Contents
zggevx - compute for a pair of N-by-N complex nonsymmetric
matrices (A,B) the generalized eigenvalues, and optionally,
the left and/or right generalized eigenvectors
SUBROUTINE ZGGEVX(BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM,
BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO)
CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE
DOUBLE COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*),
VL(LDVL,*), VR(LDVR,*), WORK(*)
INTEGER N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, INFO
INTEGER IWORK(*)
LOGICAL BWORK(*)
DOUBLE PRECISION ABNRM, BBNRM
DOUBLE PRECISION LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*),
RWORK(*)
SUBROUTINE ZGGEVX_64(BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM,
BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO)
CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE
DOUBLE COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*),
VL(LDVL,*), VR(LDVR,*), WORK(*)
INTEGER*8 N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, INFO
INTEGER*8 IWORK(*)
LOGICAL*8 BWORK(*)
DOUBLE PRECISION ABNRM, BBNRM
DOUBLE PRECISION LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*),
RWORK(*)
F95 INTERFACE
SUBROUTINE GGEVX(BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], B, [LDB],
ALPHA, BETA, VL, [LDVL], VR, [LDVR], ILO, IHI, LSCALE, RSCALE,
ABNRM, BBNRM, RCONDE, RCONDV, [WORK], [LWORK], [RWORK], [IWORK],
[BWORK], [INFO])
CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE
COMPLEX(8), DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, VL, VR
INTEGER :: N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL, DIMENSION(:) :: BWORK
REAL(8) :: ABNRM, BBNRM
REAL(8), DIMENSION(:) :: LSCALE, RSCALE, RCONDE, RCONDV,
RWORK
SUBROUTINE GGEVX_64(BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], B,
[LDB], ALPHA, BETA, VL, [LDVL], VR, [LDVR], ILO, IHI, LSCALE,
RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, [WORK], [LWORK], [RWORK],
[IWORK], [BWORK], [INFO])
CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE
COMPLEX(8), DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX(8), DIMENSION(:,:) :: A, B, VL, VR
INTEGER(8) :: N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK
LOGICAL(8), DIMENSION(:) :: BWORK
REAL(8) :: ABNRM, BBNRM
REAL(8), DIMENSION(:) :: LSCALE, RSCALE, RCONDE, RCONDV,
RWORK
C INTERFACE
#include <sunperf.h>
void zggevx(char balanc, char jobvl, char jobvr, char sense,
int n, doublecomplex *a, int lda, doublecomplex
*b, int ldb, doublecomplex *alpha, doublecomplex
*beta, doublecomplex *vl, int ldvl, doublecomplex
*vr, int ldvr, int *ilo, int *ihi, double *lscale,
double *rscale, double *abnrm, double *bbnrm, dou-
ble *rconde, double *rcondv, int *info);
void zggevx_64(char balanc, char jobvl, char jobvr, char
sense, long n, doublecomplex *a, long lda, doub-
lecomplex *b, long ldb, doublecomplex *alpha,
doublecomplex *beta, doublecomplex *vl, long ldvl,
doublecomplex *vr, long ldvr, long *ilo, long
*ihi, double *lscale, double *rscale, double
*abnrm, double *bbnrm, double *rconde, double
*rcondv, long *info);
zggevx computes for a pair of N-by-N complex nonsymmetric
matrices (A,B) the generalized eigenvalues, and optionally,
the left and/or right generalized eigenvectors.
Optionally, it also computes a balancing transformation to
improve the conditioning of the eigenvalues and eigenvectors
(ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal
condition numbers for the eigenvalues (RCONDE), and recipro-
cal condition numbers for the right eigenvectors (RCONDV).
A generalized eigenvalue for a pair of matrices (A,B) is a
scalar lambda or a ratio alpha/beta = lambda, such that A -
lambda*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.
The right eigenvector v(j) corresponding to the eigenvalue
lambda(j) of (A,B) satisfies
A * v(j) = lambda(j) * B * v(j) .
The left eigenvector u(j) corresponding to the eigenvalue
lambda(j) of (A,B) satisfies
u(j)**H * A = lambda(j) * u(j)**H * B.
where u(j)**H is the conjugate-transpose of u(j).
BALANC (input)
Specifies the balance option to be performed:
= 'N': do not diagonally scale or permute;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale. Computed recipro-
cal condition numbers will be for the matrices
after permuting and/or balancing. Permuting does
not change condition numbers (in exact arith-
metic), but balancing does.
JOBVL (input)
= 'N': do not compute the left generalized eigen-
vectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input)
= 'N': do not compute the right generalized
eigenvectors;
= 'V': compute the right generalized eigenvec-
tors.
SENSE (input)
Determines which reciprocal condition numbers are
computed. = 'N': none are computed;
= 'E': computed for eigenvalues only;
= 'V': computed for eigenvectors only;
= 'B': computed for eigenvalues and eigenvectors.
N (input) The order of the matrices A, B, VL, and VR. N >=
0.
A (input/output)
On entry, the matrix A in the pair (A,B). On
exit, A has been overwritten. If JOBVL='V' or
JOBVR='V' or both, then A contains the first part
of the complex Schur form of the "balanced" ver-
sions of the input A and B.
LDA (input)
The leading dimension of A. LDA >= max(1,N).
B (input/output)
On entry, the matrix B in the pair (A,B). On
exit, B has been overwritten. If JOBVL='V' or
JOBVR='V' or both, then B contains the second part
of the complex Schur form of the "balanced" ver-
sions of the input A and B.
LDB (input)
The leading dimension of B. LDB >= max(1,N).
ALPHA (output)
On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
generalized eigenvalues.
Note: the quotient 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).
BETA (output)
See description of ALPHA.
VL (output)
If JOBVL = 'V', the left generalized eigenvectors
u(j) are stored one after another in the columns
of VL, in the same order as their eigenvalues.
Each eigenvector will be scaled so the largest
component will have abs(real part) + abs(imag.
part) = 1. Not referenced if JOBVL = 'N'.
LDVL (input)
The leading dimension of the matrix VL. LDVL >= 1,
and if JOBVL = 'V', LDVL >= N.
VR (output)
If JOBVR = 'V', the right generalized eigenvectors
v(j) are stored one after another in the columns
of VR, in the same order as their eigenvalues.
Each eigenvector will be scaled so the largest
component will have abs(real part) + abs(imag.
part) = 1. Not referenced if JOBVR = 'N'.
LDVR (input)
The leading dimension of the matrix VR. LDVR >= 1,
and if JOBVR = 'V', LDVR >= N.
ILO (output)
ILO is an integer value such that on exit A(i,j) =
0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i
= IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1
and IHI = N.
IHI (output)
IHI is an integer value such that on exit A(i,j) =
0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i
= IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1
and IHI = N.
LSCALE (output)
Details of the permutations and scaling factors
applied to the left side of A and B. If PL(j) is
the index of the row interchanged with row j, and
DL(j) is the scaling factor applied to row j, then
LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j)
for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N.
The order in which the interchanges are made is N
to IHI+1, then 1 to ILO-1.
RSCALE (output)
Details of the permutations and scaling factors
applied to the right side of A and B. If PR(j) is
the index of the column interchanged with column
j, and DR(j) is the scaling factor applied to
column j, then RSCALE(j) = PR(j) for j =
1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j)
for j = IHI+1,...,N The order in which the inter-
changes are made is N to IHI+1, then 1 to ILO-1.
ABNRM (output)
The one-norm of the balanced matrix A.
BBNRM (output)
The one-norm of the balanced matrix B.
RCONDE (output)
If SENSE = 'E' or 'B', the reciprocal condition
numbers of the selected eigenvalues, stored in
consecutive elements of the array. If SENSE =
'V', RCONDE is not referenced.
RCONDV (output)
If JOB = 'V' or 'B', the estimated reciprocal con-
dition numbers of the selected eigenvectors,
stored in consecutive elements of the array. If
the eigenvalues cannot be reordered to compute
RCONDV(j), RCONDV(j) is set to 0; this can only
occur when the true value would be very small any-
way. If SENSE = 'E', RCONDV is not referenced.
Not referenced if JOB = 'E'.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >=
max(1,2*N). If SENSE = 'N' or 'E', LWORK >= 2*N.
If SENSE = 'V' or 'B', LWORK >= 2*N*N+2*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.
RWORK (workspace)
dimension(6*N) Real workspace.
IWORK (workspace)
dimension(N+2) If SENSE = 'E', IWORK is not refer-
enced.
BWORK (workspace)
dimension(N) If SENSE = 'N', BWORK is not refer-
enced.
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal value.
= 1,...,N: The QZ iteration failed. No eigenvec-
tors have been calculated, but ALPHA(j) and
BETA(j) should be correct for j=INFO+1,...,N. >
N: =N+1: other than QZ iteration failed in
CHGEQZ.
=N+2: error return from CTGEVC.
Balancing a matrix pair (A,B) includes, first, permuting
rows and columns to isolate eigenvalues, second, applying
diagonal similarity transformation to the rows and columns
to make the rows and columns as close in norm as possible.
The computed reciprocal condition numbers correspond to the
balanced matrix. Permuting rows and columns will not change
the condition numbers (in exact arithmetic) but diagonal
scaling will. For further explanation of balancing, see
section 4.11.1.2 of LAPACK Users' Guide.
An approximate error bound on the chordal distance between
the i-th computed generalized eigenvalue w and the
corresponding exact eigenvalue lambda is
hord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
An approximate error bound for the angle between the i-th
computed eigenvector VL(i) or VR(i) is given by
PS * norm(ABNRM, BBNRM) / DIF(i).
For further explanation of the reciprocal condition numbers
RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide.