Contents
sggevx - compute for a pair of N-by-N real nonsymmetric
matrices (A,B)
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);
sggevx 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 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 real 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 real Schur form of the "balanced" 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 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. For a complex
conjugate pair of eigenvalues two consecutive ele-
ments 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 consecutive elements of
RCONDV are set to the same value. If the eigen-
values 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 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 ALPHAR(j),
ALPHAI(j), and BETA(j) should be correct for
j=INFO+1,...,N. > N: =N+1: other than QZ itera-
tion 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.