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 Topics FrontPage mathematical algorithms FriCAS Algebra BiCartesianTensor <-- You are here. BiCartesianTensor(dim,R) provides Cartesian tensors with components belonging to a commutative ring R. These tensors can have any number of covariant and contravariant indices. Each index takes values from -dim..1 or 1..dim. spad)abbrev domain BITENS BiCartesianTensor BiCartesianTensor(dim, R): Exports == Implementation where NNI ==> NonNegativeInteger I ==> PositiveInteger DP ==> DirectProduct SM ==> SquareMatrix dim: NNI R: CommutativeRing Exports ==> Join(GradedAlgebra(R, DP(2,NNI)), GradedModule(Integer, DP(2,NNI))) with coerce: DP(dim, R) -> % ++ coerce(v) views a vector over the ring R as a (0,1)-tensor. coerce: SM(dim, R) -> % ++ coerce(m) views a matrix on the ring R as a (2,0)-tensor. coerce: List R -> % ++ coerce([r_1,...,r_dim]) views a list of ring R elements ++ as a (1,0)-tensor coerce: List % -> % ++ coerce([t_1,...,t_dim]) allows tensors to be constructed ++ using lists. rank: % -> DP(2,NNI) ++ rank(t) returns the tensorial rank of t (that is, the ++ number of indices). This is the same as the graded module ++ degree. elt: (%) -> R ++ elt(t) gives the component of a rank 0 tensor. elt: (%, List List I) -> R ++ elt(t,[[i1,...,in],[ji,...,jm]]) gives a component of a (n,m)-tensor. -- This specializes the documentation from GradedAlgebra. product: (%,%) -> % ++ product(s,t) is the outer product of the tensors s and t. ++ For example, if \spad{r = product(s,t)} for rank 2 tensors s and t, ++ then \spad{r} is a rank 4 tensor given by ++ \spad{r(i,j,k,l) = s(i,j)*t(k,l)}. "*": (%, %) -> % ++ s*t is the inner product of the tensors s and t which contracts ++ the last index of s with the first index of t, i.e. ++ \spad{t*s = contract(t,rank t, s, 1)} ++ \spad{t*s = sum(k=1..N, t[i1,..,iN,k]*s[k,j1,..,jM])} ++ This is compatible with the use of \spad{M*v} to denote ++ the matrix-vector inner product. contract: (%, Integer, %, Integer) -> % ++ contract(t,i,s,j) is the inner product of tenors s and t ++ which sums along the \spad{k1}-th index of ++ t and the \spad{k2}-th index of s. ++ For example, if \spad{r = contract(s,2,t,1)} for rank 3 tensors ++ rank 3 tensors \spad{s} and \spad{t}, then \spad{r} is ++ the rank 4 \spad{(= 3 + 3 - 2)} tensor given by ++ \spad{r(i,j,k,l) = sum(h=1..dim,s(i,h,j)*t(h,k,l))}. contract: (%, Integer, Integer) -> % ++ contract(t,i,j) is the contraction of tensor t which ++ sums along the \spad{i}-th and \spad{j}-th indices. ++ For example, if ++ \spad{r = contract(t,1,3)} for a rank 4 tensor t, then ++ \spad{r} is the rank 2 \spad{(= 4 - 2)} tensor given by ++ \spad{r(i,j) = sum(h=1..dim,t(h,i,h,j))}. transpose: % -> % ++ transpose(t) exchanges the first and last indices of t. ++ For example, if \spad{r = transpose(t)} for a rank 4 tensor t, then ++ \spad{r} is the rank 4 tensor given by ++ \spad{r(i,j,k,l) = t(l,j,k,i)}. transpose: (%, Integer, Integer) -> % ++ transpose(t,i,j) exchanges the \spad{i}-th and \spad{j}-th indices of t. ++ For example, if \spad{r = transpose(t,2,3)} for a rank 4 tensor t, then ++ \spad{r} is the rank 4 tensor given by ++ \spad{r(i,j,k,l) = t(i,k,j,l)}. reindex: (%, List Integer) -> % ++ reindex(t,[i1,...,idim]) permutes the indices of t. ++ For example, if \spad{r = reindex(t, [4,1,2,3])} ++ for a rank 4 tensor t, ++ then \spad{r} is the rank for tensor given by ++ \spad{r(i,j,k,l) = t(l,i,j,k)}. kroneckerDelta: () -> % ++ kroneckerDelta() is the (1,1)-tensor defined by ++ \spad{kroneckerDelta()(i,j)} ++ \spad{= 1 if i = j} ++ \spad{= 0 if i \~= j} leviCivitaSymbol: () -> % ++ leviCivitaSymbol() is the (\spad{dim},0)-tensor defined by ++ \spad{leviCivitaSymbol()(i1,...idim) = +1/0/-1} ++ if \spad{i1,...,idim} is an even/is nota /is an odd permutation ++ of \spad{minix,...,minix+dim-1}. ravel: % -> List R ++ ravel(t) produces a list of (n+m)*dim components from a ++ (n,m)-tensor such that \spad{unravel(ravel(t)) = t}. unravel: List R -> % ++ unravel(t) produces a (#t/dim,0)-tensor from a list of ++ components such that ++ \spad{unravel(ravel(t)) = t}. sample: () -> % ++ sample() returns an object of type %. Implementation ==> add PERM ==> List Vector I -- permutation of 1..n and 1..m INDEX ==> List Vector I -- vector of n indices and m indices in 1..dim Rep := IndexedVector(R,0) get ==> elt$Rep set_! ==> setelt$Rep -- Use row-major order: -- x[[h],[i,j]] <-> x[(h-1)*dim^2+(i-1)*dim+(j-1)] n: Integer r,s: R x,y,z: % ---- Local stuff dim2: NNI := dim^2 dim3: NNI := dim^3 dim4: NNI := dim^4 sample()==kroneckerDelta()$% int2index(n: Integer, indv: INDEX): INDEX == n < 0 => error "Index error (too small)" rnk := #(indv.1) for i in 1..rnk repeat qr := divide(n, dim) n := qr.quotient indv.1.((rnk-i+1) pretend I) := qr.remainder + 1 rnk := #(indv.2) for i in 1..rnk repeat qr := divide(n, dim) n := qr.quotient indv.2.((rnk-i+1) pretend I) := qr.remainder + 1 n ~= 0 => error "Index error (too big)" indv index2int(indv: INDEX): Integer == n: I := 0 for i in 1..#(indv.1) repeat ix := indv.1.i - 1 ix<0 or ix>dim-1 => error "Index error (out of range)" n := dim*n + ix for i in 1..#(indv.2) repeat ix := indv.2.i - 1 ix<0 or ix>dim-1 => error "Index error (out of range)" n := dim*n + ix n lengthRankOrElse(v: Integer): NNI == v = 1 => 0 v = dim => 1 v = dim2 => 2 v = dim3 => 3 v = dim4 => 4 rx := 0 while v ~= 0 repeat qr := divide(v, dim) v := qr.quotient if v ~= 0 then qr.remainder ~= 0 => error "Rank is not a whole number" rx := rx + 1 rx -- l must be a list of the numbers 1..#l mkPerm(n: NNI, l: List Integer): PERM == #l ~= n => error "The list is not a permutation." p: PERM := new(n, 0) seen: Vector Boolean := new(n, false) for i in 1..n for e in l repeat e < 1 or e > n => error "The list is not a permutation." p.i := e seen.e := true for e in 1..n repeat not seen.e => error "The list is not a permutation." p -- permute s according to p into result t. permute_!(t: INDEX, s: INDEX, p: PERM): INDEX == for i in 1..#p repeat t.i := s.(p.i) t -- permsign!(v) = 1, 0, or -1 according as -- v is an even, is not, or is an odd permutation of minix..minix+#v-1. permsign_!(v: INDEX): Integer == -- sum minix..minix+#v-1. maxix := minix+#v-1 psum := (((maxix+1)*maxix - minix*(minix-1)) exquo 2)::Integer -- +/v ~= psum => 0 n := 0 for i in 1..#v repeat n := n + v.i n ~= psum => 0 -- Bubble sort! This is pretty grotesque. totTrans: Integer := 0 nTrans: Integer := 1 while nTrans ~= 0 repeat nTrans := 0 for i in 1..#v-1 for j in 2..#v repeat if v.i > v.j then nTrans := nTrans + 1 e := v.i; v.i := v.j; v.j := e totTrans := totTrans + nTrans for i in 1..dim repeat if v.i ~= minix+i-1 then return 0 odd? totTrans => -1 1 ---- Exported functions ravel x == [get(x,i) for i in 0..#x-1] unravel l == -- lengthRankOrElse #l gives sytnax error nz: NNI := # l lengthRankOrElse nz z := new(nz, 0) for i in 0..nz-1 for r in l repeat set_!(z, i, r) z kroneckerDelta() == z := new(dim2, 0) for i in 1..dim for zi in 0.. by (dim+1) repeat set_!(z, zi, 1) z leviCivitaSymbol() == nz := dim^dim z := new(nz, 0) indv: INDEX := new(dim, 0) for i in 0..nz-1 repeat set_!(z, i, permsign_!(int2index(i, indv))::R) z -- from GradedModule degree x == rank x rank x == n := #x lengthRankOrElse n elt(x) == #x ~= 1 => error "Index error (the rank is not 0)" get(x,0) elt(x, i: I) == #x ~= dim => error "Index error (the rank is not 1)" get(x,(i-minix)) elt(x, i: I, j: I) == #x ~= dim2 => error "Index error (the rank is not 2)" get(x,(dim*(i-minix) + (j-minix))) elt(x, i: I, j: I, k: I) == #x ~= dim3 => error "Index error (the rank is not 3)" get(x,(dim2*(i-minix) + dim*(j-minix) + (k-minix))) elt(x, i: I, j: I, k: I, l: I) == #x ~= dim4 => error "Index error (the rank is not 4)" get(x,(dim3*(i-minix) + dim2*(j-minix) + dim*(k-minix) + (l-minix))) elt(x, i: List I) == #i ~= rank x => error "Index error (wrong rank)" n: I := 0 for ii in i repeat ix := ii - minix ix<0 or ix>dim-1 => error "Index error (out of range)" n := dim*n + ix get(x,n) coerce(lr: List R): % == #lr ~= dim => error "Incorrect number of components" z := new(dim, 0) for r in lr for i in 0..dim-1 repeat set_!(z, i, r) z coerce(lx: List %): % == #lx ~= dim => error "Incorrect number of slices" rx := rank first lx for x in lx repeat rank x ~= rx => error "Inhomogeneous slice ranks" nx := # first lx z := new(dim * nx, 0) for x in lx for offz in 0.. by nx repeat for i in 0..nx-1 repeat set_!(z, offz + i, get(x,i)) z retractIfCan(x:%):Union(R,"failed") == zero? rank(x) => x() "failed" Outf ==> OutputForm mkOutf(x:%, i0:I, rnk:NNI): Outf == odd? rnk => rnk1 := (rnk-1) pretend NNI nskip := dim^rnk1 [mkOutf(x, i0+nskip*i, rnk1) for i in 0..dim-1]::Outf rnk = 0 => get(x,i0)::Outf rnk1 := (rnk-2) pretend NNI nskip := dim^rnk1 matrix [[mkOutf(x, i0+nskip*(dim*i + j), rnk1) for j in 0..dim-1] for i in 0..dim-1] coerce(x): Outf == mkOutf(x, 0, rank x) 0 == 0$R::Rep 1 == 1$R::Rep --coerce(n: I): % == new(1, n::R) coerce(r: R): % == new(1,r) coerce(v: DP(dim,R)): % == z := new(dim, 0) for i in 0..dim-1 for j in minIndex v .. maxIndex v repeat set_!(z, i, v.j) z coerce(m: SM(dim,R)): % == z := new(dim^2, 0) offz := 0 for i in 0..dim-1 repeat for j in 0..dim-1 repeat set_!(z, offz + j, m(i+1,j+1)) offz := offz + dim z x = y == #x ~= #y => false for i in 0..#x-1 repeat if get(x,i) ~= get(y,i) then return false true x + y == #x ~= #y => error "Rank mismatch" -- z := [xi + yi for xi in x for yi in y] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, get(x,i) + get(y,i)) z x - y == #x ~= #y => error "Rank mismatch" -- [xi - yi for xi in x for yi in y] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, get(x,i) - get(y,i)) z - x == -- [-xi for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, -get(x,i)) z n * x == -- [n * xi for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, n * get(x,i)) z x * n == -- [n * xi for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, n* get(x,i)) -- Commutative!! z r * x == -- [r * xi for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, r * get(x,i)) z x * r == -- [xi*r for xi in x] z := new(#x, 0) for i in 0..#x-1 repeat set_!(z, i, r* get(x,i)) -- Commutative!! z product(x, y) == nx := #x; ny := #y z := new(nx * ny, 0) for i in 0..nx-1 for ioff in 0.. by ny repeat for j in 0..ny-1 repeat set_!(z, ioff + j, get(x,i) * get(y,j)) z x * y == rx := rank x ry := rank y rx = 0 => get(x,0) * y ry = 0 => x * get(y,0) contract(x, rx, y, 1) contract(x, i, j) == rx := rank x i < 1 or i > rx or j < 1 or j > rx or i = j => error "Improper index for contraction" if i > j then (i,j) := (j,i) rl:= (rx- j) pretend NNI; nl:= dim^rl; zol:= 1; xol:= zol rm:= (j-i-1) pretend NNI; nm:= dim^rm; zom:= nl; xom:= zom*dim rh:= (i - 1) pretend NNI; nh:= dim^rh; zoh:= nl*nm xoh:= zoh*dim^2 xok := nl*(1 + nm*dim) z := new(nl*nm*nh, 0) for h in 1..nh _ for xh in 0.. by xoh for zh in 0.. by zoh repeat for m in 1..nm _ for xm in xh.. by xom for zm in zh.. by zom repeat for l in 1..nl _ for xl in xm.. by xol for zl in zm.. by zol repeat set_!(z, zl, 0) for k in 1..dim for xk in xl.. by xok repeat set_!(z, zl, get(z,zl) + get(x,xk)) z contract(x, i, y, j) == rx := rank x ry := rank y i < 1 or i > rx or j < 1 or j > ry => error "Improper index for contraction" rly:= (ry-j) pretend NNI; nly:= dim^rly; oly:= 1; zoly:= 1 rhy:= (j -1) pretend NNI; nhy:= dim^rhy ohy:= nly*dim; zohy:= zoly*nly rlx:= (rx-i) pretend NNI; nlx:= dim^rlx olx:= 1; zolx:= zohy*nhy rhx:= (i -1) pretend NNI; nhx:= dim^rhx ohx:= nlx*dim; zohx:= zolx*nlx z := new(nlx*nhx*nly*nhy, 0) for dxh in 1..nhx _ for xh in 0.. by ohx for zhx in 0.. by zohx repeat for dxl in 1..nlx _ for xl in xh.. by olx for zlx in zhx.. by zolx repeat for dyh in 1..nhy _ for yh in 0.. by ohy for zhy in zlx.. by zohy repeat for dyl in 1..nly _ for yl in yh.. by oly for zly in zhy.. by zoly repeat set_!(z, zly, 0) for k in 1..dim _ for xk in xl.. by nlx for yk in yl.. by nly repeat set_!(z, zly, get(z,zly)+get(x,xk)*get(y,yk)) z transpose x == transpose(x, 1, rank x) transpose(x, i, j) == rx := rank x i < 1 or i > rx or j < 1 or j > rx or i = j => error "Improper indicies for transposition" if i > j then (i,j) := (j,i) rl:= (rx- j) pretend NNI; nl:= dim^rl; zol:= 1; zoi := zol*nl rm:= (j-i-1) pretend NNI; nm:= dim^rm; zom:= nl*dim; zoj := zom*nm rh:= (i - 1) pretend NNI; nh:= dim^rh; zoh:= nl*nm*dim^2 z := new(#x, 0) for h in 1..nh for zh in 0.. by zoh repeat _ for m in 1..nm for zm in zh.. by zom repeat _ for l in 1..nl for zl in zm.. by zol repeat _ for p in 1..dim _ for zp in zl.. by zoi for xp in zl.. by zoj repeat for q in 1..dim _ for zq in zp.. by zoj for xq in xp.. by zoi repeat set_!(z, zq, get(x,xq)) z reindex(x, l) == nx := #x z: % := new(nx, 0) rx := rank x p := mkPerm(rx, l) xiv: INDEX := new(rx, 0) ziv: INDEX := new(rx, 0) -- Use permutation for i in 0..#x-1 repeat pi := index2int(permute_!(ziv, int2index(i,xiv),p)) set_!(z, pi, get(x,i)) z spad Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7319274953584694234-25px001.spad using old system compiler. BITENS abbreviates domain BiCartesianTensor ------------------------------------------------------------------------ initializing NRLIB BITENS for BiCartesianTensor compiling into NRLIB BITENS processing macro definition PERM ==> List Vector PositiveInteger processing macro definition INDEX ==> List Vector PositiveInteger processing macro definition get ==> elt(Rep,elt) processing macro definition set_! ==> elt(Rep,setelt) compiling exported sample : () ->$ Time: 0.06 SEC. compiling local int2index : (Integer,List Vector PositiveInteger) -> List Vector PositiveInteger ****** comp fails at level 6 with expression: ****** error in function int2index (SEQ (LET |qr| (|divide| |n| |dim|)) (LET |n| (|qr| |quotient|)) (|exit| 1 (LET ((|indv| 1) (|pretend| (+ (- |rnk| |i|) 1) (|PositiveInteger|))) (+ (|qr| |remainder|) 1)))) ****** level 6 ****** $x:= (elt indv (One))$m:= (Record (: quotient (Integer)) (: remainder (Integer))) \$f:= ((((|n| # # #) (|qr| #) (|remainder| #) (|quotient| #) ...))) >> Apparent user error: Cannot coerce indv of mode (List (Vector (PositiveInteger))) to mode (IndexedVector R (Zero))

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