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)abbrev package PSEUDLIN PseudoLinearNormalForm
++ Normal forms of pseudo-linear operators
++ Author: Bruno Zuercher
++ Date Created: November 1993
++ Date Last Updated: 12 April 1994
++ Description:
++ PseudoLinearNormalForm provides a function for computing a block-companion
++ form for pseudo-linear operators.
PseudoLinearNormalForm(K:Field): Exports == Implementation where
ER ==> Record(C: Matrix K, g: Vector K)
REC ==> Record(R: Matrix K, A: Matrix K, Ainv: Matrix K)
Exports ==> with
normalForm: (Matrix K, Automorphism K, K -> K) -> REC
++ normalForm(M, sig, der) returns \spad{[R, A, A^{-1}]} such that
++ the pseudo-linear operator whose matrix in the basis \spad{y} is
++ \spad{M} had matrix \spad{R} in the basis \spad{z = A y}.
++ \spad{der} is a \spad{sig}-derivation.
changeBase: (Matrix K, Matrix K, Automorphism K, K -> K) -> Matrix K
++ changeBase(M, A, sig, der): computes the new matrix of a pseudo-linear
++ transform given by the matrix M under the change of base A
companionBlocks: (Matrix K, Vector K) -> List ER
++ companionBlocks(m, v) returns \spad{[[C_1, g_1],...,[C_k, g_k]]}
++ such that each \spad{C_i} is a companion block and
++ \spad{m = diagonal(C_1,...,C_k)}.
Implementation ==> add
normalForm0: (Matrix K, Automorphism K, Automorphism K, K -> K) -> REC
mulMatrix: (Integer, Integer, K) -> Matrix K
-- mulMatrix(N, i, a): under a change of base with the resulting matrix of
-- size N*N the following operations are performed:
-- D1: column i will be multiplied by sig(a)
-- D2: row i will be multiplied by 1/a
-- D3: addition of der(a)/a to the element at position (i,i)
addMatrix: (Integer, Integer, Integer, K) -> Matrix K
-- addMatrix(N, i, k, a): under a change of base with the resulting matrix
-- of size N*N the following operations are performed:
-- C1: addition of column i multiplied by sig(a) to column k
-- C2: addition of row k multiplied by -a to row i
-- C3: addition of -a*der(a) to the element at position (i,k)
permutationMatrix: (Integer, Integer, Integer) -> Matrix K
-- permutationMatrix(N, i, k): under a change of base with the resulting
-- permutation matrix of size N*N the following operations are performed:
-- P1: columns i and k will be exchanged
-- P2: rows i and k will be exchanged
inv: Matrix K -> Matrix K
-- inv(M): computes the inverse of a invertable matrix M.
-- avoids possible type conflicts
inv m == inverse(m) :: Matrix K
changeBase(M, A, sig, der) == inv(A) * (M * map(sig #1, A) + map(der, A))
normalForm(M, sig, der) == normalForm0(M, sig, inv sig, der)
companionBlocks(R, w) ==
-- decomposes the rational matrix R into single companion blocks
-- and the inhomogenity w as well
i:Integer := 1
n := nrows R
l:List(ER) := empty()
while i <= n repeat
j := i
while j+1 <= n and R(j,j+1) = 1 repeat j := j+1
--split block now
v:Vector K := new((j-i+1)::NonNegativeInteger, 0)
for k in i..j repeat v(k-i+1) := w k
l := concat([subMatrix(R,i,j,i,j), v], l)
i := j+1
l
normalForm0(M, sig, siginv, der) ==
-- the changes of base will be incremented in B and Binv,
-- where B**(-1)=Binv; E defines an elementary matrix
B, Binv, E : Matrix K
recOfMatrices : REC
N := nrows M
B := diagonalMatrix [1 for k in 1..N]
Binv := copy B
-- avoid unnecessary recursion
if diagonal?(M) then return [M, B, Binv]
i : Integer := 1
while i < N repeat
j := i + 1
while j <= N and M(i, j) = 0 repeat j := j + 1
if j <= N then
-- expand companionblock by lemma 5
if j ~= i+1 then
-- perform first a permutation
E := permutationMatrix(N, i+1, j)
M := changeBase(M, E, sig, der)
B := B*E
Binv := E*Binv
-- now is M(i, i+1) ~= 0
E := mulMatrix(N, i+1, siginv inv M(i,i+1))
M := changeBase(M, E, sig, der)
B := B*E
Binv := inv(E)*Binv
for j: local in 1..N repeat
if j ~= i+1 then
E := addMatrix(N, i+1, j, siginv(-M(i,j)))
M := changeBase(M, E, sig, der)
B := B*E
Binv := inv(E)*Binv
i := i + 1
else
-- apply lemma 6
for j: local in i..2 by -1 repeat
for k in (i+1)..N repeat
E := addMatrix(N, k, j-1, M(k,j))
M := changeBase(M, E, sig, der)
B := B*E
Binv := inv(E)*Binv
j := i + 1
while j <= N and M(j,1) = 0 repeat j := j + 1
if j <= N then
-- expand companionblock by lemma 8
E := permutationMatrix(N, 1, j)
M := changeBase(M, E, sig, der)
B := B*E
Binv := E*Binv
-- start again to establish rational form
i := 1
else
-- split a direct factor
recOfMatrices :=
normalForm(subMatrix(M, i+1, N, i+1, N), sig, der)
setsubMatrix!(M, i+1, i+1, recOfMatrices.R)
E := diagonalMatrix [1 for k in 1..N]
setsubMatrix!(E, i+1, i+1, recOfMatrices.A)
B := B*E
setsubMatrix!(E, i+1, i+1, recOfMatrices.Ainv)
Binv := E*Binv
-- M in blockdiagonalform, stop program
i := N
[M, B, Binv]
mulMatrix(N, i, a) ==
M : Matrix K := diagonalMatrix [1 for j in 1..N]
M(i, i) := a
M
addMatrix(N, i, k, a) ==
A : Matrix K := diagonalMatrix [1 for j in 1..N]
A(i, k) := a
A
permutationMatrix(N, i, k) ==
P : Matrix K := diagonalMatrix [1 for j in 1..N]
P(i, i) := P(k, k) := 0
P(i, k) := P(k, i) := 1
P
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