17 …ctor space themselves, Alt^k V.  The dimension of Alt^k V, if V is N dimensional, is N choose k.  …
18    shows that for an N dimensional space, only 0 <= k <= N are valid form degrees.)
19 …scDTAltV, has one basis k-form for each ordered subset of k coordinates of the N dimensional space:
27 …If j + k <= N, then a j-form f and a k-form g can be multiplied to create a (j+k)-form using the w…
43 …When there is a linear map L: V -> W from an N dimensional vector space to an M dimensional vector…
47 …Alt^k V and Alt^(N-k) V have the same dimension, and the Hodge star operator maps between them.  W…
58 …PetscDTAltVApply - Apply an a k-form (an alternating k-linear map) to a set of k N-dimensional vec…
61 + N - the dimension of the vector space, N >= 0
62 . k - the degree k of the k-form w, 0 <= k <= N
63 . w - a k-form, size [N choose k] (each degree of freedom of a k-form is associated with a subset o…
65 - v - a set of k vectors of size N, size [k x N], each vector stored contiguously
75 PetscErrorCode PetscDTAltVApply(PetscInt N, PetscInt k, const PetscReal *w, const PetscReal *v, Pet…  in PetscDTAltVApply()  argument
78   PetscCheck(N >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid dimension");  in PetscDTAltVApply()
79   PetscCheck(k >= 0 && k <= N, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid form degree");  in PetscDTAltVApply()
80   if (N <= 3) {  in PetscDTAltVApply()
84       if (N == 1) {  in PetscDTAltVApply()
86       } else if (N == 2) {  in PetscDTAltVApply()
109     PetscCall(PetscDTBinomialInt(N, k, &Nk));  in PetscDTAltVApply()
114       PetscCall(PetscDTEnumSubset(N, k, i, subset));  in PetscDTAltVApply()
121         for (l = 0; l < k; l++) prod *= v[perm[l] * N + subset[l]];  in PetscDTAltVApply()
136 + N - the dimension of the vector space, N >= 0
137 . j - the degree j of the j-form a, 0 <= j <= N
138 . k - the degree k of the k-form b, 0 <= k <= N and 0 <= j+k <= N
139 . a - a j-form, size [N choose j]
140 - b - a k-form, size [N choose k]
143 . awedgeb - the (j+k)-form a wedge b, size [N choose (j+k)]: (a wedge b)(v_1,...,v_{j+k}) = \sum_{s…
150 PetscErrorCode PetscDTAltVWedge(PetscInt N, PetscInt j, PetscInt k, const PetscReal *a, const Petsc…  in PetscDTAltVWedge()  argument
155   PetscCheck(N >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid dimension");  in PetscDTAltVWedge()
157   PetscCheck(j + k <= N, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Wedge greater than dimension");  in PetscDTAltVWedge()
158   if (N <= 3) {  in PetscDTAltVWedge()
161     PetscCall(PetscDTBinomialInt(N, j + k, &Njk));  in PetscDTAltVWedge()
167       if (N == 2) {  in PetscDTAltVWedge()
185     PetscCall(PetscDTBinomialInt(N, j + k, &Njk));  in PetscDTAltVWedge()
192       PetscCall(PetscDTEnumSubset(N, j + k, i, subset));  in PetscDTAltVWedge()
200         PetscCall(PetscDTSubsetIndex(N, j, subsetj, &jInd));  in PetscDTAltVWedge()
201         PetscCall(PetscDTSubsetIndex(N, k, subsetk, &kInd));  in PetscDTAltVWedge()
215 + N - the dimension of the vector space, N >= 0
216 . j - the degree j of the j-form a, 0 <= j <= N
217 . k - the degree k of the k-forms that (a wedge) will be applied to, 0 <= k <= N and 0 <= j+k <= N
218 - a - a j-form, size [N choose j]
221 . awedgeMat - (a wedge), an [(N choose j+k) x (N choose k)] matrix in row-major order, such that (a…
227 PetscErrorCode PetscDTAltVWedgeMatrix(PetscInt N, PetscInt j, PetscInt k, const PetscReal *a, Petsc…  in PetscDTAltVWedgeMatrix()  argument
232   PetscCheck(N >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid dimension");  in PetscDTAltVWedgeMatrix()
234   PetscCheck(j + k <= N, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Wedge greater than dimension");  in PetscDTAltVWedgeMatrix()
235   if (N <= 3) {  in PetscDTAltVWedgeMatrix()
238     PetscCall(PetscDTBinomialInt(N, j + k, &Njk));  in PetscDTAltVWedgeMatrix()
245       if (N == 2) {  in PetscDTAltVWedgeMatrix()
272     PetscCall(PetscDTBinomialInt(N, k, &Nk));  in PetscDTAltVWedgeMatrix()
273     PetscCall(PetscDTBinomialInt(N, j + k, &Njk));  in PetscDTAltVWedgeMatrix()
280       PetscCall(PetscDTEnumSubset(N, j + k, i, subset));  in PetscDTAltVWedgeMatrix()
288         PetscCall(PetscDTSubsetIndex(N, j, subsetj, &jInd));  in PetscDTAltVWedgeMatrix()
289         PetscCall(PetscDTSubsetIndex(N, k, subsetk, &kInd));  in PetscDTAltVWedgeMatrix()
302 + N - the dimension of the origin vector space of the linear transformation, M >= 0
303 . M - the dimension of the image vector space of the linear transformation, N >= 0
304 . L - a linear transformation, an [M x N] matrix in row-major format
305 . k - the *signed* degree k of the |k|-form w, -(min(M,N)) <= k <= min(M,N).  A negative form degre…
310 . Lstarw - the pullback of w to a |k|-form in the origin space, size [N choose |k|]: (Lstarw)(v_1,.…
323 PetscErrorCode PetscDTAltVPullback(PetscInt N, PetscInt M, const PetscReal *L, PetscInt k, const Pe…  in PetscDTAltVPullback()  argument
328   PetscCheck(N >= 0 && M >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid dimensions");  in PetscDTAltVPullback()
329 …PetscCheck(PetscAbsInt(k) <= N && PetscAbsInt(k) <= M, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, …  in PetscDTAltVPullback()
330   if (N <= 3 && M <= 3) {  in PetscDTAltVPullback()
332     PetscCall(PetscDTBinomialInt(N, PetscAbsInt(k), &Nk));  in PetscDTAltVPullback()
360 …um += (L[pairs[j][0] * N + pairs[i][0]] * L[pairs[j][1] * N + pairs[i][1]] - L[pairs[j][1] * N + p…  in PetscDTAltVPullback()
369       PetscInt offi = (N == 2) ? 2 : 0;  in PetscDTAltVPullback()
375 …* N + pairs[offi + i][0]] * L[pairs[offj + j][1] * N + pairs[offi + i][1]] - L[pairs[offj + j][1] …  in PetscDTAltVPullback()
393     PetscCall(PetscDTBinomialInt(N, PetscAbsInt(k), &Nk));  in PetscDTAltVPullback()
404     PetscCall(PetscMalloc5(k, &subsetw, k, &subsetv, k, &perm, N * k, &Lw, k * k, &Lwv));  in PetscDTAltVPullback()
409         PetscCall(PetscDTEnumSubset(N, k, j, subsetv));  in PetscDTAltVPullback()
416           for (l = 0; l < k; l++) prod *= L[subsetw[perm[l]] * N + subsetv[l]];  in PetscDTAltVPullback()
425       PetscCall(PetscDTAltVStar(N, N - k, -1, Lstarw, sLsw));  in PetscDTAltVPullback()
439 + N - the dimension of the origin vector space of the linear transformation, N >= 0
441 . L - a linear transformation, an [M x N] matrix in row-major format
442 - k - the *signed* degree k of the |k|-forms on which Lstar acts, -(min(M,N)) <= k <= min(M,N).
446 . Lstar - the pullback matrix, an [(N choose |k|) x (M choose |k|)] matrix in row-major format such…
452 PetscErrorCode PetscDTAltVPullbackMatrix(PetscInt N, PetscInt M, const PetscReal *L, PetscInt k, Pe…  in PetscDTAltVPullbackMatrix()  argument
461   PetscCheck(N >= 0 && M >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid dimensions");  in PetscDTAltVPullbackMatrix()
462 …PetscCheck(PetscAbsInt(k) <= N && PetscAbsInt(k) <= M, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, …  in PetscDTAltVPullbackMatrix()
463   if (N <= 3 && M <= 3) {  in PetscDTAltVPullbackMatrix()
467     PetscCall(PetscDTBinomialInt(N, PetscAbsInt(k), &Nk));  in PetscDTAltVPullbackMatrix()
486 …+ j] = L[pairs[j][0] * N + pairs[i][0]] * L[pairs[j][1] * N + pairs[i][1]] - L[pairs[j][1] * N + p…  in PetscDTAltVPullbackMatrix()
494       PetscInt offi = (N == 2) ? 2 : 0;  in PetscDTAltVPullbackMatrix()
498 …* N + pairs[offi + i][0]] * L[pairs[offj + j][1] * N + pairs[offi + i][1]] - L[pairs[offj + j][1] …  in PetscDTAltVPullbackMatrix()
511     PetscCall(PetscDTBinomialInt(N, PetscAbsInt(k), &Nk));  in PetscDTAltVPullbackMatrix()
513     PetscCall(PetscMalloc5(M, &subsetw, N, &subsetv, k, &perm, N * k, &Lw, k * k, &Lwv));  in PetscDTAltVPullbackMatrix()
525         PetscCall(PetscDTEnumSplit(N, k, j, subsetv, &jOdd));  in PetscDTAltVPullbackMatrix()
527         jOdd = negative ? (PetscBool)(iOdd ^ jOdd ^ ((k * (N - k)) & 1)) : PETSC_FALSE;  in PetscDTAltVPullbackMatrix()
535           for (l = 0; l < k; l++) prod *= L[subsetw[perm[l]] * N + subsetv[l]];  in PetscDTAltVPullbackMatrix()
549 + N - the dimension of the vector space, N >= 0
550 . k - the degree k of the k-form w, 0 <= k <= N
551 . w - a k-form, size [N choose k]
552 - v - an N dimensional vector
555 . wIntv - the (k-1)-form (w int v), size [N choose (k-1)]: (w int v) is defined by its action on (k…
561 PetscErrorCode PetscDTAltVInterior(PetscInt N, PetscInt k, const PetscReal *w, const PetscReal *v, …  in PetscDTAltVInterior()  argument
566   PetscCheck(k > 0 && k <= N, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid form degree");  in PetscDTAltVInterior()
567   PetscCall(PetscDTBinomialInt(N, k, &Nk));  in PetscDTAltVInterior()
568   PetscCall(PetscDTBinomialInt(N, k - 1, &Nkm));  in PetscDTAltVInterior()
569   if (N <= 3) {  in PetscDTAltVInterior()
573       for (i = 0; i < N; i++) sum += w[i] * v[i];  in PetscDTAltVInterior()
575     } else if (k == N) {  in PetscDTAltVInterior()
578       for (i = 0; i < N; i++) wIntv[N - 1 - i] = w[0] * v[i] * mult[i];  in PetscDTAltVInterior()
592       PetscCall(PetscDTEnumSubset(N, k, i, subset));  in PetscDTAltVInterior()
600         PetscCall(PetscDTSubsetIndex(N, k - 1, work, &idx));  in PetscDTAltVInterior()
613 + N - the dimension of the vector space, N >= 0
614 . k - the degree k of the k-forms on which intvMat acts, 0 <= k <= N
615 - v - an N dimensional vector
618 . intvMat - an [(N choose (k-1)) x (N choose k)] matrix, row-major: (intvMat) * w = (w int v)
624 PetscErrorCode PetscDTAltVInteriorMatrix(PetscInt N, PetscInt k, const PetscReal *v, PetscReal *int…  in PetscDTAltVInteriorMatrix()  argument
629   PetscCheck(k > 0 && k <= N, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid form degree");  in PetscDTAltVInteriorMatrix()
630   PetscCall(PetscDTBinomialInt(N, k, &Nk));  in PetscDTAltVInteriorMatrix()
631   PetscCall(PetscDTBinomialInt(N, k - 1, &Nkm));  in PetscDTAltVInteriorMatrix()
632   if (N <= 3) {  in PetscDTAltVInteriorMatrix()
634       for (i = 0; i < N; i++) intvMat[i] = v[i];  in PetscDTAltVInteriorMatrix()
635     } else if (k == N) {  in PetscDTAltVInteriorMatrix()
638       for (i = 0; i < N; i++) intvMat[N - 1 - i] = v[i] * mult[i];  in PetscDTAltVInteriorMatrix()
658       PetscCall(PetscDTEnumSubset(N, k, i, subset));  in PetscDTAltVInteriorMatrix()
666         PetscCall(PetscDTSubsetIndex(N, k - 1, work, &idx));  in PetscDTAltVInteriorMatrix()
679 + N - the dimension of the vector space, $N \ge 0$
680 … degree of the k-forms on which `intvMat` from `PetscDTAltVInteriorMatrix()` acts, $ 0 le k le N $.
683 …The interior product matrix `intvMat` has dimensions [(N choose (k-1)) x (N choose k)] and has (N …
694 PetscErrorCode PetscDTAltVInteriorPattern(PetscInt N, PetscInt k, PetscInt (*indices)[3])  in PetscDTAltVInteriorPattern()  argument
699   PetscCheck(k > 0 && k <= N, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid form degree");  in PetscDTAltVInteriorPattern()
700   PetscCall(PetscDTBinomialInt(N, k, &Nk));  in PetscDTAltVInteriorPattern()
701   PetscCall(PetscDTBinomialInt(N, k - 1, &Nkm));  in PetscDTAltVInteriorPattern()
702   if (N <= 3) {  in PetscDTAltVInteriorPattern()
704       for (i = 0; i < N; i++) {  in PetscDTAltVInteriorPattern()
709     } else if (k == N) {  in PetscDTAltVInteriorPattern()
712       for (i = 0; i < N; i++) {  in PetscDTAltVInteriorPattern()
713         indices[i][0] = N - 1 - i;  in PetscDTAltVInteriorPattern()
744       PetscCall(PetscDTEnumSubset(N, k, i, subset));  in PetscDTAltVInteriorPattern()
752         PetscCall(PetscDTSubsetIndex(N, k - 1, work, &idx));  in PetscDTAltVInteriorPattern()
764 …PetscDTAltVStar - Apply a power of the Hodge star operator, which maps k-forms to (N-k) forms, to …
767 + N   - the dimension of the vector space, N >= 0
768 . k   - the degree k of the k-form w, 0 <= k <= N
770 - w   - a k-form, size [N choose k]
778   Each degree of freedom of a k-form is associated with a subset S of k coordinates of the N
782   implies (star)^2 w = (-1)^{k(N-k)} w, and (star)^4 w = w.
786 PetscErrorCode PetscDTAltVStar(PetscInt N, PetscInt k, PetscInt pow, const PetscReal *w, PetscReal …  in PetscDTAltVStar()  argument
791   PetscCheck(k >= 0 && k <= N, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "invalid form degree");  in PetscDTAltVStar()
792   PetscCall(PetscDTBinomialInt(N, k, &Nk));  in PetscDTAltVStar()
796   if (N <= 3) {  in PetscDTAltVStar()
804     if (pow > 1 && ((k * (N - k)) & 1)) {  in PetscDTAltVStar()
810     PetscCall(PetscMalloc1(N, &subset));  in PetscDTAltVStar()
812       PetscInt l = (pow == 1) ? k : N - k;  in PetscDTAltVStar()
817         PetscCall(PetscDTEnumSplit(N, l, i, subset, &sOdd));  in PetscDTAltVStar()
818         PetscCall(PetscDTSubsetIndex(N, l, subset, &idx));  in PetscDTAltVStar()
819         PetscCall(PetscDTSubsetIndex(N, N - l, &subset[l], &j));  in PetscDTAltVStar()
825     /* star^2 = -1^(k * (N - k)) */  in PetscDTAltVStar()
826     if (pow > 1 && (k * (N - k)) % 2) {  in PetscDTAltVStar()