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)abbrev package SMITH SmithNormalForm
++ Author: Patrizia Gianni
++ Date Created: October 1992
++ Date Last Updated:
++ Basic Operations:
++ Related Domains: Matrix(R)
++ Also See:
++ AMS Classifications:
++ Keywords: matrix, canonical forms, linear algebra
++ Examples:
++ References:
++ Description:
++ \spadtype{SmithNormalForm} is a package
++ which provides some standard canonical forms for matrices.
SmithNormalForm(R,Row,Col,M) : Exports == Implementation where
R : EuclideanDomain
Row : FiniteLinearAggregate R
Col : FiniteLinearAggregate R
M : MatrixCategory(R,Row,Col)
I ==> Integer
NNI ==> NonNegativeInteger
HermiteForm ==> Record(Hermite:M,eqMat:M)
SmithForm ==> Record(Smith : M, leftEqMat : M, rightEqMat : M)
PartialV ==> Union(Col, "failed")
Both ==> Record(particular: PartialV, basis: List Col)
Exports == with
hermite : M -> M
++ \spad{hermite(m)} returns the Hermite normal form of the
++ matrix m.
completeHermite : M -> HermiteForm
++ \spad{completeHermite} returns a record that contains
++ the Hermite normal form H of the matrix and the equivalence matrix
++ U such that U*m = H
smith : M -> M
++ \spad{smith(m)} returns the Smith Normal form of the matrix m.
completeSmith : M -> SmithForm
++ \spad{completeSmith} returns a record that contains
++ the Smith normal form H of the matrix and the left and right
++ equivalence matrices U and V such that U*m*v = H
diophantineSystem : (M,Col) -> Both
++ \spad{diophantineSystem(A,B)} returns a particular integer solution and
++ an integer basis of the equation \spad{AX = B}.
Implementation == add
MATCAT1 ==> MatrixCategoryFunctions2(R,Row,Col,M,QF,Row2,Col2,M2)
MATCAT2 ==> MatrixCategoryFunctions2(QF,Row2,Col2,M2,R,Row,Col,M)
QF ==> Fraction R
Row2 ==> Vector QF
Col2 ==> Vector QF
M2 ==> Matrix QF
------ Local Functions -----
elRow1 : (M,I,I) -> M
elRow2 : (M,R,I,I) -> M
elColumn2 : (M,R,I,I) -> M
isDiagonal? : M -> Boolean
ijDivide : (SmithForm ,I,I) -> SmithForm
lastStep : SmithForm -> SmithForm
test1 : (M,Col,NNI) -> Union(NNI, "failed")
test2 : (M, Col,NNI,NNI) -> Union( Col, "failed")
-- inconsistent system : case 0 = c --
test1(sm:M,b:Col,m1 : NNI) : Union(NNI , "failed") ==
km:=m1
while zero? sm(km,km) repeat
if not zero?(b(km)) then return "failed"
km:= (km - 1) :: NNI
km
if Col has shallowlyMutable then
test2(sm : M ,b : Col, n1:NNI,dk:NNI) : Union( Col, "failed") ==
-- test divisibility --
sol:Col := new(n1,0)
for k in 1..dk repeat
if (c:=(b(k) exquo sm(k,k))) case "failed" then return "failed"
sol(k):= c::R
sol
-- test if the matrix is diagonal or pseudo-diagonal --
isDiagonal?(m : M) : Boolean ==
m1:= nrows m
n1:= ncols m
for i in 1..m1 repeat
for j in 1..n1 | (j ~= i) repeat
if not zero?(m(i,j)) then return false
true
-- elementary operation of first kind: exchange two rows --
elRow1(m:M,i:I,j:I) : M ==
vec:=row(m,i)
setRow!(m,i,row(m,j))
setRow!(m,j,vec)
m
-- elementary operation of second kind: add to row i--
-- a*row j (i~=j) --
elRow2(m : M,a:R,i:I,j:I) : M ==
vec:= map(a*#1,row(m,j))
vec:=map("+",row(m,i),vec)
setRow!(m,i,vec)
m
-- elementary operation of second kind: add to column i --
-- a*column j (i~=j) --
elColumn2(m : M,a:R,i:I,j:I) : M ==
vec:= map(a*#1,column(m,j))
vec:=map("+",column(m,i),vec)
setColumn!(m,i,vec)
m
-- modify SmithForm in such a way that the term m(i,i) --
-- divides the term m(j,j). m is diagonal --
ijDivide(sf : SmithForm , i : I,j : I) : SmithForm ==
m:=sf.Smith
mii:=m(i,i)
mjj:=m(j,j)
extGcd:=extendedEuclidean(mii,mjj)
d := extGcd.generator
mii:=(mii exquo d)::R
mjj := (mjj exquo d) :: R
-- add to row j extGcd.coef1*row i --
lMat:=elRow2(sf.leftEqMat,extGcd.coef1,j,i)
-- switch rows i and j --
lMat:=elRow1(lMat,i,j)
-- add to row j -mii*row i --
lMat := elRow2(lMat,-mii,j,i)
-- lMat := ijModify(mii,mjj,extGcd.coef1,extGcd.coef2,sf.leftEqMat,i,j)
m(j,j):= m(i,i) * mjj
m(i,i):= d
-- add to column i extGcd.coef2 * column j --
rMat := elColumn2(sf.rightEqMat,extGcd.coef2,i,j)
-- add to column j -mjj*column i --
rMat:=elColumn2(rMat,-mjj,j,i)
-- multiply by -1 column j --
setColumn!(rMat,j,map(-1 * #1,column(rMat,j)))
[m,lMat,rMat]
-- given a diagonal matrix compute its Smith form --
lastStep(sf : SmithForm) : SmithForm ==
m:=sf.Smith
m1:=min(nrows m,ncols m)
for i in 1..m1 while not zero?(mii:=m(i,i)) repeat
for j in i+1..m1 repeat
if (m(j,j) exquo mii) case "failed" then return
lastStep(ijDivide(sf,i,j))
sf
-- given m and t row-equivalent matrices, with t in upper triangular --
-- form compute the matrix u such that u*m=t --
findEqMat(m : M,t : M) : Record(Hermite : M, eqMat : M) ==
m1:=nrows m
n1:=ncols m
"and"/[zero? t(m1,j) for j in 1..n1] => -- there are 0 rows
if "and"/[zero? t(1,j) for j in 1..n1]
then return [m,scalarMatrix(m1,1)] -- m is the zero matrix
mm:=horizConcat(m,scalarMatrix(m1,1))
mmh:=rowEchelon mm
[subMatrix(mmh,1,m1,1,n1), subMatrix(mmh,1,m1,n1+1,n1+m1)]
u:M:=zero(m1,m1)
j: Integer :=1
while t(1,j)=0 repeat j:=j+1 -- there are 0 columns
t1:=copy t
mm:=copy m
if j>1 then
t1:=subMatrix(t,1,m1,j,n1)
mm:=subMatrix(m,1,m1,j,n1)
t11:=t1(1,1)
for i in 1..m1 repeat
u(i,1) := (mm(i,1) exquo t11) :: R
for j: local in 2..m1 repeat
j0:=j
tjj : R
while zero?(tjj:=t1(j,j0)) repeat j0:=j0+1
u(i,j) :=((mm(i,j0) - ("+"/[u(i,k) * t1(k,j0) for k in 1..(j-1)])) exquo
tjj) :: R
u1:M2:= map(#1 :: QF,u)$MATCAT1
[t,map(retract$QF,(inverse u1)::M2)$MATCAT2]
--- Hermite normal form of m ---
hermite(m:M) : M == rowEchelon m
-- Hermite normal form and equivalence matrix --
completeHermite(m : M) : Record(Hermite : M, eqMat : M) ==
findEqMat(m,rowEchelon m)
smith(m : M) : M == completeSmith(m).Smith
completeSmith(m : M) : Record(Smith : M, leftEqMat : M, rightEqMat : M) ==
cm1:=completeHermite m
leftm:=cm1.eqMat
m1:=cm1.Hermite
isDiagonal? m1 => lastStep([m1,leftm,scalarMatrix(ncols m,1)])
nr:=nrows m
cm1:=completeHermite transpose m1
rightm:= transpose cm1.eqMat
m1:=cm1.Hermite
isDiagonal? m1 =>
cm2:=lastStep([m1,leftm,rightm])
nrows(m:=cm2.Smith) = nr => cm2
[transpose m,cm2.leftEqMat, cm2.rightEqMat]
cm2:=completeSmith m1
cm2:=lastStep([cm2.Smith,transpose(cm2.rightEqMat)*leftm,
rightm*transpose(cm2.leftEqMat)])
nrows(m:=cm2.Smith) = nr => cm2
[transpose m, cm2.leftEqMat, cm2.rightEqMat]
-- Find the solution in R of the linear system mX = b --
diophantineSystem(m : M, b : Col) : Both ==
sf:=completeSmith m
sm:=sf.Smith
m1:=nrows sm
lm:=sf.leftEqMat
b1:Col:= lm* b
(t1:=test1(sm,b1,m1)) case "failed" => ["failed",empty()]
dk:=t1 :: NNI
n1:=ncols sm
(t2:=test2(sm,b1,n1,dk)) case "failed" => ["failed",empty()]
rm := sf.rightEqMat
sol:=rm*(t2 :: Col) -- particular solution
dk = n1 => [sol,list new(n1,0)]
lsol:List Col := [column(rm,i) for i in (dk+1)..n1]
[sol,lsol]
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