ZGEGV(3F) ZGEGV(3F)
ZGEGV - compute for a pair of N-by-N complex nonsymmetric matrices A and
B, the generalized eigenvalues (alpha, beta), and optionally,
SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL,
VR, LDVR, WORK, LWORK, RWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
LDVL, * ), VR( LDVR, * ), WORK( * )
ZGEGV computes for a pair of N-by-N complex nonsymmetric matrices A and
B, the generalized eigenvalues (alpha, beta), and optionally, the left
and/or right generalized eigenvectors (VL and VR).
A generalized eigenvalue for a pair of matrices (A,B) is, roughly
speaking, a scalar w or a ratio alpha/beta = w, such that A - w*B is
singular. It is usually represented as the pair (alpha,beta), as there
is a reasonable interpretation for beta=0, and even for both being zero.
A good beginning reference is the book, "Matrix Computations", by G.
Golub & C. van Loan (Johns Hopkins U. Press)
A right generalized eigenvector corresponding to a generalized eigenvalue
w for a pair of matrices (A,B) is a vector r such that (A - w B) r =
0 . A left generalized eigenvector is a vector l such that l**H * (A - w
B) = 0, where l**H is the
conjugate-transpose of l.
Note: this routine performs "full balancing" on A and B -- see "Further
Details", below.
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
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ZGEGV(3F) ZGEGV(3F)
A (input/output) COMPLEX*16 array, dimension (LDA, N)
On entry, the first of the pair of matrices whose generalized
eigenvalues and (optionally) generalized eigenvectors are to be
computed. On exit, the contents will have been destroyed. (For
a description of the contents of A on exit, see "Further
Details", below.)
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the second of the pair of matrices whose generalized
eigenvalues and (optionally) generalized eigenvectors are to be
computed. On exit, the contents will have been destroyed. (For
a description of the contents of B on exit, see "Further
Details", below.)
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N) On exit,
ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues.
Note: the quotients ALPHA(j)/BETA(j) may easily over- or
underflow, and BETA(j) may even be zero. Thus, the user should
avoid naively computing the ratio alpha/beta. However, ALPHA
will be always less than and usually comparable with norm(A) in
magnitude, and BETA always less than and usually comparable with
norm(B).
VL (output) COMPLEX*16 array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors. (See
"Purpose", above.) Each eigenvector will be scaled so the
largest component will have abs(real part) + abs(imag. part) = 1,
*except* that for eigenvalues with alpha=beta=0, a zero vector
will be returned as the corresponding eigenvector. Not
referenced if JOBVL = 'N'.
LDVL (input) INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL =
'V', LDVL >= N.
VR (output) COMPLEX*16 array, dimension (LDVR,N)
If JOBVL = 'V', the right generalized eigenvectors. (See
"Purpose", above.) Each eigenvector will be scaled so the
largest component will have abs(real part) + abs(imag. part) = 1,
*except* that for eigenvalues with alpha=beta=0, a zero vector
will be returned as the corresponding eigenvector. Not
referenced if JOBVR = 'N'.
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ZGEGV(3F) ZGEGV(3F)
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR =
'V', LDVR >= N.
WORK (workspace/output) COMPLEX*16 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(1,2*N). For good
performance, LWORK must generally be larger. To compute the
optimal value of LWORK, call ILAENV to get blocksizes (for
ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: NB -- MAX of the
blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; The optimal LWORK is
MAX( 2*N, N*(NB+1) ).
RWORK (workspace/output) DOUBLE PRECISION array, dimension (8*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
=1,...,N: The QZ iteration failed. No eigenvectors have been
calculated, but ALPHA(j) and BETA(j) should be correct for
j=INFO+1,...,N. > N: errors that usually indicate LAPACK
problems:
=N+1: error return from ZGGBAL
=N+2: error return from ZGEQRF
=N+3: error return from ZUNMQR
=N+4: error return from ZUNGQR
=N+5: error return from ZGGHRD
=N+6: error return from ZHGEQZ (other than failed iteration)
=N+7: error return from ZTGEVC
=N+8: error return from ZGGBAK (computing VL)
=N+9: error return from ZGGBAK (computing VR)
=N+10: error return from ZLASCL (various calls)
FURTHER DETAILS
Balancing
---------
This driver calls ZGGBAL to both permute and scale rows and columns of A
and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R
will be upper triangular except for the diagonal blocks A(i:j,i:j) and
B(i:j,i:j), with i and j as close together as possible. The diagonal
scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR,
DL*PL*B*PR*DR have elements close to one (except for the elements that
start out zero.)
After the eigenvalues and eigenvectors of the balanced matrices have been
computed, ZGGBAK transforms the eigenvectors back to what they would have
been (in perfect arithmetic) if they had not been balanced.
Contents of A and B on Exit
Page 3
ZGEGV(3F) ZGEGV(3F)
-------- -- - --- - -- ----
If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both),
then on exit the arrays A and B will contain the complex Schur form[*] of
the "balanced" versions of A and B. If no eigenvectors are computed,
then only the diagonal blocks will be correct.
[*] In other words, upper triangular form.
ZGEGV(3F) ZGEGV(3F)
ZGEGV - compute for a pair of N-by-N complex nonsymmetric matrices A and
B, the generalized eigenvalues (alpha, beta), and optionally,
SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL,
VR, LDVR, WORK, LWORK, RWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
LDVL, * ), VR( LDVR, * ), WORK( * )
ZGEGV computes for a pair of N-by-N complex nonsymmetric matrices A and
B, the generalized eigenvalues (alpha, beta), and optionally, the left
and/or right generalized eigenvectors (VL and VR).
A generalized eigenvalue for a pair of matrices (A,B) is, roughly
speaking, a scalar w or a ratio alpha/beta = w, such that A - w*B is
singular. It is usually represented as the pair (alpha,beta), as there
is a reasonable interpretation for beta=0, and even for both being zero.
A good beginning reference is the book, "Matrix Computations", by G.
Golub & C. van Loan (Johns Hopkins U. Press)
A right generalized eigenvector corresponding to a generalized eigenvalue
w for a pair of matrices (A,B) is a vector r such that (A - w B) r =
0 . A left generalized eigenvector is a vector l such that l**H * (A - w
B) = 0, where l**H is the
conjugate-transpose of l.
Note: this routine performs "full balancing" on A and B -- see "Further
Details", below.
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
Page 1
ZGEGV(3F) ZGEGV(3F)
A (input/output) COMPLEX*16 array, dimension (LDA, N)
On entry, the first of the pair of matrices whose generalized
eigenvalues and (optionally) generalized eigenvectors are to be
computed. On exit, the contents will have been destroyed. (For
a description of the contents of A on exit, see "Further
Details", below.)
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the second of the pair of matrices whose generalized
eigenvalues and (optionally) generalized eigenvectors are to be
computed. On exit, the contents will have been destroyed. (For
a description of the contents of B on exit, see "Further
Details", below.)
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N) On exit,
ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues.
Note: the quotients ALPHA(j)/BETA(j) may easily over- or
underflow, and BETA(j) may even be zero. Thus, the user should
avoid naively computing the ratio alpha/beta. However, ALPHA
will be always less than and usually comparable with norm(A) in
magnitude, and BETA always less than and usually comparable with
norm(B).
VL (output) COMPLEX*16 array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors. (See
"Purpose", above.) Each eigenvector will be scaled so the
largest component will have abs(real part) + abs(imag. part) = 1,
*except* that for eigenvalues with alpha=beta=0, a zero vector
will be returned as the corresponding eigenvector. Not
referenced if JOBVL = 'N'.
LDVL (input) INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL =
'V', LDVL >= N.
VR (output) COMPLEX*16 array, dimension (LDVR,N)
If JOBVL = 'V', the right generalized eigenvectors. (See
"Purpose", above.) Each eigenvector will be scaled so the
largest component will have abs(real part) + abs(imag. part) = 1,
*except* that for eigenvalues with alpha=beta=0, a zero vector
will be returned as the corresponding eigenvector. Not
referenced if JOBVR = 'N'.
Page 2
ZGEGV(3F) ZGEGV(3F)
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR =
'V', LDVR >= N.
WORK (workspace/output) COMPLEX*16 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(1,2*N). For good
performance, LWORK must generally be larger. To compute the
optimal value of LWORK, call ILAENV to get blocksizes (for
ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: NB -- MAX of the
blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; The optimal LWORK is
MAX( 2*N, N*(NB+1) ).
RWORK (workspace/output) DOUBLE PRECISION array, dimension (8*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
=1,...,N: The QZ iteration failed. No eigenvectors have been
calculated, but ALPHA(j) and BETA(j) should be correct for
j=INFO+1,...,N. > N: errors that usually indicate LAPACK
problems:
=N+1: error return from ZGGBAL
=N+2: error return from ZGEQRF
=N+3: error return from ZUNMQR
=N+4: error return from ZUNGQR
=N+5: error return from ZGGHRD
=N+6: error return from ZHGEQZ (other than failed iteration)
=N+7: error return from ZTGEVC
=N+8: error return from ZGGBAK (computing VL)
=N+9: error return from ZGGBAK (computing VR)
=N+10: error return from ZLASCL (various calls)
FURTHER DETAILS
Balancing
---------
This driver calls ZGGBAL to both permute and scale rows and columns of A
and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R
will be upper triangular except for the diagonal blocks A(i:j,i:j) and
B(i:j,i:j), with i and j as close together as possible. The diagonal
scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR,
DL*PL*B*PR*DR have elements close to one (except for the elements that
start out zero.)
After the eigenvalues and eigenvectors of the balanced matrices have been
computed, ZGGBAK transforms the eigenvectors back to what they would have
been (in perfect arithmetic) if they had not been balanced.
Contents of A and B on Exit
Page 3
ZGEGV(3F) ZGEGV(3F)
-------- -- - --- - -- ----
If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both),
then on exit the arrays A and B will contain the complex Schur form[*] of
the "balanced" versions of A and B. If no eigenvectors are computed,
then only the diagonal blocks will be correct.
[*] In other words, upper triangular form.
PPPPaaaaggggeeee 4444 [ Back ]
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