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))