cggevx(3P) Sun Performance Library cggevx(3P)NAMEcggevx - 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
SYNOPSIS
SUBROUTINE CGGEVX(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
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(*)
REAL ABNRM, BBNRM
REAL LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*), RWORK(*)
SUBROUTINE CGGEVX_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
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(*)
REAL ABNRM, BBNRM
REAL 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, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, VL, VR
INTEGER :: N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL, DIMENSION(:) :: BWORK
REAL :: ABNRM, BBNRM
REAL, 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, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, 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 :: ABNRM, BBNRM
REAL, DIMENSION(:) :: LSCALE, RSCALE, RCONDE, RCONDV, RWORK
C INTERFACE
#include <sunperf.h>
void cggevx(char balanc, char jobvl, char jobvr, char sense, int n,
complex *a, int lda, complex *b, int ldb, complex *alpha,
complex *beta, complex *vl, int ldvl, complex *vr, int ldvr,
int *ilo, int *ihi, float *lscale, float *rscale, float
*abnrm, float *bbnrm, float *rconde, float *rcondv, int
*info);
void cggevx_64(char balanc, char jobvl, char jobvr, char sense, long n,
complex *a, long lda, complex *b, long ldb, complex *alpha,
complex *beta, complex *vl, long ldvl, complex *vr, long
ldvr, long *ilo, long *ihi, float *lscale, float *rscale,
float *abnrm, float *bbnrm, float *rconde, float *rcondv,
long *info);
PURPOSEcggevx 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 eigen‐
values (RCONDE), and reciprocal condition numbers for the right eigen‐
vectors (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 singu‐
lar. 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).
ARGUMENTS
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 reciprocal condi‐
tion numbers will be for the matrices after permuting and/or
balancing. Permuting does not change condition numbers (in
exact arithmetic), but balancing does.
JOBVL (input)
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input)
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
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" versions 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" versions 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. How‐
ever, ALPHA will be always less than and usually comparable
with norm(A) in magnitude, and BETA always less than and usu‐
ally 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 col‐
umn interchanged with column j, and DR(j) is the scaling fac‐
tor 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 interchanges 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 condition num‐
bers of the selected eigenvectors, stored in consecutive ele‐
ments 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 anyway. 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. If SENSE = 'N' or 'E',
LWORK >= 2*N+1. If SENSE = 'V' or 'B', LWORK > 2*N*N+2*N+1.
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 referenced.
BWORK (workspace)
dimension(N) If SENSE = 'N', BWORK is not referenced.
INFO (output)
= 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: =N+1: other than QZ iteration
failed in CHGEQZ.
=N+2: error return from CTGEVC.
FURTHER DETAILS
Balancing a matrix pair (A,B) includes, first, permuting rows and col‐
umns 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 eigenval‐
ue 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.
6 Mar 2009 cggevx(3P)
