dtrevc(3P) Sun Performance Library dtrevc(3P)NAMEdtrevc - compute some or all of the right and/or left eigenvectors of a
real upper quasi-triangular matrix T
SYNOPSIS
SUBROUTINE DTREVC(SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
LDVR, MM, M, WORK, INFO)
CHARACTER * 1 SIDE, HOWMNY
INTEGER N, LDT, LDVL, LDVR, MM, M, INFO
LOGICAL SELECT(*)
DOUBLE PRECISION T(LDT,*), VL(LDVL,*), VR(LDVR,*), WORK(*)
SUBROUTINE DTREVC_64(SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
LDVR, MM, M, WORK, INFO)
CHARACTER * 1 SIDE, HOWMNY
INTEGER*8 N, LDT, LDVL, LDVR, MM, M, INFO
LOGICAL*8 SELECT(*)
DOUBLE PRECISION T(LDT,*), VL(LDVL,*), VR(LDVR,*), WORK(*)
F95 INTERFACE
SUBROUTINE TREVC(SIDE, HOWMNY, SELECT, N, T, [LDT], VL, [LDVL], VR,
[LDVR], MM, M, [WORK], [INFO])
CHARACTER(LEN=1) :: SIDE, HOWMNY
INTEGER :: N, LDT, LDVL, LDVR, MM, M, INFO
LOGICAL, DIMENSION(:) :: SELECT
REAL(8), DIMENSION(:) :: WORK
REAL(8), DIMENSION(:,:) :: T, VL, VR
SUBROUTINE TREVC_64(SIDE, HOWMNY, SELECT, N, T, [LDT], VL, [LDVL],
VR, [LDVR], MM, M, [WORK], [INFO])
CHARACTER(LEN=1) :: SIDE, HOWMNY
INTEGER(8) :: N, LDT, LDVL, LDVR, MM, M, INFO
LOGICAL(8), DIMENSION(:) :: SELECT
REAL(8), DIMENSION(:) :: WORK
REAL(8), DIMENSION(:,:) :: T, VL, VR
C INTERFACE
#include <sunperf.h>
void dtrevc(char side, char howmny, int *select, int n, double *t, int
ldt, double *vl, int ldvl, double *vr, int ldvr, int mm, int
*m, int *info);
void dtrevc_64(char side, char howmny, long *select, long n, double *t,
long ldt, double *vl, long ldvl, double *vr, long ldvr, long
mm, long *m, long *info);
PURPOSEdtrevc computes some or all of the right and/or left eigenvectors of a
real upper quasi-triangular matrix T.
The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:
T*x = w*x, y'*T = w*y'
where y' denotes the conjugate transpose of the vector y.
If all eigenvectors are requested, the routine may either return the
matrices X and/or Y of right or left eigenvectors of T, or the products
Q*X and/or Q*Y, where Q is an input orthogonal
matrix. If T was obtained from the real-Schur factorization of an orig‐
inal matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of right or
left eigenvectors of A.
T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its off-diag‐
onal elements of opposite sign. Corresponding to each 2-by-2 diagonal
block is a complex conjugate pair of eigenvalues and eigenvectors; only
one eigenvector of the pair is computed, namely the one corresponding
to the eigenvalue with positive imaginary part.
ARGUMENTS
SIDE (input)
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
HOWMNY (input)
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors, and back‐
transform them using the input matrices supplied in VR and/or
VL; = 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT.
SELECT (input/output)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be com‐
puted. If HOWMNY = 'A' or 'B', SELECT is not referenced. To
select the real eigenvector corresponding to a real eigenval‐
ue w(j), SELECT(j) must be set to .TRUE.. To select the com‐
plex eigenvector corresponding to a complex conjugate pair
w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must be set
to .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1)
is .FALSE..
N (input) The order of the matrix T. N >= 0.
T (input/output)
The upper quasi-triangular matrix T in Schur canonical form.
LDT (input)
The leading dimension of the array T. LDT >= max(1,N).
VL (input/output)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must con‐
tain an N-by-N matrix Q (usually the orthogonal matrix Q of
Schur vectors returned by SHSEQR). On exit, if SIDE = 'L' or
'B', VL contains: if HOWMNY = 'A', the matrix Y of left
eigenvectors of T; VL has the same quasi-lower triangular
form as T'. If T(i,i) is a real eigenvalue, then the i-th
column VL(i) of VL is its corresponding eigenvector. If
T(i:i+1,i:i+1) is a 2-by-2 block whose eigenvalues are com‐
plex-conjugate eigenvalues of T, then VL(i)+sqrt(-1)*VL(i+1)
is the complex eigenvector corresponding to the eigenvalue
with positive real part. if HOWMNY = 'B', the matrix Q*Y; if
HOWMNY = 'S', the left eigenvectors of T specified by SELECT,
stored consecutively in the columns of VL, in the same order
as their eigenvalues. A complex eigenvector corresponding to
a complex eigenvalue is stored in two consecutive columns,
the first holding the real part, and the second the imaginary
part. If SIDE = 'R', VL is not referenced.
LDVL (input)
The leading dimension of the array VL. LDVL >= max(1,N) if
SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
VR (input/output)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must con‐
tain an N-by-N matrix Q (usually the orthogonal matrix Q of
Schur vectors returned by SHSEQR). On exit, if SIDE = 'R' or
'B', VR contains: if HOWMNY = 'A', the matrix X of right
eigenvectors of T; VR has the same quasi-upper triangular
form as T. If T(i,i) is a real eigenvalue, then the i-th col‐
umn VR(i) of VR is its corresponding eigenvector. If
T(i:i+1,i:i+1) is a 2-by-2 block whose eigenvalues are com‐
plex-conjugate eigenvalues of T, then VR(i)+sqrt(-1)*VR(i+1)
is the complex eigenvector corresponding to the eigenvalue
with positive real part. if HOWMNY = 'B', the matrix Q*X; if
HOWMNY = 'S', the right eigenvectors of T specified by
SELECT, stored consecutively in the columns of VR, in the
same order as their eigenvalues. A complex eigenvector cor‐
responding to a complex eigenvalue is stored in two consecu‐
tive columns, the first holding the real part and the second
the imaginary part. If SIDE = 'L', VR is not referenced.
LDVR (input)
The leading dimension of the array VR. LDVR >= max(1,N) if
SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
MM (input)
The number of columns in the arrays VL and/or VR. MM >= M.
M (output)
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is
set to N. Each selected real eigenvector occupies one column
and each selected complex eigenvector occupies two columns.
WORK (workspace)
dimension(3*N)
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
FURTHER DETAILS
The algorithm used in this program is basically backward (forward) sub‐
stitution, with scaling to make the the code robust against possible
overflow.
Each eigenvector is normalized so that the element of largest magnitude
has magnitude 1; here the magnitude of a complex number (x,y) is taken
to be |x| + |y|.
6 Mar 2009 dtrevc(3P)
