
DTREVC(l) ) DTREVC(l)
NAME
DTREVC  compute some or all of the right and/or left eigenvectors of a real upper quasi
triangular matrix T
SYNOPSIS
SUBROUTINE DTREVC( 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( * )
DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
DTREVC 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 correspond
ing 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 realSchur 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 DHSEQR), that is, block upper triangular
with 1by1 and 2by2 diagonal blocks; each 2by2 diagonal block has its diagonal ele
ments equal and its offdiagonal elements of opposite sign. Corresponding to each 2by2
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 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 correspond
ing 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) DOUBLE PRECISION array, dimension (LDT,N)
The upper quasitriangular matrix T in Schur canonical form.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an NbyN matrix
Q (usually the orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit,
if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvec
tors of T; VL has the same quasilower triangular form as T'. If T(i,i) is a real
eigenvalue, then the ith column VL(i) of VL is its corresponding eigenvector. If
T(i:i+1,i:i+1) is a 2by2 block whose eigenvalues are complexconjugate eigenval
ues 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 eigenvec
tor 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) DOUBLE PRECISION array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an NbyN matrix
Q (usually the orthogonal matrix Q of Schur vectors returned by DHSEQR). 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 quasiupper triangular form as T. If T(i,i) is a
real eigenvalue, then the ith column VR(i) of VR is its corresponding eigenvec
tor. If T(i:i+1,i:i+1) is a 2by2 block whose eigenvalues are complexconjugate
eigenvalues of T, then VR(i)+sqrt(1)*VR(i+1) is the complex eigenvector corre
sponding 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 com
plex eigenvector corresponding to a complex eigenvalue is stored in two consecu
tive 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 eigen
vectors. 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) DOUBLE PRECISION array, dimension (3*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith 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 DTREVC(l) 
