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NAME
sgeevx - compute for an N-by-N real nonsymmetric matrix A,
the eigenvalues and, optionally, the left and/or right
eigenvectors
SYNOPSIS
SUBROUTINE SGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA,
WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE,
ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
REAL ABNRM
INTEGER IWORK( * )
REAL A( LDA, * ), RCONDE( * ), RCONDV( * ), SCALE( * ), VL(
LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ), WR(
* )
#include <sunperf.h>
void sgeevx(char balanc, char jobvl, char jobvr, char sense,
int n, float *sa, int lda, float *wr, float *wi,
float * vl, int ldvl, float *vr, int ldvr, int
*ilo, int *ihi, float *sscale, float *abnrm, float
*rconde, float *rcondv, int *info);
PURPOSE
SGEEVX computes for an N-by-N real nonsymmetric matrix A,
the eigenvalues and, optionally, the left and/or right
eigenvectors.
Optionally also, it computes a balancing transformation to
improve the conditioning of the eigenvalues and eigenvectors
(ILO, IHI, SCALE, and ABNRM), reciprocal condition numbers
for the eigenvalues (RCONDE), and reciprocal condition
numbers for the right
eigenvectors (RCONDV).
The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).
The computed eigenvectors are normalized to have Euclidean
norm equal to 1 and largest component real.
Balancing a matrix means permuting the rows and columns to
make it more nearly upper triangular, and applying a diago-
nal similarity transformation D * A * D**(-1), where D is a
diagonal matrix, to make its rows and columns closer in norm
and the condition numbers of its eigenvalues and eigenvec-
tors smaller. 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 explana-
tion of balancing, see section 4.10.2 of the LAPACK Users'
Guide.
ARGUMENTS
BALANC (input) CHARACTER*1
Indicates how the input matrix should be diago-
nally scaled and/or permuted to improve the condi-
tioning of its eigenvalues. = 'N': Do not diago-
nally scale or permute;
= 'P': Perform permutations to make the matrix
more nearly upper triangular. Do not diagonally
scale; = 'S': Diagonally scale the matrix, i.e.
replace A by D*A*D**(-1), where D is a diagonal
matrix chosen to make the rows and columns of A
more equal in norm. Do not permute; = 'B': Both
diagonally scale and permute A.
Computed reciprocal condition numbers will be for
the matrix after balancing and/or permuting. Per-
muting does not change condition numbers (in exact
arithmetic), but balancing does.
JOBVL (input) CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed. If
SENSE = 'E' or 'B', JOBVL must = 'V'.
JOBVR (input) CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed. If
SENSE = 'E' or 'B', JOBVR must = 'V'.
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are
computed. = 'N': None are computed;
= 'E': Computed for eigenvalues only;
= 'V': Computed for right eigenvectors only;
= 'B': Computed for eigenvalues and right eigen-
vectors.
If SENSE = 'E' or 'B', both left and right eigen-
vectors must also be computed (JOBVL = 'V' and
JOBVR = 'V').
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A. On exit, A has
been overwritten. If JOBVL = 'V' or JOBVR = 'V',
A contains the real Schur form of the balanced
version of the input matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >=
max(1,N).
WR (output) REAL array, dimension (N)
WI (output) REAL array, dimension (N) WR and
WI contain the real and imaginary parts, respec-
tively, of the computed eigenvalues. Complex con-
jugate pairs of eigenvalues will appear consecu-
tively with the eigenvalue having the positive
imaginary part first.
VL (output) REAL array, dimension (LDVL,N)
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 JOBVL =
'N', VL is not referenced. 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)-st 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).
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1;
if JOBVL = 'V', LDVL >= N.
VR (output) REAL array, dimension (LDVR,N)
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 JOBVR =
'N', VR is not referenced. 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)-st 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).
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1,
and if JOBVR = 'V', LDVR >= N.
ILO,IHI (output) INTEGER ILO and IHI are integer
values determined when A was balanced. The bal-
anced A(i,j) = 0 if I > J and J = 1,...,ILO-1 or I
= IHI+1,...,N.
SCALE (output) REAL array, dimension (N)
Details of the permutations and scaling factors
applied when balancing A. If P(j) is the index of
the row and column interchanged with row and
column j, and D(j) is the scaling factor applied
to row and column j, then SCALE(J) = P(J), for
J = 1,...,ILO-1 = D(J), for J = ILO,...,IHI =
P(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) REAL
The one-norm of the balanced matrix (the maximum
of the sum of absolute values of elements of any
column).
RCONDE (output) REAL array, dimension (N)
RCONDE(j) is the reciprocal condition number of
the j-th eigenvalue.
RCONDV (output) REAL array, dimension (N)
RCONDV(j) is the reciprocal condition number of
the j-th right eigenvector.
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. If SENSE = 'N'
or 'E', LWORK >= max(1,2*N), and if JOBVL = 'V' or
JOBVR = 'V', LWORK >= 3*N. If SENSE = 'V' or 'B',
LWORK >= N*(N+6). For good performance, LWORK
must generally be larger.
IWORK (workspace) INTEGER array, dimension (2*N-2)
If SENSE = 'N' or 'E', not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal value.
> 0: if INFO = i, the QR algorithm failed to com-
pute all the eigenvalues, and no eigenvectors or
condition numbers have been computed; elements
1:ILO-1 and i+1:N of WR and WI contain eigenvalues
which have converged.
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