SHGEQZ(l) ) SHGEQZ(l)NAME
SHGEQZ - implement a single-/double-shift version of the QZ method for
finding the generalized eigenvalues w(j)=(ALPHAR(j) +
i*ALPHAI(j))/BETAR(j) of the equation det( A - w(i) B ) = 0 In addi‐
tion, the pair A,B may be reduced to generalized Schur form
SYNOPSIS
SUBROUTINE SHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
INFO )
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
PURPOSE
SHGEQZ implements a single-/double-shift version of the QZ method for
finding the generalized eigenvalues w(j)=(ALPHAR(j) +
i*ALPHAI(j))/BETAR(j) of the equation det( A - w(i) B ) = 0 In addi‐
tion, the pair A,B may be reduced to generalized Schur form: B is upper
triangular, and A is block upper triangular, where the diagonal blocks
are either 1-by-1 or 2-by-2, the 2-by-2 blocks having complex general‐
ized eigenvalues (see the description of the argument JOB.)
If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form
by applying one orthogonal tranformation (usually called Q) on the left
and another (usually called Z) on the right. The 2-by-2 upper-triangu‐
lar diagonal blocks of B corresponding to 2-by-2 blocks of A will be
reduced to positive diagonal matrices. (I.e., if A(j+1,j) is non-zero,
then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be positive.)
If JOB='E', then at each iteration, the same transformations are com‐
puted, but they are only applied to those parts of A and B which are
needed to compute ALPHAR, ALPHAI, and BETAR.
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal
transformations used to reduce (A,B) are accumulated into the arrays Q
and Z s.t.:
Q(in)A(in)Z(in)* = Q(out)A(out)Z(out)*
Q(in)B(in)Z(in)* = Q(out)B(out)Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
ARGUMENTS
JOB (input) CHARACTER*1
= 'E': compute only ALPHAR, ALPHAI, and BETA. A and B will not
necessarily be put into generalized Schur form. = 'S': put A
and B into generalized Schur form, as well as computing ALPHAR,
ALPHAI, and BETA.
COMPQ (input) CHARACTER*1
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the transpose of
the orthogonal tranformation that is applied to the left side
of A and B to reduce them to Schur form. = 'I': like
COMPQ='V', except that Q will be initialized to the identity
first.
COMPZ (input) CHARACTER*1
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the orthogonal
tranformation that is applied to the right side of A and B to
reduce them to Schur form. = 'I': like COMPZ='V', except that
Z will be initialized to the identity first.
N (input) INTEGER
The order of the matrices A, B, Q, and Z. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A is already upper
triangular in rows and columns 1:ILO-1 and IHI+1:N. 1 <= ILO
<= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the N-by-N upper Hessenberg matrix A. Elements below
the subdiagonal must be zero. If JOB='S', then on exit A and B
will have been simultaneously reduced to generalized Schur
form. If JOB='E', then on exit A will have been destroyed.
The diagonal blocks will be correct, but the off-diagonal por‐
tion will be meaningless.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B. Elements below
the diagonal must be zero. 2-by-2 blocks in B corresponding to
2-by-2 blocks in A will be reduced to positive diagonal form.
(I.e., if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and
B(j,j) and B(j+1,j+1) will be positive.) If JOB='S', then on
exit A and B will have been simultaneously reduced to Schur
form. If JOB='E', then on exit B will have been destroyed.
Elements corresponding to diagonal blocks of A will be correct,
but the off-diagonal portion will be meaningless.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHAR (output) REAL array, dimension (N)
ALPHAR(1:N) will be set to real parts of the diagonal elements
of A that would result from reducing A and B to Schur form and
then further reducing them both to triangular form using uni‐
tary transformations s.t. the diagonal of B was non-negative
real. Thus, if A(j,j) is in a 1-by-1 block (i.e.,
A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=A(j,j). Note that the
(real or complex) values (ALPHAR(j) + i*ALPHAI(j))/BETA(j),
j=1,...,N, are the generalized eigenvalues of the matrix pencil
A - wB.
ALPHAI (output) REAL array, dimension (N)
ALPHAI(1:N) will be set to imaginary parts of the diagonal ele‐
ments of A that would result from reducing A and B to Schur
form and then further reducing them both to triangular form
using unitary transformations s.t. the diagonal of B was non-
negative real. Thus, if A(j,j) is in a 1-by-1 block (i.e.,
A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0. Note that the (real or
complex) values (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N,
are the generalized eigenvalues of the matrix pencil A - wB.
BETA (output) REAL array, dimension (N)
BETA(1:N) will be set to the (real) diagonal elements of B that
would result from reducing A and B to Schur form and then fur‐
ther reducing them both to triangular form using unitary trans‐
formations s.t. the diagonal of B was non-negative real. Thus,
if A(j,j) is in a 1-by-1 block (i.e., A(j+1,j)=A(j,j+1)=0),
then BETA(j)=B(j,j). Note that the (real or complex) values
(ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the general‐
ized eigenvalues of the matrix pencil A - wB. (Note that
BETA(1:N) will always be non-negative, and no BETAI is neces‐
sary.)
Q (input/output) REAL array, dimension (LDQ, N)
If COMPQ='N', then Q will not be referenced. If COMPQ='V' or
'I', then the transpose of the orthogonal transformations which
are applied to A and B on the left will be applied to the array
Q on the right.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If COMPQ='V'
or 'I', then LDQ >= N.
Z (input/output) REAL array, dimension (LDZ, N)
If COMPZ='N', then Z will not be referenced. If COMPZ='V' or
'I', then the orthogonal transformations which are applied to A
and B on the right will be applied to the array Z on the right.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1. If COMPZ='V'
or 'I', then LDZ >= N.
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (A,B) is not in
Schur form, but ALPHAR(i), ALPHAI(i), and BETA(i),
i=INFO+1,...,N should be correct. = N+1,...,2*N: the shift
calculation failed. (A,B) is not in Schur form, but ALPHAR(i),
ALPHAI(i), and BETA(i), i=INFO-N+1,...,N should be correct. >
2*N: various "impossible" errors.
FURTHER DETAILS
Iteration counters:
JITER -- counts iterations.
IITER -- counts iterations run since ILAST was last
changed. This is therefore reset only when a 1-by-1 or
2-by-2 block deflates off the bottom.
LAPACK version 3.0 15 June 2000 SHGEQZ(l)
