STREVC - compute 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.
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 eigenvec
tors 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 factoriza
tion of an original 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 diago
nal elements equal and its off-diagonal elements of oppo
site 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 imagi
nary part.
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
= 'B': compute all right and/or left eigenvec
tors, and backtransform 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) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors
to be computed. If HOWMNY = 'A' or 'B', SELECT is
not referenced. To select the real eigenvector
corresponding to a real eigenvalue w(j), SELECT(j)
must be set to .TRUE.. To select the complex
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) 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 contain 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 con
tains: 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 eigen
value, 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 complex-conju
gate 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 con
secutively in the columns of VL, in the same order
as their eigenvalues. A complex eigenvector cor
responding 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) INTEGER
The leading dimension of the array VL. LDVL >=
wise.
VR (input/output) REAL array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B',
VR must contain 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 con
tains: 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 eigen
value, then the i-th column 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 complex-conju
gate 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 con
secutively 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 consecutive columns, the first holding the
real part and the second the imaginary part. If
SIDE = 'L', VR is not referenced.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >=
max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 other
wise.
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 ille
gal value
FURTHER DETAILS
The algorithm used in this program is basically backward
(forward) substitution, with scaling to make the the code
robust against possible overflow.
largest magnitude has magnitude 1; here the magnitude of a
complex number (x,y) is taken to be |x| + |y|.
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