STREVC(1) LAPACK routine (version 3.2) STREVC(1)NAME
STREVC - computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T
SYNOPSIS
SUBROUTINE STREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR,
MM, M, WORK, INFO )
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
LOGICAL SELECT( * )
REAL T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
STREVC computes some or all of the right and/or left eigenvectors of a
real upper quasi-triangular matrix T. Matrices of this type are pro‐
duced by the Schur factorization of a real general matrix: A =
Q*T*Q**T, as computed by SHSEQR.
The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:
T*x = w*x, (y**H)*T = w*(y**H)
where y**H denotes the conjugate transpose of y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.
This routine returns the matrices X and/or Y of right and left eigen‐
vectors of T, or the products Q*X and/or Q*Y, where Q is an input
matrix. If Q is the orthogonal factor that reduces a matrix A to Schur
form T, then Q*X and Q*Y are the matrices of right and left eigenvec‐
tors of A.
ARGUMENTS
SIDE (input) CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
HOWMNY (input) CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors, backtrans‐
formed by the matrices in VR and/or VL; = 'S': compute
selected right and/or left eigenvectors, as indicated by the
logical array SELECT.
SELECT (input/output) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be com‐
puted. If w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE.. If w(j) and
w(j+1) are the real and imaginary parts of a complex eigenval‐
ue, the corresponding complex eigenvector is computed if either
SELECT(j) or SELECT(j+1) is .TRUE., and on exit SELECT(j) is
set to .TRUE. and SELECT(j+1) is set to .FALSE.. Not refer‐
enced if HOWMNY = 'A' or 'B'.
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input) REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL (input/output) REAL array, dimension (LDVL,MM)
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 eigen‐
vectors of T; if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S',
the left eigenvectors of T specified by SELECT, stored consecu‐
tively in the columns of VL, in the same order as their eigen‐
values. A complex eigenvector corresponding to a complex ei‐
genvalue is stored in two consecutive columns, the first hold‐
ing the real part, and the second the imaginary part. Not ref‐
erenced if SIDE = 'R'.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1, and if SIDE
= 'L' or 'B', LDVL >= N.
VR (input/output) REAL array, dimension (LDVR,MM)
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 eigen‐
vectors of T; if HOWMNY = 'B', the matrix Q*X; if HOWMNY = 'S',
the right eigenvectors of T specified by SELECT, stored consec‐
utively in the columns of VR, in the same order as their eigen‐
values. A complex eigenvector corresponding to a complex ei‐
genvalue is stored in two consecutive columns, the first hold‐
ing the real part and the second the imaginary part. Not ref‐
erenced if SIDE = 'L'.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1, and if SIDE
= 'R' or 'B', LDVR >= N.
MM (input) INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
M (output) INTEGER
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) REAL array, dimension (3*N)
INFO (output) INTEGER
= 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|.
LAPACK routine (version 3.2) November 2008 STREVC(1)
