11 …A k-form w (k is called the "form degree" of w) is an alternating k-linear map acting on tuples (v…
13 …y two vectors in a tuple reverses the sign of the result, e.g. w(v_1, v_2, ..., v_k) = -w(v_2, v_1…
14 …- k-linear: w acts linear in each vector separately, e.g. w(a*v + b*y, v_2, ..., v_k) = a*w(v,v_2,…
36 …If k > 0, a k-form w and a vector v can combine to make a (k-1)-form through the interior product,…
37    defined by (w int v)(v_1,...,v_{k-1}) = w(v,v_1,...,v_{k-1}).
39 …The interior product is implemented for either two inputs (w and v) in PetscDTAltVInterior, for on…
40    matrix to multiply against multiple choices of w) in `PetscDTAltVInteriorMatrix()`,
43 …When there is a linear map L: V -> W from an N dimensional vector space to an M dimensional vector…
44 …it induces the linear pullback map L^* : Alt^k W -> Alt^k V, defined by L^* w(v_1,...,v_k) = w(L v…
45 …The pullback is implemented as `PetscDTAltVPullback()` (acting on a known w) and `PetscDTAltVPullb…
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…
68 . wv - w(v_1,...,v_k) = \sum_i w_i * det(V_i): the degree of freedom w_i is associated with coordin…
75 PetscErrorCode PetscDTAltVApply(PetscInt N, PetscInt k, const PetscReal *w, const PetscReal *v, Pet…  in PetscDTAltVApply()  argument
82       *wv = w[0];  in PetscDTAltVApply()
85         *wv = w[0] * v[0];  in PetscDTAltVApply()
88           *wv = w[0] * v[0] + w[1] * v[1];  in PetscDTAltVApply()
90           *wv = w[0] * (v[0] * v[3] - v[1] * v[2]);  in PetscDTAltVApply()
94           *wv = w[0] * v[0] + w[1] * v[1] + w[2] * v[2];  in PetscDTAltVApply()
96 …*wv = w[0] * (v[0] * v[4] - v[1] * v[3]) + w[1] * (v[0] * v[5] - v[2] * v[3]) + w[2] * (v[1] * v[5…  in PetscDTAltVApply()
98 …*wv = w[0] * (v[0] * (v[4] * v[8] - v[5] * v[7]) + v[1] * (v[5] * v[6] - v[3] * v[8]) + v[2] * (v[…  in PetscDTAltVApply()
124       sum += w[i] * subsum;  in PetscDTAltVApply()
305 . k - the *signed* degree k of the |k|-form w, -(min(M,N)) <= k <= min(M,N).  A negative form degre…
307 - w - a |k|-form in the image space, size [M choose |k|]
310 . Lstarw - the pullback of w to a |k|-form in the origin space, size [N choose |k|]: (Lstarw)(v_1,.…
323 …back(PetscInt N, PetscInt M, const PetscReal *L, PetscInt k, const PetscReal *w, PetscReal *Lstarw)  in PetscDTAltVPullback()  argument
334       Lstarw[0] = w[0];  in PetscDTAltVPullback()
339         for (j = 0; j < Mk; j++) sum += L[j * Nk + i] * w[j];  in PetscDTAltVPullback()
348         for (j = 0; j < Mk; j++) sum += L[(Mk - 1 - j) * Nk + (Nk - 1 - i)] * w[j] * mult[j];  in PetscDTAltVPullback()
360 …] * N + pairs[i][1]] - L[pairs[j][1] * N + pairs[i][0]] * L[pairs[j][0] * N + pairs[i][1]]) * w[j];  in PetscDTAltVPullback()
375 …rs[offj + j][1] * N + pairs[offi + i][0]] * L[pairs[offj + j][0] * N + pairs[offi + i][1]]) * w[j];  in PetscDTAltVPullback()
381       for (i = 0; i < Nk; i++) Lstarw[i] = detL * w[i];  in PetscDTAltVPullback()
399       PetscCall(PetscDTAltVStar(M, M - k, 1, w, walloc));  in PetscDTAltVPullback()
402       ww = w;  in PetscDTAltVPullback()
446 …matrix, an [(N choose |k|) x (M choose |k|)] matrix in row-major format such that Lstar * w = L^* w
550 . k - the degree k of the k-form w, 0 <= k <= N
551 . w - a k-form, size [N choose k]
555 …form (w int v), size [N choose (k-1)]: (w int v) is defined by its action on (k-1) vectors {v_1, .…
561 PetscErrorCode PetscDTAltVInterior(PetscInt N, PetscInt k, const PetscReal *w, const PetscReal *v, …  in PetscDTAltVInterior()  argument
573       for (i = 0; i < N; i++) sum += w[i] * v[i];  in PetscDTAltVInterior()
578       for (i = 0; i < N; i++) wIntv[N - 1 - i] = w[0] * v[i] * mult[i];  in PetscDTAltVInterior()
580       wIntv[0] = -w[0] * v[1] - w[1] * v[2];  in PetscDTAltVInterior()
581       wIntv[1] = w[0] * v[0] - w[2] * v[2];  in PetscDTAltVInterior()
582       wIntv[2] = w[1] * v[0] + w[2] * v[1];  in PetscDTAltVInterior()
601         wIntv[idx] += flip ? -(w[i] * v[subset[j]]) : (w[i] * v[subset[j]]);  in PetscDTAltVInterior()
618 . intvMat - an [(N choose (k-1)) x (N choose k)] matrix, row-major: (intvMat) * w = (w int v)
768 . k   - the degree k of the k-form w, 0 <= k <= N
770 - w   - a k-form, size [N choose k]
773 . starw - (star)^pow w
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
800       for (i = 0; i < Nk; i++) starw[Nk - 1 - i] = w[i] * mult[i];  in PetscDTAltVStar()
802       for (i = 0; i < Nk; i++) starw[i] = w[i];  in PetscDTAltVStar()
820         starw[j] = sOdd ? -w[idx] : w[idx];  in PetscDTAltVStar()
823       for (i = 0; i < Nk; i++) starw[i] = w[i];  in PetscDTAltVStar()