dggevx - compute for a pair of N-by-N real nonsymmetric matrices (A,B)
SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, * ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, * RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, BWORK, * INFO) CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE INTEGER N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, INFO INTEGER IWORK(*) LOGICAL BWORK(*) DOUBLE PRECISION ABNRM, BBNRM DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*), VR(LDVR,*), LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*), WORK(*)
SUBROUTINE DGGEVX_64( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, * LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, * RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, BWORK, * INFO) CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE INTEGER*8 N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, INFO INTEGER*8 IWORK(*) LOGICAL*8 BWORK(*) DOUBLE PRECISION ABNRM, BBNRM DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*), VR(LDVR,*), LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*), WORK(*)
SUBROUTINE GGEVX( BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], B, * [LDB], ALPHAR, ALPHAI, BETA, VL, [LDVL], VR, [LDVR], ILO, IHI, * LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, [WORK], [LWORK], * [IWORK], [BWORK], [INFO]) CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE INTEGER :: N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, INFO INTEGER, DIMENSION(:) :: IWORK LOGICAL, DIMENSION(:) :: BWORK REAL(8) :: ABNRM, BBNRM REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, LSCALE, RSCALE, RCONDE, RCONDV, WORK REAL(8), DIMENSION(:,:) :: A, B, VL, VR
SUBROUTINE GGEVX_64( BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], B, * [LDB], ALPHAR, ALPHAI, BETA, VL, [LDVL], VR, [LDVR], ILO, IHI, * LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, [WORK], [LWORK], * [IWORK], [BWORK], [INFO]) CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE 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(:) :: ALPHAR, ALPHAI, BETA, LSCALE, RSCALE, RCONDE, RCONDV, WORK REAL(8), DIMENSION(:,:) :: A, B, VL, VR
#include <sunperf.h>
void dggevx(char balanc, char jobvl, char jobvr, char sense, int n, double *a, int lda, double *b, int ldb, double *alphar, double *alphai, double *beta, double *vl, int ldvl, double *vr, int ldvr, int *ilo, int *ihi, double *lscale, double *rscale, double *abnrm, double *bbnrm, double *rconde, double *rcondv, int *info);
void dggevx_64(char balanc, char jobvl, char jobvr, char sense, long n, double *a, long lda, double *b, long ldb, double *alphar, double *alphai, double *beta, double *vl, long ldvl, double *vr, long ldvr, long *ilo, long *ihi, double *lscale, double *rscale, double *abnrm, double *bbnrm, double *rconde, double *rcondv, long *info);
dggevx computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.
Optionally also, it 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).
= '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.
ALPHAI(j)
is zero, then
the j-th eigenvalue is real; if positive, then the j-th and
(j+1)-st eigenvalues are a complex conjugate pair, with
ALPHAI(j+1)
negative.
Note: the quotients ALPHAR(j)/BETA(j)
and ALPHAI(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, ALPHAR and ALPHAI 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. If the j-th eigenvalue is real, then
u(j)
= VL(:,j), the j-th column of VL. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
u(j)
= VL(:,j)+i*VL(:,j+1)
and u(j+1)
= VL(:,j)-i*VL(:,j+1).
Each eigenvector will be scaled so the largest component 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. If the j-th eigenvalue is real, then
v(j)
= VR(:,j), the j-th column of VR. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
v(j)
= VR(:,j)+i*VR(:,j+1)
and v(j+1)
= VR(:,j)-i*VR(:,j+1).
Each eigenvector will be scaled so the largest component 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.
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.
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(N+6)
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 ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j =INFO+1,...,N. > N: =N+1: other than QZ iteration failed in SHGEQZ.
=N+2: error return from STGEVC.
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.