Date: 2022-03-04T17:38:46+00:00

Actual source code: petscdt.h

      1: /*
      2:   Common tools for constructing discretizations
      3: */
      4: #if !defined(PETSCDT_H)
      5: #define PETSCDT_H

      7: #include <petscsys.h>

      9: PETSC_EXTERN PetscClassId PETSCQUADRATURE_CLASSID;

     11: /*S
     12:   PetscQuadrature - Quadrature rule for integration.

     14:   Level: beginner

     16: .seealso:  PetscQuadratureCreate(), PetscQuadratureDestroy()
     17: S*/
     18: typedef struct _p_PetscQuadrature *PetscQuadrature;

     20: /*E
     21:   PetscGaussLobattoLegendreCreateType - algorithm used to compute the Gauss-Lobatto-Legendre nodes and weights

     23:   Level: intermediate

     25: $  PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA - compute the nodes via linear algebra
     26: $  PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON - compute the nodes by solving a nonlinear equation with Newton's method

     28: E*/
     29: typedef enum {PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA,PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON} PetscGaussLobattoLegendreCreateType;

     31: /*E
     32:   PetscDTNodeType - A description of strategies for generating nodes (both
     33:   quadrature nodes and nodes for Lagrange polynomials)

     35:   Level: intermediate

     37: $  PETSCDTNODES_DEFAULT - Nodes chosen by PETSc
     38: $  PETSCDTNODES_GAUSSJACOBI - Nodes at either Gauss-Jacobi or Gauss-Lobatto-Jacobi quadrature points
     39: $  PETSCDTNODES_EQUISPACED - Nodes equispaced either including the endpoints or excluding them
     40: $  PETSCDTNODES_TANHSINH - Nodes at Tanh-Sinh quadrature points

     42:   Note: a PetscDTNodeType can be paired with a PetscBool to indicate whether
     43:   the nodes include endpoints or not, and in the case of PETSCDT_GAUSSJACOBI
     44:   with exponents for the weight function.

     46: E*/
     47: typedef enum {PETSCDTNODES_DEFAULT=-1, PETSCDTNODES_GAUSSJACOBI, PETSCDTNODES_EQUISPACED, PETSCDTNODES_TANHSINH} PetscDTNodeType;

     49: PETSC_EXTERN const char *const PetscDTNodeTypes[];

     51: PETSC_EXTERN PetscErrorCode PetscQuadratureCreate(MPI_Comm, PetscQuadrature *);
     52: PETSC_EXTERN PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature, PetscQuadrature *);
     53: PETSC_EXTERN PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature, PetscInt*);
     54: PETSC_EXTERN PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature, PetscInt);
     55: PETSC_EXTERN PetscErrorCode PetscQuadratureGetNumComponents(PetscQuadrature, PetscInt*);
     56: PETSC_EXTERN PetscErrorCode PetscQuadratureSetNumComponents(PetscQuadrature, PetscInt);
     57: PETSC_EXTERN PetscErrorCode PetscQuadratureGetData(PetscQuadrature, PetscInt*, PetscInt*, PetscInt*, const PetscReal *[], const PetscReal *[]);
     58: PETSC_EXTERN PetscErrorCode PetscQuadratureSetData(PetscQuadrature, PetscInt, PetscInt, PetscInt, const PetscReal [], const PetscReal []);
     59: PETSC_EXTERN PetscErrorCode PetscQuadratureView(PetscQuadrature, PetscViewer);
     60: PETSC_EXTERN PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *);

     62: PETSC_EXTERN PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature, PetscInt, const PetscReal[], const PetscReal[], PetscQuadrature *);

     64: PETSC_EXTERN PetscErrorCode PetscQuadraturePushForward(PetscQuadrature, PetscInt, const PetscReal[], const PetscReal[], const PetscReal[], PetscInt, PetscQuadrature *);

     66: PETSC_EXTERN PetscErrorCode PetscDTLegendreEval(PetscInt,const PetscReal*,PetscInt,const PetscInt*,PetscReal*,PetscReal*,PetscReal*);
     67: PETSC_EXTERN PetscErrorCode PetscDTJacobiNorm(PetscReal,PetscReal,PetscInt,PetscReal *);
     68: PETSC_EXTERN PetscErrorCode PetscDTJacobiEval(PetscInt,PetscReal,PetscReal,const PetscReal*,PetscInt,const PetscInt*,PetscReal*,PetscReal*,PetscReal*);
     69: PETSC_EXTERN PetscErrorCode PetscDTJacobiEvalJet(PetscReal,PetscReal,PetscInt,const PetscReal[],PetscInt,PetscInt,PetscReal[]);
     70: PETSC_EXTERN PetscErrorCode PetscDTPKDEvalJet(PetscInt,PetscInt,const PetscReal[],PetscInt,PetscInt,PetscReal[]);
     71: PETSC_EXTERN PetscErrorCode PetscDTGaussQuadrature(PetscInt,PetscReal,PetscReal,PetscReal*,PetscReal*);
     72: PETSC_EXTERN PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt,PetscReal,PetscReal,PetscReal,PetscReal,PetscReal*,PetscReal*);
     73: PETSC_EXTERN PetscErrorCode PetscDTGaussLobattoJacobiQuadrature(PetscInt,PetscReal,PetscReal,PetscReal,PetscReal,PetscReal*,PetscReal*);
     74: PETSC_EXTERN PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt,PetscGaussLobattoLegendreCreateType,PetscReal*,PetscReal*);
     75: PETSC_EXTERN PetscErrorCode PetscDTReconstructPoly(PetscInt,PetscInt,const PetscReal*,PetscInt,const PetscReal*,PetscReal*);
     76: PETSC_EXTERN PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt,PetscInt,PetscInt,PetscReal,PetscReal,PetscQuadrature*);
     77: PETSC_EXTERN PetscErrorCode PetscDTStroudConicalQuadrature(PetscInt,PetscInt,PetscInt,PetscReal,PetscReal,PetscQuadrature*);

     79: PETSC_EXTERN PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt, PetscInt, PetscReal, PetscReal, PetscQuadrature *);
     80: PETSC_EXTERN PetscErrorCode PetscDTTanhSinhIntegrate(void (*)(PetscReal, PetscReal *), PetscReal, PetscReal, PetscInt, PetscReal *);
     81: PETSC_EXTERN PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*)(PetscReal, PetscReal *), PetscReal, PetscReal, PetscInt, PetscReal *);

     83: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt, PetscReal *, PetscReal *, const PetscReal *, PetscReal *);
     84: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
     85: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
     86: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt, PetscReal *, PetscReal *, PetscReal ***, PetscReal ***);
     87: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt, PetscReal *, PetscReal *, PetscReal ***, PetscReal ***);
     88: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
     89: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
     90: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
     91: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt, PetscReal *, PetscReal *, PetscReal ***);

     93: PETSC_EXTERN PetscErrorCode PetscDTAltVApply(PetscInt, PetscInt, const PetscReal *, const PetscReal *, PetscReal *);
     94: PETSC_EXTERN PetscErrorCode PetscDTAltVWedge(PetscInt, PetscInt, PetscInt, const PetscReal *, const PetscReal *, PetscReal *);
     95: PETSC_EXTERN PetscErrorCode PetscDTAltVWedgeMatrix(PetscInt, PetscInt, PetscInt, const PetscReal *, PetscReal *);
     96: PETSC_EXTERN PetscErrorCode PetscDTAltVPullback(PetscInt, PetscInt, const PetscReal *, PetscInt, const PetscReal *, PetscReal *);
     97: PETSC_EXTERN PetscErrorCode PetscDTAltVPullbackMatrix(PetscInt, PetscInt, const PetscReal *, PetscInt, PetscReal *);
     98: PETSC_EXTERN PetscErrorCode PetscDTAltVInterior(PetscInt, PetscInt, const PetscReal *, const PetscReal *, PetscReal *);
     99: PETSC_EXTERN PetscErrorCode PetscDTAltVInteriorMatrix(PetscInt, PetscInt, const PetscReal *, PetscReal *);
    100: PETSC_EXTERN PetscErrorCode PetscDTAltVInteriorPattern(PetscInt, PetscInt, PetscInt (*)[3]);
    101: PETSC_EXTERN PetscErrorCode PetscDTAltVStar(PetscInt, PetscInt, PetscInt, const PetscReal *, PetscReal *);

    103: PETSC_EXTERN PetscErrorCode PetscDTBaryToIndex(PetscInt,PetscInt,const PetscInt[],PetscInt*);
    104: PETSC_EXTERN PetscErrorCode PetscDTIndexToBary(PetscInt,PetscInt,PetscInt,PetscInt[]);
    105: PETSC_EXTERN PetscErrorCode PetscDTGradedOrderToIndex(PetscInt,const PetscInt[],PetscInt*);
    106: PETSC_EXTERN PetscErrorCode PetscDTIndexToGradedOrder(PetscInt,PetscInt,PetscInt[]);

    108: #if defined(PETSC_USE_64BIT_INDICES)
    109: #define PETSC_FACTORIAL_MAX 20
    110: #define PETSC_BINOMIAL_MAX  61
    111: #else
    112: #define PETSC_FACTORIAL_MAX 12
    113: #define PETSC_BINOMIAL_MAX  29
    114: #endif

    116: /*MC
    117:    PetscDTFactorial - Approximate n! as a real number

    119:    Input Parameter:
    120: .  n - a non-negative integer

    122:    Output Parameter:
    123: .  factorial - n!

    125:    Level: beginner
    126: M*/
    127: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorial(PetscInt n, PetscReal *factorial)
    128: {
    129:   PetscReal f = 1.0;
    130:   PetscInt  i;

    133:   *factorial = -1.0;
    134:   if (n < 0) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Factorial called with negative number %D\n", n);
    135:   for (i = 1; i < n+1; ++i) f *= (PetscReal)i;
    136:   *factorial = f;
    137:   return(0);
    138: }

    140: /*MC
    141:    PetscDTFactorialInt - Compute n! as an integer

    143:    Input Parameter:
    144: .  n - a non-negative integer

    146:    Output Parameter:
    147: .  factorial - n!

    149:    Level: beginner

    151:    Note: this is limited to n such that n! can be represented by PetscInt, which is 12 if PetscInt is a signed 32-bit integer and 20 if PetscInt is a signed 64-bit integer.
    152: M*/
    153: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorialInt(PetscInt n, PetscInt *factorial)
    154: {
    155:   PetscInt facLookup[13] = {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600};

    158:   *factorial = -1;
    159:   if (n < 0 || n > PETSC_FACTORIAL_MAX) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of elements %D is not in supported range [0,%D]\n",n,PETSC_FACTORIAL_MAX);
    160:   if (n <= 12) {
    161:     *factorial = facLookup[n];
    162:   } else {
    163:     PetscInt f = facLookup[12];
    164:     PetscInt i;

    166:     for (i = 13; i < n+1; ++i) f *= i;
    167:     *factorial = f;
    168:   }
    169:   return(0);
    170: }

    172: /*MC
    173:    PetscDTBinomial - Approximate the binomial coefficient "n choose k"

    175:    Input Parameters:
    176: +  n - a non-negative integer
    177: -  k - an integer between 0 and n, inclusive

    179:    Output Parameter:
    180: .  binomial - approximation of the binomial coefficient n choose k

    182:    Level: beginner
    183: M*/
    184: PETSC_STATIC_INLINE PetscErrorCode PetscDTBinomial(PetscInt n, PetscInt k, PetscReal *binomial)
    185: {
    187:   *binomial = -1.0;
    188:   if (n < 0 || k < 0 || k > n) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial arguments (%D %D) must be non-negative, k <= n\n", n, k);
    189:   if (n <= 3) {
    190:     PetscInt binomLookup[4][4] = {{1, 0, 0, 0}, {1, 1, 0, 0}, {1, 2, 1, 0}, {1, 3, 3, 1}};

    192:     *binomial = (PetscReal)binomLookup[n][k];
    193:   } else {
    194:     PetscReal binom = 1.0;
    195:     PetscInt  i;

    197:     k = PetscMin(k, n - k);
    198:     for (i = 0; i < k; i++) binom = (binom * (PetscReal)(n - i)) / (PetscReal)(i + 1);
    199:     *binomial = binom;
    200:   }
    201:   return(0);
    202: }

    204: /*MC
    205:    PetscDTBinomialInt - Compute the binomial coefficient "n choose k"

    207:    Input Parameters:
    208: +  n - a non-negative integer
    209: -  k - an integer between 0 and n, inclusive

    211:    Output Parameter:
    212: .  binomial - the binomial coefficient n choose k

    214:    Note: this is limited by integers that can be represented by PetscInt

    216:    Level: beginner
    217: M*/
    218: PETSC_STATIC_INLINE PetscErrorCode PetscDTBinomialInt(PetscInt n, PetscInt k, PetscInt *binomial)
    219: {
    220:   PetscInt bin;

    223:   *binomial = -1;
    224:   if (n < 0 || k < 0 || k > n) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial arguments (%D %D) must be non-negative, k <= n\n", n, k);
    225:   if (n > PETSC_BINOMIAL_MAX) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial elements %D is larger than max for PetscInt, %D\n", n, PETSC_BINOMIAL_MAX);
    226:   if (n <= 3) {
    227:     PetscInt binomLookup[4][4] = {{1, 0, 0, 0}, {1, 1, 0, 0}, {1, 2, 1, 0}, {1, 3, 3, 1}};

    229:     bin = binomLookup[n][k];
    230:   } else {
    231:     PetscInt  binom = 1;
    232:     PetscInt  i;

    234:     k = PetscMin(k, n - k);
    235:     for (i = 0; i < k; i++) binom = (binom * (n - i)) / (i + 1);
    236:     bin = binom;
    237:   }
    238:   *binomial = bin;
    239:   return(0);
    240: }

    242: /*MC
    243:    PetscDTEnumPerm - Get a permutation of n integers from its encoding into the integers [0, n!) as a sequence of swaps.

    245:    A permutation can be described by the operations that convert the lists [0, 1, ..., n-1] into the permutation,
    246:    by a sequence of swaps, where the ith step swaps whatever number is in ith position with a number that is in
    247:    some position j >= i.  This swap is encoded as the difference (j - i).  The difference d_i at step i is less than
    248:    (n - i).  This sequence of n-1 differences [d_0, ..., d_{n-2}] is encoded as the number
    249:    (n-1)! * d_0 + (n-2)! * d_1 + ... + 1! * d_{n-2}.

    251:    Input Parameters:
    252: +  n - a non-negative integer (see note about limits below)
    253: -  k - an integer in [0, n!)

    255:    Output Parameters:
    256: +  perm - the permuted list of the integers [0, ..., n-1]
    257: -  isOdd - if not NULL, returns wether the permutation used an even or odd number of swaps.

    259:    Note: this is limited to n such that n! can be represented by PetscInt, which is 12 if PetscInt is a signed 32-bit integer and 20 if PetscInt is a signed 64-bit integer.

    261:    Level: beginner
    262: M*/
    263: PETSC_STATIC_INLINE PetscErrorCode PetscDTEnumPerm(PetscInt n, PetscInt k, PetscInt *perm, PetscBool *isOdd)
    264: {
    265:   PetscInt  odd = 0;
    266:   PetscInt  i;
    267:   PetscInt  work[PETSC_FACTORIAL_MAX];
    268:   PetscInt *w;

    271:   if (isOdd) *isOdd = PETSC_FALSE;
    272:   if (n < 0 || n > PETSC_FACTORIAL_MAX) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of elements %D is not in supported range [0,%D]\n",n,PETSC_FACTORIAL_MAX);
    273:   w = &work[n - 2];
    274:   for (i = 2; i <= n; i++) {
    275:     *(w--) = k % i;
    276:     k /= i;
    277:   }
    278:   for (i = 0; i < n; i++) perm[i] = i;
    279:   for (i = 0; i < n - 1; i++) {
    280:     PetscInt s = work[i];
    281:     PetscInt swap = perm[i];

    283:     perm[i] = perm[i + s];
    284:     perm[i + s] = swap;
    285:     odd ^= (!!s);
    286:   }
    287:   if (isOdd) *isOdd = odd ? PETSC_TRUE : PETSC_FALSE;
    288:   return(0);
    289: }

    291: /*MC
    292:    PetscDTPermIndex - Encode a permutation of n into an integer in [0, n!).  This inverts PetscDTEnumPerm.

    294:    Input Parameters:
    295: +  n - a non-negative integer (see note about limits below)
    296: -  perm - the permuted list of the integers [0, ..., n-1]

    298:    Output Parameters:
    299: +  k - an integer in [0, n!)
    300: -  isOdd - if not NULL, returns wether the permutation used an even or odd number of swaps.

    302:    Note: this is limited to n such that n! can be represented by PetscInt, which is 12 if PetscInt is a signed 32-bit integer and 20 if PetscInt is a signed 64-bit integer.

    304:    Level: beginner
    305: M*/
    306: PETSC_STATIC_INLINE PetscErrorCode PetscDTPermIndex(PetscInt n, const PetscInt *perm, PetscInt *k, PetscBool *isOdd)
    307: {
    308:   PetscInt  odd = 0;
    309:   PetscInt  i, idx;
    310:   PetscInt  work[PETSC_FACTORIAL_MAX];
    311:   PetscInt  iwork[PETSC_FACTORIAL_MAX];

    314:   *k = -1;
    315:   if (isOdd) *isOdd = PETSC_FALSE;
    316:   if (n < 0 || n > PETSC_FACTORIAL_MAX) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of elements %D is not in supported range [0,%D]\n",n,PETSC_FACTORIAL_MAX);
    317:   for (i = 0; i < n; i++) work[i] = i;  /* partial permutation */
    318:   for (i = 0; i < n; i++) iwork[i] = i; /* partial permutation inverse */
    319:   for (idx = 0, i = 0; i < n - 1; i++) {
    320:     PetscInt j = perm[i];
    321:     PetscInt icur = work[i];
    322:     PetscInt jloc = iwork[j];
    323:     PetscInt diff = jloc - i;

    325:     idx = idx * (n - i) + diff;
    326:     /* swap (i, jloc) */
    327:     work[i] = j;
    328:     work[jloc] = icur;
    329:     iwork[j] = i;
    330:     iwork[icur] = jloc;
    331:     odd ^= (!!diff);
    332:   }
    333:   *k = idx;
    334:   if (isOdd) *isOdd = odd ? PETSC_TRUE : PETSC_FALSE;
    335:   return(0);
    336: }

    338: /*MC
    339:    PetscDTEnumSubset - Get an ordered subset of the integers [0, ..., n - 1] from its encoding as an integers in [0, n choose k).
    340:    The encoding is in lexicographic order.

    342:    Input Parameters:
    343: +  n - a non-negative integer (see note about limits below)
    344: .  k - an integer in [0, n]
    345: -  j - an index in [0, n choose k)

    347:    Output Parameter:
    348: .  subset - the jth subset of size k of the integers [0, ..., n - 1]

    350:    Note: this is limited by arguments such that n choose k can be represented by PetscInt

    352:    Level: beginner

    354: .seealso: PetscDTSubsetIndex()
    355: M*/
    356: PETSC_STATIC_INLINE PetscErrorCode PetscDTEnumSubset(PetscInt n, PetscInt k, PetscInt j, PetscInt *subset)
    357: {
    358:   PetscInt       Nk, i, l;

    362:   PetscDTBinomialInt(n, k, &Nk);
    363:   for (i = 0, l = 0; i < n && l < k; i++) {
    364:     PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);
    365:     PetscInt Nminusk = Nk - Nminuskminus;

    367:     if (j < Nminuskminus) {
    368:       subset[l++] = i;
    369:       Nk = Nminuskminus;
    370:     } else {
    371:       j -= Nminuskminus;
    372:       Nk = Nminusk;
    373:     }
    374:   }
    375:   return(0);
    376: }

    378: /*MC
    379:    PetscDTSubsetIndex - Convert an ordered subset of k integers from the set [0, ..., n - 1] to its encoding as an integers in [0, n choose k) in lexicographic order.  This is the inverse of PetscDTEnumSubset.

    381:    Input Parameters:
    382: +  n - a non-negative integer (see note about limits below)
    383: .  k - an integer in [0, n]
    384: -  subset - an ordered subset of the integers [0, ..., n - 1]

    386:    Output Parameter:
    387: .  index - the rank of the subset in lexicographic order

    389:    Note: this is limited by arguments such that n choose k can be represented by PetscInt

    391:    Level: beginner

    393: .seealso: PetscDTEnumSubset()
    394: M*/
    395: PETSC_STATIC_INLINE PetscErrorCode PetscDTSubsetIndex(PetscInt n, PetscInt k, const PetscInt *subset, PetscInt *index)
    396: {
    397:   PetscInt       i, j = 0, l, Nk;

    401:   *index = -1;
    402:   PetscDTBinomialInt(n, k, &Nk);
    403:   for (i = 0, l = 0; i < n && l < k; i++) {
    404:     PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);
    405:     PetscInt Nminusk = Nk - Nminuskminus;

    407:     if (subset[l] == i) {
    408:       l++;
    409:       Nk = Nminuskminus;
    410:     } else {
    411:       j += Nminuskminus;
    412:       Nk = Nminusk;
    413:     }
    414:   }
    415:   *index = j;
    416:   return(0);
    417: }

    419: /*MC
    420:    PetscDTEnumSubset - Split the integers [0, ..., n - 1] into two complementary ordered subsets, the first subset of size k and being the jth subset of that size in lexicographic order.

    422:    Input Parameters:
    423: +  n - a non-negative integer (see note about limits below)
    424: .  k - an integer in [0, n]
    425: -  j - an index in [0, n choose k)

    427:    Output Parameters:
    428: +  perm - the jth subset of size k of the integers [0, ..., n - 1], followed by its complementary set.
    429: -  isOdd - if not NULL, return whether perm is an even or odd permutation.

    431:    Note: this is limited by arguments such that n choose k can be represented by PetscInt

    433:    Level: beginner

    435: .seealso: PetscDTEnumSubset(), PetscDTSubsetIndex()
    436: M*/
    437: PETSC_STATIC_INLINE PetscErrorCode PetscDTEnumSplit(PetscInt n, PetscInt k, PetscInt j, PetscInt *perm, PetscBool *isOdd)
    438: {
    439:   PetscInt       i, l, m, *subcomp, Nk;
    440:   PetscInt       odd;

    444:   if (isOdd) *isOdd = PETSC_FALSE;
    445:   PetscDTBinomialInt(n, k, &Nk);
    446:   odd = 0;
    447:   subcomp = &perm[k];
    448:   for (i = 0, l = 0, m = 0; i < n && l < k; i++) {
    449:     PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);
    450:     PetscInt Nminusk = Nk - Nminuskminus;

    452:     if (j < Nminuskminus) {
    453:       perm[l++] = i;
    454:       Nk = Nminuskminus;
    455:     } else {
    456:       subcomp[m++] = i;
    457:       j -= Nminuskminus;
    458:       odd ^= ((k - l) & 1);
    459:       Nk = Nminusk;
    460:     }
    461:   }
    462:   for (; i < n; i++) {
    463:     subcomp[m++] = i;
    464:   }
    465:   if (isOdd) *isOdd = odd ? PETSC_TRUE : PETSC_FALSE;
    466:   return(0);
    467: }

    469: struct _p_PetscTabulation {
    470:   PetscInt    K;    /* Indicates a k-jet, namely tabulated derivatives up to order k */
    471:   PetscInt    Nr;   /* The number of tabulation replicas (often 1) */
    472:   PetscInt    Np;   /* The number of tabulation points in a replica */
    473:   PetscInt    Nb;   /* The number of functions tabulated */
    474:   PetscInt    Nc;   /* The number of function components */
    475:   PetscInt    cdim; /* The coordinate dimension */
    476:   PetscReal **T;    /* The tabulation T[K] of functions and their derivatives
    477:                        T[0] = B[Nr*Np][Nb][Nc]:             The basis function values at quadrature points
    478:                        T[1] = D[Nr*Np][Nb][Nc][cdim]:       The basis function derivatives at quadrature points
    479:                        T[2] = H[Nr*Np][Nb][Nc][cdim][cdim]: The basis function second derivatives at quadrature points */
    480: };
    481: typedef struct _p_PetscTabulation *PetscTabulation;

    483: #endif