1419:     PetscFree6(px,wx,py,wy,pz,wz);
    1420:     break;
    1421:   default:
    1422:     SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot construct quadrature rule for dimension %d", dim);
    1423:   }
    1424:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    1425:   PetscQuadratureSetOrder(*q, 2*npoints-1);
    1426:   PetscQuadratureSetData(*q, dim, Nc, totpoints, x, w);
    1427:   PetscObjectChangeTypeName((PetscObject)*q,"GaussJacobi");
    1428:   return(0);
    1429: }

    1431: /*@
    1432:   PetscDTTanhSinhTensorQuadrature - create tanh-sinh quadrature for a tensor product cell

    1434:   Not Collective

    1436:   Input Arguments:
    1437: + dim   - The cell dimension
    1438: . level - The number of points in one dimension, 2^l
    1439: . a     - left end of interval (often-1)
    1440: - b     - right end of interval (often +1)

    1442:   Output Argument:
    1443: . q - A PetscQuadrature object

    1445:   Level: intermediate

    1447: .seealso: PetscDTGaussTensorQuadrature()
    1448: @*/
    1449: PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt dim, PetscInt level, PetscReal a, PetscReal b, PetscQuadrature *q)
    1450: {
    1451:   const PetscInt  p     = 16;                        /* Digits of precision in the evaluation */
    1452:   const PetscReal alpha = (b-a)/2.;                  /* Half-width of the integration interval */
    1453:   const PetscReal beta  = (b+a)/2.;                  /* Center of the integration interval */
    1454:   const PetscReal h     = PetscPowReal(2.0, -level); /* Step size, length between x_k */
    1455:   PetscReal       xk;                                /* Quadrature point x_k on reference domain [-1, 1] */
    1456:   PetscReal       wk    = 0.5*PETSC_PI;              /* Quadrature weight at x_k */
    1457:   PetscReal      *x, *w;
    1458:   PetscInt        K, k, npoints;
    1459:   PetscErrorCode  ierr;

    1462:   if (dim > 1) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_SUP, "Dimension %d not yet implemented", dim);
    1463:   if (!level) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a number of significant digits");
    1464:   /* Find K such that the weights are < 32 digits of precision */
    1465:   for (K = 1; PetscAbsReal(PetscLog10Real(wk)) < 2*p; ++K) {
    1466:     wk = 0.5*h*PETSC_PI*PetscCoshReal(K*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(K*h)));
    1467:   }
    1468:   PetscQuadratureCreate(PETSC_COMM_SELF, q);
    1469:   PetscQuadratureSetOrder(*q, 2*K+1);
    1470:   npoints = 2*K-1;
    1471:   PetscMalloc1(npoints*dim, &x);
    1472:   PetscMalloc1(npoints, &w);
    1473:   /* Center term */
    1474:   x[0] = beta;
    1475:   w[0] = 0.5*alpha*PETSC_PI;
    1476:   for (k = 1; k < K; ++k) {
    1477:     wk = 0.5*alpha*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    1478:     xk = PetscTanhReal(0.5*PETSC_PI*PetscSinhReal(k*h));
    1479:     x[2*k-1] = -alpha*xk+beta;
    1480:     w[2*k-1] = wk;
    1481:     x[2*k+0] =  alpha*xk+beta;
    1482:     w[2*k+0] = wk;
    1483:   }
    1484:   PetscQuadratureSetData(*q, dim, 1, npoints, x, w);
    1485:   return(0);
    1486: }

    1488: PetscErrorCode PetscDTTanhSinhIntegrate(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    1489: {
    1490:   const PetscInt  p     = 16;        /* Digits of precision in the evaluation */
    1491:   const PetscReal alpha = (b-a)/2.;  /* Half-width of the integration interval */
    1492:   const PetscReal beta  = (b+a)/2.;  /* Center of the integration interval */
    1493:   PetscReal       h     = 1.0;       /* Step size, length between x_k */
    1494:   PetscInt        l     = 0;         /* Level of refinement, h = 2^{-l} */
    1495:   PetscReal       osum  = 0.0;       /* Integral on last level */
    1496:   PetscReal       psum  = 0.0;       /* Integral on the level before the last level */
    1497:   PetscReal       sum;               /* Integral on current level */
    1498:   PetscReal       yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
    1499:   PetscReal       lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
    1500:   PetscReal       wk;                /* Quadrature weight at x_k */
    1501:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
    1502:   PetscInt        d;                 /* Digits of precision in the integral */

    1505:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
    1506:   /* Center term */
    1507:   func(beta, &lval);
    1508:   sum = 0.5*alpha*PETSC_PI*lval;
    1509:   /* */
    1510:   do {
    1511:     PetscReal lterm, rterm, maxTerm = 0.0, d1, d2, d3, d4;
    1512:     PetscInt  k = 1;

    1514:     ++l;
    1515:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
    1516:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
    1517:     psum = osum;
    1518:     osum = sum;
    1519:     h   *= 0.5;
    1520:     sum *= 0.5;
    1521:     do {
    1522:       wk = 0.5*h*PETSC_PI*PetscCoshReal(k*h)/PetscSqr(PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    1523:       yk = 1.0/(PetscExpReal(0.5*PETSC_PI*PetscSinhReal(k*h)) * PetscCoshReal(0.5*PETSC_PI*PetscSinhReal(k*h)));
    1524:       lx = -alpha*(1.0 - yk)+beta;
    1525:       rx =  alpha*(1.0 - yk)+beta;
    1526:       func(lx, &lval);
    1527:       func(rx, &rval);
    1528:       lterm   = alpha*wk*lval;
    1529:       maxTerm = PetscMax(PetscAbsReal(lterm), maxTerm);
    1530:       sum    += lterm;
    1531:       rterm   = alpha*wk*rval;
    1532:       maxTerm = PetscMax(PetscAbsReal(rterm), maxTerm);
    1533:       sum    += rterm;
    1534:       ++k;
    1535:       /* Only need to evaluate every other point on refined levels */
    1536:       if (l != 1) ++k;
    1537:     } while (PetscAbsReal(PetscLog10Real(wk)) < p); /* Only need to evaluate sum until weights are < 32 digits of precision */

    1539:     d1 = PetscLog10Real(PetscAbsReal(sum - osum));
    1540:     d2 = PetscLog10Real(PetscAbsReal(sum - psum));
    1541:     d3 = PetscLog10Real(maxTerm) - p;
    1542:     if (PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)) == 0.0) d4 = 0.0;
    1543:     else d4 = PetscLog10Real(PetscMax(PetscAbsReal(lterm), PetscAbsReal(rterm)));
    1544:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
    1545:   } while (d < digits && l < 12);
    1546:   *sol = sum;

    1548:   return(0);
    1549: }

    1551: #if defined(PETSC_HAVE_MPFR)
    1552: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    1553: {
    1554:   const PetscInt  safetyFactor = 2;  /* Calculate abcissa until 2*p digits */
    1555:   PetscInt        l            = 0;  /* Level of refinement, h = 2^{-l} */
    1556:   mpfr_t          alpha;             /* Half-width of the integration interval */
    1557:   mpfr_t          beta;              /* Center of the integration interval */
    1558:   mpfr_t          h;                 /* Step size, length between x_k */
    1559:   mpfr_t          osum;              /* Integral on last level */
    1560:   mpfr_t          psum;              /* Integral on the level before the last level */
    1561:   mpfr_t          sum;               /* Integral on current level */
    1562:   mpfr_t          yk;                /* Quadrature point 1 - x_k on reference domain [-1, 1] */
    1563:   mpfr_t          lx, rx;            /* Quadrature points to the left and right of 0 on the real domain [a, b] */
    1564:   mpfr_t          wk;                /* Quadrature weight at x_k */
    1565:   PetscReal       lval, rval;        /* Terms in the quadature sum to the left and right of 0 */
    1566:   PetscInt        d;                 /* Digits of precision in the integral */
    1567:   mpfr_t          pi2, kh, msinh, mcosh, maxTerm, curTerm, tmp;

    1570:   if (digits <= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must give a positive number of significant digits");
    1571:   /* Create high precision storage */
    1572:   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);
    1573:   /* Initialization */
    1574:   mpfr_set_d(alpha, 0.5*(b-a), MPFR_RNDN);
    1575:   mpfr_set_d(beta,  0.5*(b+a), MPFR_RNDN);
    1576:   mpfr_set_d(osum,  0.0,       MPFR_RNDN);
    1577:   mpfr_set_d(psum,  0.0,       MPFR_RNDN);
    1578:   mpfr_set_d(h,     1.0,       MPFR_RNDN);
    1579:   mpfr_const_pi(pi2, MPFR_RNDN);
    1580:   mpfr_mul_d(pi2, pi2, 0.5, MPFR_RNDN);
    1581:   /* Center term */
    1582:   func(0.5*(b+a), &lval);
    1583:   mpfr_set(sum, pi2, MPFR_RNDN);
    1584:   mpfr_mul(sum, sum, alpha, MPFR_RNDN);
    1585:   mpfr_mul_d(sum, sum, lval, MPFR_RNDN);
    1586:   /* */
    1587:   do {
    1588:     PetscReal d1, d2, d3, d4;
    1589:     PetscInt  k = 1;

    1591:     ++l;
    1592:     mpfr_set_d(maxTerm, 0.0, MPFR_RNDN);
    1593:     /* PetscPrintf(PETSC_COMM_SELF, "LEVEL %D sum: %15.15f\n", l, sum); */
    1594:     /* At each level of refinement, h --> h/2 and sum --> sum/2 */
    1595:     mpfr_set(psum, osum, MPFR_RNDN);
    1596:     mpfr_set(osum,  sum, MPFR_RNDN);
    1597:     mpfr_mul_d(h,   h,   0.5, MPFR_RNDN);
    1598:     mpfr_mul_d(sum, sum, 0.5, MPFR_RNDN);
    1599:     do {
    1600:       mpfr_set_si(kh, k, MPFR_RNDN);
    1601:       mpfr_mul(kh, kh, h, MPFR_RNDN);
    1602:       /* Weight */
    1603:       mpfr_set(wk, h, MPFR_RNDN);
    1604:       mpfr_sinh_cosh(msinh, mcosh, kh, MPFR_RNDN);
    1605:       mpfr_mul(msinh, msinh, pi2, MPFR_RNDN);
    1606:       mpfr_mul(mcosh, mcosh, pi2, MPFR_RNDN);
    1607:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
    1608:       mpfr_sqr(tmp, tmp, MPFR_RNDN);
    1609:       mpfr_mul(wk, wk, mcosh, MPFR_RNDN);
    1610:       mpfr_div(wk, wk, tmp, MPFR_RNDN);
    1611:       /* Abscissa */
    1612:       mpfr_set_d(yk, 1.0, MPFR_RNDZ);
    1613:       mpfr_cosh(tmp, msinh, MPFR_RNDN);
    1614:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
    1615:       mpfr_exp(tmp, msinh, MPFR_RNDN);
    1616:       mpfr_div(yk, yk, tmp, MPFR_RNDZ);
    1617:       /* Quadrature points */
    1618:       mpfr_sub_d(lx, yk, 1.0, MPFR_RNDZ);
    1619:       mpfr_mul(lx, lx, alpha, MPFR_RNDU);
    1620:       mpfr_add(lx, lx, beta, MPFR_RNDU);
    1621:       mpfr_d_sub(rx, 1.0, yk, MPFR_RNDZ);
    1622:       mpfr_mul(rx, rx, alpha, MPFR_RNDD);
    1623:       mpfr_add(rx, rx, beta, MPFR_RNDD);
    1624:       /* Evaluation */
    1625:       func(mpfr_get_d(lx, MPFR_RNDU), &lval);
    1626:       func(mpfr_get_d(rx, MPFR_RNDD), &rval);
    1627:       /* Update */
    1628:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
    1629:       mpfr_mul_d(tmp, tmp, lval, MPFR_RNDN);
    1630:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
    1631:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    1632:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
    1633:       mpfr_set(curTerm, tmp, MPFR_RNDN);
    1634:       mpfr_mul(tmp, wk, alpha, MPFR_RNDN);
    1635:       mpfr_mul_d(tmp, tmp, rval, MPFR_RNDN);
    1636:       mpfr_add(sum, sum, tmp, MPFR_RNDN);
    1637:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    1638:       mpfr_max(maxTerm, maxTerm, tmp, MPFR_RNDN);
    1639:       mpfr_max(curTerm, curTerm, tmp, MPFR_RNDN);
    1640:       ++k;
    1641:       /* Only need to evaluate every other point on refined levels */
    1642:       if (l != 1) ++k;
    1643:       mpfr_log10(tmp, wk, MPFR_RNDN);
    1644:       mpfr_abs(tmp, tmp, MPFR_RNDN);
    1645:     } while (mpfr_get_d(tmp, MPFR_RNDN) < safetyFactor*digits); /* Only need to evaluate sum until weights are < 32 digits of precision */
    1646:     mpfr_sub(tmp, sum, osum, MPFR_RNDN);
    1647:     mpfr_abs(tmp, tmp, MPFR_RNDN);
    1648:     mpfr_log10(tmp, tmp, MPFR_RNDN);
    1649:     d1 = mpfr_get_d(tmp, MPFR_RNDN);
    1650:     mpfr_sub(tmp, sum, psum, MPFR_RNDN);
    1651:     mpfr_abs(tmp, tmp, MPFR_RNDN);
    1652:     mpfr_log10(tmp, tmp, MPFR_RNDN);
    1653:     d2 = mpfr_get_d(tmp, MPFR_RNDN);
    1654:     mpfr_log10(tmp, maxTerm, MPFR_RNDN);
    1655:     d3 = mpfr_get_d(tmp, MPFR_RNDN) - digits;
    1656:     mpfr_log10(tmp, curTerm, MPFR_RNDN);
    1657:     d4 = mpfr_get_d(tmp, MPFR_RNDN);
    1658:     d  = PetscAbsInt(PetscMin(0, PetscMax(PetscMax(PetscMax(PetscSqr(d1)/d2, 2*d1), d3), d4)));
    1659:   } while (d < digits && l < 8);
    1660:   *sol = mpfr_get_d(sum, MPFR_RNDN);
    1661:   /* Cleanup */
    1662:   mpfr_clears(alpha, beta, h, sum, osum, psum, yk, wk, lx, rx, tmp, maxTerm, curTerm, pi2, kh, msinh, mcosh, NULL);
    1663:   return(0);
    1664: }
    1665: #else

    1667: PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*func)(PetscReal, PetscReal *), PetscReal a, PetscReal b, PetscInt digits, PetscReal *sol)
    1668: {
    1669:   SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "This method will not work without MPFR. Reconfigure using --download-mpfr --download-gmp");
    1670: }
    1671: #endif

    1673: /* Overwrites A. Can only handle full-rank problems with m>=n
    1674:  * A in column-major format
    1675:  * Ainv in row-major format
    1676:  * tau has length m
    1677:  * worksize must be >= max(1,n)
    1678:  */
    1679: static PetscErrorCode PetscDTPseudoInverseQR(PetscInt m,PetscInt mstride,PetscInt n,PetscReal *A_in,PetscReal *Ainv_out,PetscScalar *tau,PetscInt worksize,PetscScalar *work)
    1680: {
    1682:   PetscBLASInt   M,N,K,lda,ldb,ldwork,info;
    1683:   PetscScalar    *A,*Ainv,*R,*Q,Alpha;

    1686: #if defined(PETSC_USE_COMPLEX)
    1687:   {
    1688:     PetscInt i,j;
    1689:     PetscMalloc2(m*n,&A,m*n,&Ainv);
    1690:     for (j=0; j<n; j++) {
    1691:       for (i=0; i<m; i++) A[i+m*j] = A_in[i+mstride*j];
    1692:     }
    1693:     mstride = m;
    1694:   }
    1695: #else
    1696:   A = A_in;
    1697:   Ainv = Ainv_out;
    1698: #endif

    1700:   PetscBLASIntCast(m,&M);
    1701:   PetscBLASIntCast(n,&N);
    1702:   PetscBLASIntCast(mstride,&lda);
    1703:   PetscBLASIntCast(worksize,&ldwork);
    1704:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
    1705:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&M,&N,A,&lda,tau,work,&ldwork,&info));
    1706:   PetscFPTrapPop();
    1707:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xGEQRF error");
    1708:   R = A; /* Upper triangular part of A now contains R, the rest contains the elementary reflectors */

    1710:   /* Extract an explicit representation of Q */
    1711:   Q = Ainv;
    1712:   PetscArraycpy(Q,A,mstride*n);
    1713:   K = N;                        /* full rank */
    1714:   PetscStackCallBLAS("LAPACKorgqr",LAPACKorgqr_(&M,&N,&K,Q,&lda,tau,work,&ldwork,&info));
    1715:   if (info) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"xORGQR/xUNGQR error");

    1717:   /* Compute A^{-T} = (R^{-1} Q^T)^T = Q R^{-T} */
    1718:   Alpha = 1.0;
    1719:   ldb = lda;
    1720:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("Right","Upper","ConjugateTranspose","NotUnitTriangular",&M,&N,&Alpha,R,&lda,Q,&ldb));
    1721:   /* Ainv is Q, overwritten with inverse */

    1723: #if defined(PETSC_USE_COMPLEX)
    1724:   {
    1725:     PetscInt i;
    1726:     for (i=0; i<m*n; i++) Ainv_out[i] = PetscRealPart(Ainv[i]);
    1727:     PetscFree2(A,Ainv);
    1728:   }
    1729: #endif
    1730:   return(0);
    1731: }

    1733: /* Computes integral of L_p' over intervals {(x0,x1),(x1,x2),...} */
    1734: static PetscErrorCode PetscDTLegendreIntegrate(PetscInt ninterval,const PetscReal *x,PetscInt ndegree,const PetscInt *degrees,PetscBool Transpose,PetscReal *B)
    1735: {
    1737:   PetscReal      *Bv;
    1738:   PetscInt       i,j;

    1741:   PetscMalloc1((ninterval+1)*ndegree,&Bv);
    1742:   /* Point evaluation of L_p on all the source vertices */
    1743:   PetscDTLegendreEval(ninterval+1,x,ndegree,degrees,Bv,NULL,NULL);
    1744:   /* Integral over each interval: \int_a^b L_p' = L_p(b)-L_p(a) */
    1745:   for (i=0; i<ninterval; i++) {
    1746:     for (j=0; j<ndegree; j++) {
    1747:       if (Transpose) B[i+ninterval*j] = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
    1748:       else           B[i*ndegree+j]   = Bv[(i+1)*ndegree+j] - Bv[i*ndegree+j];
    1749:     }
    1750:   }
    1751:   PetscFree(Bv);
    1752:   return(0);
    1753: }

    1755: /*@
    1756:    PetscDTReconstructPoly - create matrix representing polynomial reconstruction using cell intervals and evaluation at target intervals

    1758:    Not Collective

    1760:    Input Arguments:
    1761: +  degree - degree of reconstruction polynomial
    1762: .  nsource - number of source intervals
    1763: .  sourcex - sorted coordinates of source cell boundaries (length nsource+1)
    1764: .  ntarget - number of target intervals
    1765: -  targetx - sorted coordinates of target cell boundaries (length ntarget+1)

    1767:    Output Arguments:
    1768: .  R - reconstruction matrix, utarget = sum_s R[t*nsource+s] * usource[s]

    1770:    Level: advanced

    1772: .seealso: PetscDTLegendreEval()
    1773: @*/
    1774: PetscErrorCode PetscDTReconstructPoly(PetscInt degree,PetscInt nsource,const PetscReal *sourcex,PetscInt ntarget,const PetscReal *targetx,PetscReal *R)
    1775: {
    1777:   PetscInt       i,j,k,*bdegrees,worksize;
    1778:   PetscReal      xmin,xmax,center,hscale,*sourcey,*targety,*Bsource,*Bsinv,*Btarget;
    1779:   PetscScalar    *tau,*work;

    1785:   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);
    1786: #if defined(PETSC_USE_DEBUG)
    1787:   for (i=0; i<nsource; i++) {
    1788:     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]);
    1789:   }
    1790:   for (i=0; i<ntarget; i++) {
    1791:     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]);
    1792:   }
    1793: #endif
    1794:   xmin = PetscMin(sourcex[0],targetx[0]);
    1795:   xmax = PetscMax(sourcex[nsource],targetx[ntarget]);
    1796:   center = (xmin + xmax)/2;
    1797:   hscale = (xmax - xmin)/2;
    1798:   worksize = nsource;
    1799:   PetscMalloc4(degree+1,&bdegrees,nsource+1,&sourcey,nsource*(degree+1),&Bsource,worksize,&work);
    1800:   PetscMalloc4(nsource,&tau,nsource*(degree+1),&Bsinv,ntarget+1,&targety,ntarget*(degree+1),&Btarget);
    1801:   for (i=0; i<=nsource; i++) sourcey[i] = (sourcex[i]-center)/hscale;
    1802:   for (i=0; i<=degree; i++) bdegrees[i] = i+1;
    1803:   PetscDTLegendreIntegrate(nsource,sourcey,degree+1,bdegrees,PETSC_TRUE,Bsource);
    1804:   PetscDTPseudoInverseQR(nsource,nsource,degree+1,Bsource,Bsinv,tau,nsource,work);
    1805:   for (i=0; i<=ntarget; i++) targety[i] = (targetx[i]-center)/hscale;
    1806:   PetscDTLegendreIntegrate(ntarget,targety,degree+1,bdegrees,PETSC_FALSE,Btarget);
    1807:   for (i=0; i<ntarget; i++) {
    1808:     PetscReal rowsum = 0;
    1809:     for (j=0; j<nsource; j++) {
    1810:       PetscReal sum = 0;
    1811:       for (k=0; k<degree+1; k++) {
    1812:         sum += Btarget[i*(degree+1)+k] * Bsinv[k*nsource+j];
    1813:       }
    1814:       R[i*nsource+j] = sum;
    1815:       rowsum += sum;
    1816:     }
    1817:     for (j=0; j<nsource; j++) R[i*nsource+j] /= rowsum; /* normalize each row */
    1818:   }
    1819:   PetscFree4(bdegrees,sourcey,Bsource,work);
    1820:   PetscFree4(tau,Bsinv,targety,Btarget);
    1821:   return(0);
    1822: }

    1824: /*@C
    1825:    PetscGaussLobattoLegendreIntegrate - Compute the L2 integral of a function on the GLL points

    1827:    Not Collective

    1829:    Input Parameter:
    1830: +  n - the number of GLL nodes
    1831: .  nodes - the GLL nodes
    1832: .  weights - the GLL weights
    1833: -  f - the function values at the nodes

    1835:    Output Parameter:
    1836: .  in - the value of the integral

    1838:    Level: beginner

    1840: .seealso: PetscDTGaussLobattoLegendreQuadrature()

    1842: @*/
    1843: PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt n,PetscReal *nodes,PetscReal *weights,const PetscReal *f,PetscReal *in)
    1844: {
    1845:   PetscInt          i;

    1848:   *in = 0.;
    1849:   for (i=0; i<n; i++) {
    1850:     *in += f[i]*f[i]*weights[i];
    1851:   }
    1852:   return(0);
    1853: }

    1855: /*@C
    1856:    PetscGaussLobattoLegendreElementLaplacianCreate - computes the Laplacian for a single 1d GLL element

    1858:    Not Collective

    1860:    Input Parameter:
    1861: +  n - the number of GLL nodes
    1862: .  nodes - the GLL nodes
    1863: -  weights - the GLL weights

    1865:    Output Parameter:
    1866: .  A - the stiffness element

    1868:    Level: beginner

    1870:    Notes:
    1871:     Destroy this with PetscGaussLobattoLegendreElementLaplacianDestroy()

    1873:    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)

    1875: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianDestroy()

    1877: @*/
    1878: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    1879: {
    1880:   PetscReal        **A;
    1881:   PetscErrorCode  ierr;
    1882:   const PetscReal  *gllnodes = nodes;
    1883:   const PetscInt   p = n-1;
    1884:   PetscReal        z0,z1,z2 = -1,x,Lpj,Lpr;
    1885:   PetscInt         i,j,nn,r;

    1888:   PetscMalloc1(n,&A);
    1889:   PetscMalloc1(n*n,&A[0]);
    1890:   for (i=1; i<n; i++) A[i] = A[i-1]+n;

    1892:   for (j=1; j<p; j++) {
    1893:     x  = gllnodes[j];
    1894:     z0 = 1.;
    1895:     z1 = x;
    1896:     for (nn=1; nn<p; nn++) {
    1897:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
    1898:       z0 = z1;
    1899:       z1 = z2;
    1900:     }
    1901:     Lpj=z2;
    1902:     for (r=1; r<p; r++) {
    1903:       if (r == j) {
    1904:         A[j][j]=2./(3.*(1.-gllnodes[j]*gllnodes[j])*Lpj*Lpj);
    1905:       } else {
    1906:         x  = gllnodes[r];
    1907:         z0 = 1.;
    1908:         z1 = x;
    1909:         for (nn=1; nn<p; nn++) {
    1910:           z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
    1911:           z0 = z1;
    1912:           z1 = z2;
    1913:         }
    1914:         Lpr     = z2;
    1915:         A[r][j] = 4./(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*Lpr*(gllnodes[j]-gllnodes[r])*(gllnodes[j]-gllnodes[r]));
    1916:       }
    1917:     }
    1918:   }
    1919:   for (j=1; j<p+1; j++) {
    1920:     x  = gllnodes[j];
    1921:     z0 = 1.;
    1922:     z1 = x;
    1923:     for (nn=1; nn<p; nn++) {
    1924:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
    1925:       z0 = z1;
    1926:       z1 = z2;
    1927:     }
    1928:     Lpj     = z2;
    1929:     A[j][0] = 4.*PetscPowRealInt(-1.,p)/(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*(1.+gllnodes[j])*(1.+gllnodes[j]));
    1930:     A[0][j] = A[j][0];
    1931:   }
    1932:   for (j=0; j<p; j++) {
    1933:     x  = gllnodes[j];
    1934:     z0 = 1.;
    1935:     z1 = x;
    1936:     for (nn=1; nn<p; nn++) {
    1937:       z2 = x*z1*(2.*((PetscReal)nn)+1.)/(((PetscReal)nn)+1.)-z0*(((PetscReal)nn)/(((PetscReal)nn)+1.));
    1938:       z0 = z1;
    1939:       z1 = z2;
    1940:     }
    1941:     Lpj=z2;

    1943:     A[p][j] = 4./(((PetscReal)p)*(((PetscReal)p)+1.)*Lpj*(1.-gllnodes[j])*(1.-gllnodes[j]));
    1944:     A[j][p] = A[p][j];
    1945:   }
    1946:   A[0][0]=0.5+(((PetscReal)p)*(((PetscReal)p)+1.)-2.)/6.;
    1947:   A[p][p]=A[0][0];
    1948:   *AA = A;
    1949:   return(0);
    1950: }

    1952: /*@C
    1953:    PetscGaussLobattoLegendreElementLaplacianDestroy - frees the Laplacian for a single 1d GLL element

    1955:    Not Collective

    1957:    Input Parameter:
    1958: +  n - the number of GLL nodes
    1959: .  nodes - the GLL nodes
    1960: .  weights - the GLL weightss
    1961: -  A - the stiffness element

    1963:    Level: beginner

    1965: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate()

    1967: @*/
    1968: PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    1969: {

    1973:   PetscFree((*AA)[0]);
    1974:   PetscFree(*AA);
    1975:   *AA  = NULL;
    1976:   return(0);
    1977: }

    1979: /*@C
    1980:    PetscGaussLobattoLegendreElementGradientCreate - computes the gradient for a single 1d GLL element

    1982:    Not Collective

    1984:    Input Parameter:
    1985: +  n - the number of GLL nodes
    1986: .  nodes - the GLL nodes
    1987: .  weights - the GLL weights

    1989:    Output Parameter:
    1990: .  AA - the stiffness element
    1991: -  AAT - the transpose of AA (pass in NULL if you do not need this array)

    1993:    Level: beginner

    1995:    Notes:
    1996:     Destroy this with PetscGaussLobattoLegendreElementGradientDestroy()

    1998:    You can access entries in these arrays with AA[i][j] but in memory it is stored in contiguous memory, row oriented

    2000: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianDestroy()

    2002: @*/
    2003: PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA,PetscReal ***AAT)
    2004: {
    2005:   PetscReal        **A, **AT = NULL;
    2006:   PetscErrorCode  ierr;
    2007:   const PetscReal  *gllnodes = nodes;
    2008:   const PetscInt   p = n-1;
    2009:   PetscReal        Li, Lj,d0;
    2010:   PetscInt         i,j;

    2013:   PetscMalloc1(n,&A);
    2014:   PetscMalloc1(n*n,&A[0]);
    2015:   for (i=1; i<n; i++) A[i] = A[i-1]+n;

    2017:   if (AAT) {
    2018:     PetscMalloc1(n,&AT);
    2019:     PetscMalloc1(n*n,&AT[0]);
    2020:     for (i=1; i<n; i++) AT[i] = AT[i-1]+n;
    2021:   }

    2023:   if (n==1) {A[0][0] = 0.;}
    2024:   d0 = (PetscReal)p*((PetscReal)p+1.)/4.;
    2025:   for  (i=0; i<n; i++) {
    2026:     for  (j=0; j<n; j++) {
    2027:       A[i][j] = 0.;
    2028:       PetscDTComputeJacobi(0., 0., p, gllnodes[i], &Li);
    2029:       PetscDTComputeJacobi(0., 0., p, gllnodes[j], &Lj);
    2030:       if (i!=j)             A[i][j] = Li/(Lj*(gllnodes[i]-gllnodes[j]));
    2031:       if ((j==i) && (i==0)) A[i][j] = -d0;
    2032:       if (j==i && i==p)     A[i][j] = d0;
    2033:       if (AT) AT[j][i] = A[i][j];
    2034:     }
    2035:   }
    2036:   if (AAT) *AAT = AT;
    2037:   *AA  = A;
    2038:   return(0);
    2039: }

    2041: /*@C
    2042:    PetscGaussLobattoLegendreElementGradientDestroy - frees the gradient for a single 1d GLL element obtained with PetscGaussLobattoLegendreElementGradientCreate()

    2044:    Not Collective

    2046:    Input Parameter:
    2047: +  n - the number of GLL nodes
    2048: .  nodes - the GLL nodes
    2049: .  weights - the GLL weights
    2050: .  AA - the stiffness element
    2051: -  AAT - the transpose of the element

    2053:    Level: beginner

    2055: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate(), PetscGaussLobattoLegendreElementAdvectionCreate()

    2057: @*/
    2058: PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA,PetscReal ***AAT)
    2059: {

    2063:   PetscFree((*AA)[0]);
    2064:   PetscFree(*AA);
    2065:   *AA  = NULL;
    2066:   if (*AAT) {
    2067:     PetscFree((*AAT)[0]);
    2068:     PetscFree(*AAT);
    2069:     *AAT  = NULL;
    2070:   }
    2071:   return(0);
    2072: }

    2074: /*@C
    2075:    PetscGaussLobattoLegendreElementAdvectionCreate - computes the advection operator for a single 1d GLL element

    2077:    Not Collective

    2079:    Input Parameter:
    2080: +  n - the number of GLL nodes
    2081: .  nodes - the GLL nodes
    2082: -  weights - the GLL weightss

    2084:    Output Parameter:
    2085: .  AA - the stiffness element

    2087:    Level: beginner

    2089:    Notes:
    2090:     Destroy this with PetscGaussLobattoLegendreElementAdvectionDestroy()

    2092:    This is the same as the Gradient operator multiplied by the diagonal mass matrix

    2094:    You can access entries in this array with AA[i][j] but in memory it is stored in contiguous memory, row oriented

    2096: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementLaplacianCreate(), PetscGaussLobattoLegendreElementAdvectionDestroy()

    2098: @*/
    2099: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2100: {
    2101:   PetscReal       **D;
    2102:   PetscErrorCode  ierr;
    2103:   const PetscReal  *gllweights = weights;
    2104:   const PetscInt   glln = n;
    2105:   PetscInt         i,j;

    2108:   PetscGaussLobattoLegendreElementGradientCreate(n,nodes,weights,&D,NULL);
    2109:   for (i=0; i<glln; i++){
    2110:     for (j=0; j<glln; j++) {
    2111:       D[i][j] = gllweights[i]*D[i][j];
    2112:     }
    2113:   }
    2114:   *AA = D;
    2115:   return(0);
    2116: }

    2118: /*@C
    2119:    PetscGaussLobattoLegendreElementAdvectionDestroy - frees the advection stiffness for a single 1d GLL element

    2121:    Not Collective

    2123:    Input Parameter:
    2124: +  n - the number of GLL nodes
    2125: .  nodes - the GLL nodes
    2126: .  weights - the GLL weights
    2127: -  A - advection

    2129:    Level: beginner

    2131: .seealso: PetscDTGaussLobattoLegendreQuadrature(), PetscGaussLobattoLegendreElementAdvectionCreate()

    2133: @*/
    2134: PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2135: {

    2139:   PetscFree((*AA)[0]);
    2140:   PetscFree(*AA);
    2141:   *AA  = NULL;
    2142:   return(0);
    2143: }

    2145: PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2146: {
    2147:   PetscReal        **A;
    2148:   PetscErrorCode  ierr;
    2149:   const PetscReal  *gllweights = weights;
    2150:   const PetscInt   glln = n;
    2151:   PetscInt         i,j;

    2154:   PetscMalloc1(glln,&A);
    2155:   PetscMalloc1(glln*glln,&A[0]);
    2156:   for (i=1; i<glln; i++) A[i] = A[i-1]+glln;
    2157:   if (glln==1) {A[0][0] = 0.;}
    2158:   for  (i=0; i<glln; i++) {
    2159:     for  (j=0; j<glln; j++) {
    2160:       A[i][j] = 0.;
    2161:       if (j==i)     A[i][j] = gllweights[i];
    2162:     }
    2163:   }
    2164:   *AA  = A;
    2165:   return(0);
    2166: }

    2168: PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt n,PetscReal *nodes,PetscReal *weights,PetscReal ***AA)
    2169: {

    2173:   PetscFree((*AA)[0]);
    2174:   PetscFree(*AA);
    2175:   *AA  = NULL;
    2176:   return(0);
    2177: }

    2179: /*@
    2180:   PetscDTIndexToBary - convert an index into a barycentric coordinate.

    2182:   Input Parameters:
    2183: + 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)
    2184: . sum - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
    2185: - index - the index to convert: should be >= 0 and < Binomial(len - 1 + sum, sum)

    2187:   Output Parameter:
    2188: . coord - will be filled with the barycentric coordinate

    2190:   Level: beginner

    2192:   Note: the indices map to barycentric coordinates in lexicographic order, where the first index is the
    2193:   least significant and the last index is the most significant.

    2195: .seealso: PetscDTBaryToIndex
    2196: @*/
    2197: PetscErrorCode PetscDTIndexToBary(PetscInt len, PetscInt sum, PetscInt index, PetscInt coord[])
    2198: {
    2199:   PetscInt c, d, s, total, subtotal, nexttotal;

    2202:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
    2203:   if (index < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index must be non-negative");
    2204:   if (!len) {
    2205:     if (!sum && !index) return(0);
    2206:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
    2207:   }
    2208:   for (c = 1, total = 1; c <= len; c++) {
    2209:     /* total is the number of ways to have a tuple of length c with sum */
    2210:     if (index < total) break;
    2211:     total = (total * (sum + c)) / c;
    2212:   }
    2213:   if (c > len) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "index out of range");
    2214:   for (d = c; d < len; d++) coord[d] = 0;
    2215:   for (s = 0, subtotal = 1, nexttotal = 1; c > 0;) {
    2216:     /* subtotal is the number of ways to have a tuple of length c with sum s */
    2217:     /* nexttotal is the number of ways to have a tuple of length c-1 with sum s */
    2218:     if ((index + subtotal) >= total) {
    2219:       coord[--c] = sum - s;
    2220:       index -= (total - subtotal);
    2221:       sum = s;
    2222:       total = nexttotal;
    2223:       subtotal = 1;
    2224:       nexttotal = 1;
    2225:       s = 0;
    2226:     } else {
    2227:       subtotal = (subtotal * (c + s)) / (s + 1);
    2228:       nexttotal = (nexttotal * (c - 1 + s)) / (s + 1);
    2229:       s++;
    2230:     }
    2231:   }
    2232:   return(0);
    2233: }

    2235: /*@
    2236:   PetscDTBaryToIndex - convert a barycentric coordinate to an index

    2238:   Input Parameters:
    2239: + 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)
    2240: . sum - the value that the sum of the barycentric coordinates (which will be non-negative integers) should sum to
    2241: - coord - a barycentric coordinate with the given length and sum

    2243:   Output Parameter:
    2244: . index - the unique index for the coordinate, >= 0 and < Binomial(len - 1 + sum, sum)

    2246:   Level: beginner

    2248:   Note: the indices map to barycentric coordinates in lexicographic order, where the first index is the
    2249:   least significant and the last index is the most significant.

    2251: .seealso: PetscDTIndexToBary
    2252: @*/
    2253: PetscErrorCode PetscDTBaryToIndex(PetscInt len, PetscInt sum, const PetscInt coord[], PetscInt *index)
    2254: {
    2255:   PetscInt c;
    2256:   PetscInt i;
    2257:   PetscInt total;

    2260:   if (len < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "length must be non-negative");
    2261:   if (!len) {
    2262:     if (!sum) {
    2263:       *index = 0;
    2264:       return(0);
    2265:     }
    2266:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid index or sum for length 0 barycentric coordinate");
    2267:   }
    2268:   for (c = 1, total = 1; c < len; c++) total = (total * (sum + c)) / c;
    2269:   i = total - 1;
    2270:   c = len - 1;
    2271:   sum -= coord[c];
    2272:   while (sum > 0) {
    2273:     PetscInt subtotal;
    2274:     PetscInt s;

    2276:     for (s = 1, subtotal = 1; s < sum; s++) subtotal = (subtotal * (c + s)) / s;
    2277:     i   -= subtotal;
    2278:     sum -= coord[--c];
    2279:   }
    2280:   *index = i;
    2281:   return(0);
    2282: }