STREVC(l) ) STREVC(l)NAME
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 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) 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, 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) LOGICAL array, dimension (N)
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 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 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; 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 complex-conjugate eigenval‐
ues of T, then VL(i)+sqrt(-1)*VL(i+1) is the complex eigenvec‐
tor 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) INTEGER
The leading dimension of the array VL. LDVL >= max(1,N) if
SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
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; VR has the same quasi-upper triangular form as T.
If T(i,i) is a real eigenvalue, 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-conjugate eigenval‐
ues of T, then VR(i)+sqrt(-1)*VR(i+1) is the complex eigenvec‐
tor 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 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 = '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 otherwise.
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 version 3.0 15 June 2000 STREVC(l)
