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# strevc

```
NAME
STREVC - compute all or some right and/or left eigenvectors
of a real upper quasi-triangular matrix T

SYNOPSIS
SUBROUTINE STREVC( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
VR, LDVR, MM, M, WORK, INFO )

CHARACTER      HOWMNY, JOB

INTEGER        INFO, LDT, LDVL, LDVR, M, MM, N

LOGICAL        SELECT( * )

REAL           T( LDT, * ), VL( LDVL, * ), VR( LDVR, *
), WORK( * )

PURPOSE
STREVC computes all or some 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**H*T = w*y**H.

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 original
matrix A = Q*T*Q**T, then Q*X and/or 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 diag-
onal blocks; each 2-by-2 diagonal block has its diagonal
elements equal and its off-diagonal elements of opposite
sign.

ARGUMENTS
JOB     (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;
= 'O':  compute all right and/or left eigenvectors,
multiplied on the left by an input (generally
orthogonal) matrix; = 'S':  compute some 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. To select the real eigenvector
corresponding to a real eigenvalue w(j), SELECT(j)
must be set to .TRUE.. To select the complex eigen-
vector 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..  If HOWMNY = 'A'
or 'O', SELECT is not referenced.

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 canoni-
cal 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 JOB = 'L' or 'B' and HOWMNY = 'O', VL
must contain an N-by-N matrix Q (usually the orthog-
onal matrix Q of Schur vectors returned by SHSEQR).
On exit, if JOB = 'L' or 'B', VL contains: if HOWMNY
= 'A', the matrix Y of left eigenvectors of T; if
HOWMNY = 'O', 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 JOB =
'R', VL is not referenced.

LDVL    (input) INTEGER
The leading dimension of the array VL.  LDVL >=
max(1,N).

VR      (input/output) REAL array, dimension (LDVR,MM)
On entry, if JOB = 'R' or 'B' and HOWMNY = 'O', VR
must contain an N-by-N matrix Q (usually the orthog-
onal matrix Q of Schur vectors returned by SHSEQR).
On exit, if JOB = 'R' or 'B', VR contains: if HOWMNY
= 'A', the matrix X of right eigenvectors of T; if
HOWMNY = 'O', 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
corresponding to a complex eigenvalue is stored in
two consecutive columns, the first holding the real
part and the second the imaginary part.  If JOB =
'L', VR is not referenced.

LDVR    (input) INTEGER
The leading dimension of the array VR.  LDVR >=
max(1,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
required to store the eigenvectors; each selected
real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
If HOWMNY = 'A' or 'O', M is set to N.

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) substitution, with scaling to make the code robust
against possible overflow.

Each eigenvector is normalized so that the element of larg-
est magnitude has magnitude 1; here the magnitude of a com-
plex number (x,y) is taken to be |x| + |y|.
```