sgeevx(3P) Sun Performance Library sgeevx(3P)NAMEsgeevx - compute for an N-by-N real nonsymmetric matrix A, the eigen‐
values 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, RCONE, RCONV, WORK,
LDWORK, IWORK2, INFO)
CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE
INTEGER N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
INTEGER IWORK2(*)
REAL ABNRM
REAL A(LDA,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), SCALE(*),
RCONE(*), RCONV(*), WORK(*)
SUBROUTINE SGEEVX_64(BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONE, RCONV, WORK,
LDWORK, IWORK2, INFO)
CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE
INTEGER*8 N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
INTEGER*8 IWORK2(*)
REAL ABNRM
REAL A(LDA,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), SCALE(*),
RCONE(*), RCONV(*), WORK(*)
F95 INTERFACE
SUBROUTINE GEEVX(BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], WR, WI,
VL, [LDVL], VR, [LDVR], ILO, IHI, SCALE, ABNRM, RCONE, RCONV,
[WORK], [LDWORK], [IWORK2], [INFO])
CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE
INTEGER :: N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
INTEGER, DIMENSION(:) :: IWORK2
REAL :: ABNRM
REAL, DIMENSION(:) :: WR, WI, SCALE, RCONE, RCONV, WORK
REAL, DIMENSION(:,:) :: A, VL, VR
SUBROUTINE GEEVX_64(BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], WR,
WI, VL, [LDVL], VR, [LDVR], ILO, IHI, SCALE, ABNRM, RCONE, RCONV,
[WORK], [LDWORK], [IWORK2], [INFO])
CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE
INTEGER(8) :: N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK2
REAL :: ABNRM
REAL, DIMENSION(:) :: WR, WI, SCALE, RCONE, RCONV, WORK
REAL, DIMENSION(:,:) :: A, VL, VR
C INTERFACE
#include <sunperf.h>
void sgeevx(char balanc, char jobvl, char jobvr, char sense, int n,
float *a, int lda, float *wr, float *wi, float *vl, int ldvl,
float *vr, int ldvr, int *ilo, int *ihi, float *scale, float
*abnrm, float *rcone, float *rconv, int *info);
void sgeevx_64(char balanc, char jobvl, char jobvr, char sense, long n,
float *a, long lda, float *wr, float *wi, float *vl, long
ldvl, float *vr, long ldvr, long *ilo, long *ihi, float
*scale, float *abnrm, float *rcone, float *rconv, long
*info);
PURPOSEsgeevx computes for an N-by-N real nonsymmetric matrix A, the eigenval‐
ues 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 diagonal similarity transforma‐
tion 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 eigenvectors 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 explanation of balancing, see section 4.10.2 of the
LAPACK Users' Guide.
ARGUMENTS
BALANC (input)
Indicates how the input matrix should be diagonally scaled
and/or permuted to improve the conditioning of its eigenval‐
ues. = 'N': Do not diagonally 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. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.
JOBVL (input)
= '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)
= '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)
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 eigenvectors.
If SENSE = 'E' or 'B', both left and right eigenvectors must
also be computed (JOBVL = 'V' and JOBVR = 'V').
N (input) The order of the matrix A. N >= 0.
A (input/output)
On entry, the N-by-N matrix A. On exit, A has been overwrit‐
ten. If JOBVL = 'V' or JOBVR = 'V', A contains the real
Schur form of the balanced version of the input matrix A.
LDA (input)
The leading dimension of the array A. LDA >= max(1,N).
WR (output)
WR and WI contain the real and imaginary parts, respectively,
of the computed eigenvalues. Complex conjugate pairs of ei‐
genvalues will appear consecutively with the eigenvalue hav‐
ing the positive imaginary part first.
WI (output)
See the description for WR.
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 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)
The leading dimension of the array VL. LDVL >= 1; 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 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)
The leading dimension of the array VR. LDVR >= 1, and if
JOBVR = 'V', LDVR >= N.
ILO (output)
ILO and IHI are integer values determined when A was bal‐
anced. The balanced A(i,j) = 0 if I > J and J = 1,...,ILO-1
or I = IHI+1,...,N.
IHI (output)
See the description of ILO.
SCALE (output)
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)
The one-norm of the balanced matrix (the maximum of the sum
of absolute values of elements of any column).
RCONE (output)
RCONE(j) is the reciprocal condition number of the j-th ei‐
genvalue.
RCONV (output)
RCONV(j) is the reciprocal condition number of the j-th right
eigenvector.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LDWORK.
LDWORK (input)
The dimension of the array WORK. If SENSE = 'N' or 'E',
LDWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
LDWORK >= 3*N. If SENSE = 'V' or 'B', LDWORK >= N*(N+6).
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed; the rou‐
tine 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 LDWORK is issued by XERBLA.
IWORK2 (workspace)
dimension(2*N-2) If SENSE = 'N' or 'E', not referenced.
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors or condition numbers have
been computed; elements 1:ILO-1 and i+1:N of WR and WI con‐
tain eigenvalues which have converged.
6 Mar 2009 sgeevx(3P)
