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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 addition, 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 B is upper
triangular, and A is block upper triangular, where the diag-
onal blocks are either 1x1 or 2x2, the 2x2 blocks having
complex generalized eigenvalues (see the description of the
argument JOB.)
If JOB='S', then the pair (A,B) is simultaneously reduced to
Schur form using one orthogonal transformation (usually
called Q) on the left and another (usually called Z) on the
right. The 2x2 upper-triangular diagonal blocks of B
corresponding to 2x2 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 posi-
tive.)
If JOB='E', then at each iteration, the same transformations
are computed, 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 accumu-
lated 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 General-
ized 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 transformation 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 transformation 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 ini-
tialized to the identity first.
N (input) INTEGER
The number of rows and columns in the matrices A, B,
Q, and Z. N must be at least 0.
ILO (input) INTEGER
Columns 1 through ILO-1 of A and B are assumed to be
in upper triangular form already, and will not be
modified. ILO must be at least 1.
IHI (input) INTEGER
Rows IHI+1 through N of A and B are assumed to be in
upper triangular form already, and will not be
touched. IHI may not be greater than N.
A (input/output) REAL array, dimension (LDA, N)
On entry, the N x N upper Hessenberg matrix A.
Entries below the subdiagonal must be zero. If
JOB='S', then on exit A and B will have been simul-
taneously 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 portion 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 x N upper triangular matrix B.
Entries below the diagonal must be zero. 2x2 blocks
in B corresponding to 2x2 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. Entries 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 diago-
nal elements of A that would result from reducing A
and B to Schur form and then further reducing them
both to triangular form using unitary transforma-
tions s.t. the diagonal of B was non-negative real.
Thus, if A(j,j) is in a 1x1 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 elements of A that would result from reduc-
ing A and B to Schur form and then further reducing
them both to triangular form using unitary transfor-
mations s.t. the diagonal of B was non-negative
real. Thus, if A(j,j) is in a 1x1 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 ele-
ments of B 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 1x1 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 generalized
eigenvalues of the matrix pencil A - wB. (Note that
BETA(1:N) will always be non-negative, and no BETAI
is necessary.)
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 orthogo-
nal 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 must be
at least 1. If COMPQ='V' or 'I', then LDQ must also
be at least 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 transforma-
tions 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 must be
at least 1. If COMPZ='V' or 'I', then LDZ must also
be at least N.
WORK (workspace) REAL array, dimension (LWORK)
On exit, if INFO is not negative, WORK(1) will be
set to the optimal size of the array WORK.
LWORK (input) INTEGER
The number of elements in WORK. It must be at least
max( 1, N ).
INFO (output) INTEGER
< 0: if INFO = -i, the i-th argument had an illegal
value
= 0: successful exit.
= 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 1x1
or
2x2 block deflates off the bottom.