sggevx - N real nonsymmetric matrices (A,B) the generalized eigenvalues, and, optionally, the left and/or right generalized eigenvectors
SUBROUTINE SGGEVX(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(*) REAL ABNRM, BBNRM REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*), VR(LDVR,*), LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*), WORK(*) SUBROUTINE SGGEVX_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(*) REAL ABNRM, BBNRM REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*), VR(LDVR,*), LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*), WORK(*) F95 INTERFACE 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 :: ABNRM, BBNRM REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, LSCALE, RSCALE, RCONDE, RCONDV, WORK REAL, 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 :: ABNRM, BBNRM REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, LSCALE, RSCALE, RCONDE, RCONDV, WORK REAL, DIMENSION(:,:) :: A, B, VL, VR C INTERFACE #include <sunperf.h> void sggevx(char balanc, char jobvl, char jobvr, char sense, int n, float *a, int lda, float *b, int ldb, float *alphar, float *alphai, float *beta, float *vl, int ldvl, float *vr, int ldvr, int *ilo, int *ihi, float *lscale, float *rscale, float *abnrm, float *bbnrm, float *rconde, float *rcondv, int *info); void sggevx_64(char balanc, char jobvl, char jobvr, char sense, long n, float *a, long lda, float *b, long ldb, float *alphar, float *alphai, float *beta, float *vl, long ldvl, float *vr, long ldvr, long *ilo, long *ihi, float *lscale, float *rscale, float *abnrm, float *bbnrm, float *rconde, float *rcondv, long *info);
Oracle Solaris Studio Performance Library sggevx(3P)
NAME
sggevx - compute for a pair of N-by-N real nonsymmetric matrices (A,B)
the generalized eigenvalues, and, optionally, the left and/or right
generalized eigenvectors
SYNOPSIS
SUBROUTINE SGGEVX(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(*)
REAL ABNRM, BBNRM
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*),
VR(LDVR,*), LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*), WORK(*)
SUBROUTINE SGGEVX_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(*)
REAL ABNRM, BBNRM
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*),
VR(LDVR,*), LSCALE(*), RSCALE(*), RCONDE(*), RCONDV(*), WORK(*)
F95 INTERFACE
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 :: ABNRM, BBNRM
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, LSCALE, RSCALE, RCONDE,
RCONDV, WORK
REAL, 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 :: ABNRM, BBNRM
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, LSCALE, RSCALE, RCONDE,
RCONDV, WORK
REAL, DIMENSION(:,:) :: A, B, VL, VR
C INTERFACE
#include <sunperf.h>
void sggevx(char balanc, char jobvl, char jobvr, char sense, int n,
float *a, int lda, float *b, int ldb, float *alphar, float
*alphai, float *beta, float *vl, int ldvl, float *vr, int
ldvr, int *ilo, int *ihi, float *lscale, float *rscale, float
*abnrm, float *bbnrm, float *rconde, float *rcondv, int
*info);
void sggevx_64(char balanc, char jobvl, char jobvr, char sense, long n,
float *a, long lda, float *b, long ldb, float *alphar, float
*alphai, float *beta, float *vl, long ldvl, float *vr, long
ldvr, long *ilo, long *ihi, float *lscale, float *rscale,
float *abnrm, float *bbnrm, float *rconde, float *rcondv,
long *info);
PURPOSE
sggevx computes for a pair of N-by-N real nonsymmetric matrices (A,B)
the generalized eigenvalues, and optionally, the left and/or right gen-
eralized 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 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 real Schur form of the "bal-
anced" 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 real Schur form of the "bal-
anced" versions of the input A and B.
LDB (input)
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. If 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).
ALPHAI (output)
See the description of ALPHAR.
BETA (output)
See the description of ALPHAR.
VL (output)
If JOBVL = 'V', the left eigenvectors 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'.
LDVL (input)
The leading dimension of the matrix VL. LDVL >= 1, and if
JOBVL = 'V', LDVL >= N.
VR (output)
If JOBVR = 'V', the right eigenvectors 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'.
LDVR (input)
The leading dimension of the matrix VR. LDVR >= 1, and if
JOBVR = 'V', LDVR >= N.
ILO (output)
ILO and IHI are integer values 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)
See the description of ILO.
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. For a complex conjugate pair of eigenvalues two
consecutive elements of RCONDE are set to the same value.
Thus RCONDE(j), RCONDV(j), and the j-th columns of VL and VR
all correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected). If
SENSE = 'V', RCONDE is not referenced.
RCONDV (output)
If SENSE = 'V' or 'B', the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two consecu-
tive elements of RCONDV are set to the same value. If the ei-
genvalues 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.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >= max(1,6*N). If
SENSE = 'E', LWORK >= 12*N. If SENSE = 'V' or 'B', LWORK >=
2*N*N+12*N+16.
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.
IWORK (workspace)
dimension(N+6) 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 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.
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.
7 Nov 2015 sggevx(3P)