cggevx - 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 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(*)
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
#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);
cggevx 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 reciprocal 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).
= 'N': do not diagonally scale or permute;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale. Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
= 'E': computed for eigenvalues only;
= 'V': computed for eigenvectors only;
= 'B': computed for eigenvalues and eigenvectors.
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).
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'.
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'.
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.
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.
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.
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 interchanges are made is N to IHI+1,
then 1 to ILO-1.
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(1) returns the optimal LWORK.
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.
dimension(6*N)
Real workspace.
dimension(N+2)
If SENSE = 'E', IWORK is not referenced.
dimension(N)
If SENSE = 'N', BWORK is not referenced.
= 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.
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.