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 prod-
ucts 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', 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-diagonal elements of opposite sign. Corresponding to each 2-by-2 diago-
nal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corre-
sponding 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 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 vec-
tors returned by SHSEQR). On exit, if SIDE = 'L' or 'B', VL contains: 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 eigenvalue, then the i-th column VL(i) of VL is its corre-
sponding eigenvector. If T(i:i+1,i:i+1) is a 2-by-2 block whose eigenvalues are complex-conjugate 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 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 contain an N-by-N matrix Q (usually the orthogonal matrix Q of Schur vec-
tors returned by SHSEQR). On exit, if SIDE = 'R' or 'B', VR contains: 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 eigenvalue, then the i-th column VR(i) of VR is its correspond-
ing eigenvector. If T(i:i+1,i:i+1) is a 2-by-2 block whose eigenvalues are complex-conjugate 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 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) substitution, 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)