.TH DTREVC 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
DTREVC - some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T
.SH SYNOPSIS
.TP 19
SUBROUTINE DTREVC(
SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
LDVR, MM, M, WORK, INFO )
.TP 19
.ti +4
CHARACTER
HOWMNY, SIDE
.TP 19
.ti +4
INTEGER
INFO, LDT, LDVL, LDVR, M, MM, N
.TP 19
.ti +4
LOGICAL
SELECT( * )
.TP 19
.ti +4
DOUBLE
PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
WORK( * )
.SH PURPOSE
DTREVC computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix: A = Q*T*Q**T, as computed by DHSEQR.
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The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:
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T*x = w*x, (y**H)*T = w*(y**H)
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where y**H denotes the conjugate transpose of y.
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The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.
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This routine returns the matrices X and/or Y of right and left
eigenvectors 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 eigenvectors of A.
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.SH ARGUMENTS
.TP 8
SIDE (input) CHARACTER*1
= \(aqR\(aq: compute right eigenvectors only;
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= \(aqL\(aq: compute left eigenvectors only;
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= \(aqB\(aq: compute both right and left eigenvectors.
.TP 8
HOWMNY (input) CHARACTER*1
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= \(aqA\(aq: compute all right and/or left eigenvectors;
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= \(aqB\(aq: compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= \(aqS\(aq: compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.
.TP 8
SELECT (input/output) LOGICAL array, dimension (N)
If HOWMNY = \(aqS\(aq, SELECT specifies the eigenvectors to be
computed.
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 eigenvalue, 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 referenced if HOWMNY = \(aqA\(aq or \(aqB\(aq.
.TP 8
N (input) INTEGER
The order of the matrix T. N >= 0.
.TP 8
T (input) DOUBLE PRECISION array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.
.TP 8
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
.TP 8
VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
On entry, if SIDE = \(aqL\(aq or \(aqB\(aq and HOWMNY = \(aqB\(aq, VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by DHSEQR).
On exit, if SIDE = \(aqL\(aq or \(aqB\(aq, VL contains:
if HOWMNY = \(aqA\(aq, the matrix Y of left eigenvectors of T;
if HOWMNY = \(aqB\(aq, the matrix Q*Y;
if HOWMNY = \(aqS\(aq, 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.
Not referenced if SIDE = \(aqR\(aq.
.TP 8
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1, and if
SIDE = \(aqL\(aq or \(aqB\(aq, LDVL >= N.
.TP 8
VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)
On entry, if SIDE = \(aqR\(aq or \(aqB\(aq and HOWMNY = \(aqB\(aq, VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by DHSEQR).
On exit, if SIDE = \(aqR\(aq or \(aqB\(aq, VR contains:
if HOWMNY = \(aqA\(aq, the matrix X of right eigenvectors of T;
if HOWMNY = \(aqB\(aq, the matrix Q*X;
if HOWMNY = \(aqS\(aq, 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 corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = \(aqL\(aq.
.TP 8
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
SIDE = \(aqR\(aq or \(aqB\(aq, LDVR >= N.
.TP 8
MM (input) INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.
.TP 8
M (output) INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = \(aqA\(aq or \(aqB\(aq, M is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
.TP 8
WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
.TP 8
INFO (output) INTEGER
= 0: successful exit
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< 0: if INFO = -i, the i-th argument had an illegal value
.SH FURTHER DETAILS
The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.
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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|.
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