DTREVC(3F) DTREVC(3F)
DTREVC - compute some or all of the right and/or left eigenvectors of a
real upper quasi-triangular matrix T
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( * )
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 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
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 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 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.
SIDE (input) CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
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DTREVC(3F) DTREVC(3F)
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) DOUBLE PRECISION 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) DOUBLE PRECISION 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
vectors returned by DHSEQR). On exit, if SIDE = 'L' or 'B', VL
contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of
T; 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) DOUBLE PRECISION 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
vectors returned by DHSEQR). On exit, if SIDE = 'R' or 'B', VR
contains: if HOWMNY = 'A', the matrix X of right eigenvectors of
T; 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
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DTREVC(3F) DTREVC(3F)
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) DOUBLE PRECISION 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|.
DTREVC(3F) DTREVC(3F)
DTREVC - compute some or all of the right and/or left eigenvectors of a
real upper quasi-triangular matrix T
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( * )
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 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
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 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 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.
SIDE (input) CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
Page 1
DTREVC(3F) DTREVC(3F)
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) DOUBLE PRECISION 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) DOUBLE PRECISION 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
vectors returned by DHSEQR). On exit, if SIDE = 'L' or 'B', VL
contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of
T; 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) DOUBLE PRECISION 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
vectors returned by DHSEQR). On exit, if SIDE = 'R' or 'B', VR
contains: if HOWMNY = 'A', the matrix X of right eigenvectors of
T; 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
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DTREVC(3F) DTREVC(3F)
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) DOUBLE PRECISION 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|.
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