/usr/share/axiom-20170501/src/algebra/MHROWRED.spad is in axiom-source 20170501-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 | )abbrev package MHROWRED ModularHermitianRowReduction
++ Author: Manuel Bronstein
++ Date Created: 22 February 1989
++ Date Last Updated: 24 November 1993
++ Description:
++ Modular hermitian row reduction.
-- should be moved into matrix whenever possible
ModularHermitianRowReduction(R) : SIG == CODE where
R : EuclideanDomain
Z ==> Integer
V ==> Vector R
M ==> Matrix R
REC ==> Record(val:R, cl:Z, rw:Z)
SIG ==> with
rowEch : M -> M
++ rowEch(m) computes a modular row-echelon form of m, finding
++ an appropriate modulus.
rowEchelon : (M, R) -> M
++ rowEchelon(m, d) computes a modular row-echelon form mod d of
++ [d ]
++ [ d ]
++ [ . ]
++ [ d]
++ [ M ]
++ where \spad{M = m mod d}.
rowEchLocal : (M, R) -> M
++ rowEchLocal(m,p) computes a modular row-echelon form of m, finding
++ an appropriate modulus over a local ring where p is the only prime.
rowEchelonLocal : (M, R, R) -> M
++ rowEchelonLocal(m, d, p) computes the row-echelon form of m
++ concatenated with d times the identity matrix
++ over a local ring where p is the only prime.
normalizedDivide : (R, R) -> Record(quotient:R, remainder:R)
++ normalizedDivide(n,d) returns a normalized quotient and
++ remainder such that consistently unique representatives
++ for the residue class are chosen, for example, positive remainders
CODE ==> add
order : (R, R) -> Z
vconc : (M, R) -> M
non0 : (V, Z) -> Union(REC, "failed")
nonzero?: V -> Boolean
mkMat : (M, List Z) -> M
diagSubMatrix: M -> Union(Record(val:R, mat:M), "failed")
determinantOfMinor: M -> R
enumerateBinomial: (List Z, Z, Z) -> List Z
nonzero? v == any?(s +-> s ^= 0, v)
-- returns [a, i, rown] if v = [0,...,0,a,0,...,0]
-- where a <> 0 and i is the index of a, "failed" otherwise.
non0(v, rown) ==
ans:REC
allZero:Boolean := true
for i in minIndex v .. maxIndex v repeat
if qelt(v, i) ^= 0 then
if allZero then
allZero := false
ans := [qelt(v, i), i, rown]
else return "failed"
allZero => "failed"
ans
-- returns a matrix made from the non-zero rows of x whose row number
-- is not in l
mkMat(x, l) ==
empty?(ll := [parts row(x, i)
for i in minRowIndex x .. maxRowIndex x |
(not member?(i, l)) and nonzero? row(x, i)]$List(List R)) =>
zero(1, ncols x)
matrix ll
-- returns [m, d] where m = x with the zero rows and the rows of
-- the diagonal of d removed, if x has a diagonal submatrix of d's,
-- "failed" otherwise.
diagSubMatrix x ==
l := [u::REC for i in minRowIndex x .. maxRowIndex x |
(u := non0(row(x, i), i)) case REC]
for a in removeDuplicates([r.val for r in l]$List(R)) repeat
{[r.cl for r in l | r.val = a]$List(Z)}$Set(Z) =
{[z for z in minColIndex x .. maxColIndex x]$List(Z)}$Set(Z)
=> return [a, mkMat(x, [r.rw for r in l | a = r.val])]
"failed"
-- returns a non-zero determinant of a minor of x of rank equal to
-- the number of columns of x, if there is one, 0 otherwise
determinantOfMinor x ==
-- do not compute a modulus for square matrices, since this is as
-- expensive as the Hermite reduction itself
(nr := nrows x) <= (nc := ncols x) => 0
lc := [i for i in minColIndex x .. maxColIndex x]$List(Integer)
lr := [i for i in minRowIndex x .. maxRowIndex x]$List(Integer)
for i in 1..(n := binomial(nr, nc)) repeat
(d := determinant x(enumerateBinomial(lr, nc, i), lc)) ^= 0 =>
j := i + 1 + (random()$Z rem (n - i))
return gcd(d, determinant x(enumerateBinomial(lr, nc, j), lc))
0
-- returns the i-th selection of m elements of l = (a1,...,an),
-- /n\
-- where 1 <= i <= | |
-- \m/
enumerateBinomial(l, m, i) ==
m1 := minIndex l - 1
zero?(m := m - 1) => [l(m1 + i)]
for j in 1..(n := #l) repeat
i <= (b := binomial(n - j, m)) =>
return concat(l(m1 + j), enumerateBinomial(rest(l, j), m, i))
i := i - b
error "Should not happen"
rowEch x ==
(u := diagSubMatrix x) case "failed" =>
zero?(d := determinantOfMinor x) => rowEchelon x
rowEchelon(x, d)
rowEchelon(u.mat, u.val)
vconc(y, m) ==
vertConcat(diagonalMatrix new(ncols y, m)$V, map(s +-> s rem m, y))
order(m, p) ==
zero? m => -1
for i in 0.. repeat
(mm := m exquo p) case "failed" => return i
m := mm::R
if R has IntegerNumberSystem then
normalizedDivide(n:R, d:R):Record(quotient:R, remainder:R) ==
qr := divide(n, d)
qr.remainder >= 0 => qr
d > 0 =>
qr.remainder := qr.remainder + d
qr.quotient := qr.quotient - 1
qr
qr.remainder := qr.remainder - d
qr.quotient := qr.quotient + 1
qr
else
normalizedDivide(n:R, d:R):Record(quotient:R, remainder:R) ==
divide(n, d)
rowEchLocal(x,p) ==
(u := diagSubMatrix x) case "failed" =>
zero?(d := determinantOfMinor x) => rowEchelon x
rowEchelonLocal(x, d, p)
rowEchelonLocal(u.mat, u.val, p)
rowEchelonLocal(y, m, p) ==
m := p**(order(m,p)::NonNegativeInteger)
x := vconc(y, m)
nrows := maxRowIndex x
ncols := maxColIndex x
minr := i := minRowIndex x
for j in minColIndex x .. ncols repeat
if i > nrows then leave x
rown := minr - 1
pivord : Integer
npivord : Integer
for k in i .. nrows repeat
qelt(x,k,j) = 0 => "next k"
npivord := order(qelt(x,k,j),p)
(rown = minr - 1) or (npivord < pivord) =>
rown := k
pivord := npivord
rown = minr - 1 => "enuf"
x := swapRows_!(x, i, rown)
(a, b, d) := extendedEuclidean(qelt(x,i,j), m)
qsetelt_!(x,i,j,d)
pivot := d
for k in j+1 .. ncols repeat
qsetelt_!(x,i,k, a * qelt(x,i,k) rem m)
for k in i+1 .. nrows repeat
zero? qelt(x,k,j) => "next k"
q := (qelt(x,k,j) exquo pivot) :: R
for k1 in j+1 .. ncols repeat
v2 := (qelt(x,k,k1) - q * qelt(x,i,k1)) rem m
qsetelt_!(x, k, k1, v2)
qsetelt_!(x, k, j, 0)
for k in minr .. i-1 repeat
zero? qelt(x,k,j) => "enuf"
qr := normalizedDivide(qelt(x,k,j), pivot)
qsetelt_!(x,k,j, qr.remainder)
for k1 in j+1 .. ncols x repeat
qsetelt_!(x,k,k1,
(qelt(x,k,k1) - qr.quotient * qelt(x,i,k1)) rem m)
i := i+1
x
if R has Field then
rowEchelon(y, m) == rowEchelon vconc(y, m)
else
rowEchelon(y, m) ==
x := vconc(y, m)
nrows := maxRowIndex x
ncols := maxColIndex x
minr := i := minRowIndex x
for j in minColIndex x .. ncols repeat
if i > nrows then leave
rown := minr - 1
for k in i .. nrows repeat
if (qelt(x,k,j) ^= 0) and ((rown = minr - 1) or
sizeLess?(qelt(x,k,j), qelt(x,rown,j))) then rown := k
rown = minr - 1 => "next j"
x := swapRows_!(x, i, rown)
for k in i+1 .. nrows repeat
zero? qelt(x,k,j) => "next k"
(a, b, d) := extendedEuclidean(qelt(x,i,j), qelt(x,k,j))
(b1, a1) :=
((qelt(x,i,j) exquo d)::R, (qelt(x,k,j) exquo d)::R)
-- a*b1+a1*b = 1
for k1 in j+1 .. ncols repeat
v1 := (a * qelt(x,i,k1) + b * qelt(x,k,k1)) rem m
v2 := (b1 * qelt(x,k,k1) - a1 * qelt(x,i,k1)) rem m
qsetelt_!(x, i, k1, v1)
qsetelt_!(x, k, k1, v2)
qsetelt_!(x, i, j, d)
qsetelt_!(x, k, j, 0)
un := unitNormal qelt(x,i,j)
qsetelt_!(x,i,j,un.canonical)
if un.associate ^= 1 then for jj in (j+1)..ncols repeat
qsetelt_!(x,i,jj,un.associate * qelt(x,i,jj))
xij := qelt(x,i,j)
for k in minr .. i-1 repeat
zero? qelt(x,k,j) => "next k"
qr := normalizedDivide(qelt(x,k,j), xij)
qsetelt_!(x,k,j, qr.remainder)
for k1 in j+1 .. ncols x repeat
qsetelt_!(x,k,k1,
(qelt(x,k,k1) - qr.quotient * qelt(x,i,k1)) rem m)
i := i+1
x
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