DTRSNA(3F) DTRSNA(3F)
DTRSNA - estimate reciprocal condition numbers for specified eigenvalues
and/or right eigenvectors of a real upper quasi-triangular matrix T (or
of any matrix Q*T*Q**T with Q orthogonal)
SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S,
SEP, MM, M, WORK, LDWORK, IWORK, INFO )
CHARACTER HOWMNY, JOB
INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, *
), VR( LDVR, * ), WORK( LDWORK, * )
DTRSNA estimates reciprocal condition numbers for specified eigenvalues
and/or right eigenvectors of a real upper quasi-triangular matrix T (or
of any matrix Q*T*Q**T with Q orthogonal).
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.
JOB (input) CHARACTER*1
Specifies whether condition numbers are required for eigenvalues
(S) or eigenvectors (SEP):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (SEP);
= 'B': for both eigenvalues and eigenvectors (S and SEP).
HOWMNY (input) CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers for
the eigenpair corresponding to a real eigenvalue w(j), SELECT(j)
must be set to .TRUE.. To select condition numbers corresponding
to a complex conjugate pair of eigenvalues w(j) and w(j+1),
either SELECT(j) or SELECT(j+1) or both, must be set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
Page 1
DTRSNA(3F) DTRSNA(3F)
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) DOUBLE PRECISION array, dimension (LDVL,M)
If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or
of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors must
be stored in consecutive columns of VL, as returned by DHSEIN or
DTREVC. If JOB = 'V', VL is not referenced.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; and if JOB =
'E' or 'B', LDVL >= N.
VR (input) DOUBLE PRECISION array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or
of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors must
be stored in consecutive columns of VR, as returned by DHSEIN or
DTREVC. If JOB = 'V', VR is not referenced.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1; and if JOB =
'E' or 'B', LDVR >= N.
S (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'E' or 'B', the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of S are set to the same value. Thus S(j),
SEP(j), and the j-th columns of VL and VR all correspond to the
same eigenpair (but not in general the j-th eigenpair, unless all
eigenpairs are selected). If JOB = 'V', S is not referenced.
SEP (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal condition numbers
of the selected eigenvectors, stored in consecutive elements of
the array. For a complex eigenvector two consecutive elements of
SEP are set to the same value. If the eigenvalues cannot be
reordered to compute SEP(j), SEP(j) is set to 0; this can only
occur when the true value would be very small anyway. If JOB =
'E', SEP is not referenced.
MM (input) INTEGER
The number of elements in the arrays S (if JOB = 'E' or 'B')
and/or SEP (if JOB = 'V' or 'B'). MM >= M.
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DTRSNA(3F) DTRSNA(3F)
M (output) INTEGER
The number of elements of the arrays S and/or SEP actually used
to store the estimated condition numbers. If HOWMNY = 'A', M is
set to N.
WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+1)
If JOB = 'E', WORK is not referenced.
LDWORK (input) INTEGER
The leading dimension of the array WORK. LDWORK >= 1; and if JOB
= 'V' or 'B', LDWORK >= N.
IWORK (workspace) INTEGER array, dimension (N)
If JOB = 'E', IWORK is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
FURTHER DETAILS
The reciprocal of the condition number of an eigenvalue lambda is defined
as
S(lambda) = |v'*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding to
lambda; v' denotes the conjugate-transpose of v, and norm(u) denotes the
Euclidean norm. These reciprocal condition numbers always lie between
zero (very badly conditioned) and one (very well conditioned). If n = 1,
S(lambda) is defined to be 1.
An approximate error bound for a computed eigenvalue W(i) is given by
EPS * norm(T) / S(i)
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u
corresponding to lambda is defined as follows. Suppose
T = ( lambda c )
( 0 T22 )
Then the reciprocal condition number is
SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
where sigma-min denotes the smallest singular value. We approximate the
smallest singular value by the reciprocal of an estimate of the one-norm
of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to be
abs(T(1,1)).
Page 3
DTRSNA(3F) DTRSNA(3F)
An approximate error bound for a computed right eigenvector VR(i) is
given by
EPS * norm(T) / SEP(i)
DTRSNA(3F) DTRSNA(3F)
DTRSNA - estimate reciprocal condition numbers for specified eigenvalues
and/or right eigenvectors of a real upper quasi-triangular matrix T (or
of any matrix Q*T*Q**T with Q orthogonal)
SUBROUTINE DTRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S,
SEP, MM, M, WORK, LDWORK, IWORK, INFO )
CHARACTER HOWMNY, JOB
INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, *
), VR( LDVR, * ), WORK( LDWORK, * )
DTRSNA estimates reciprocal condition numbers for specified eigenvalues
and/or right eigenvectors of a real upper quasi-triangular matrix T (or
of any matrix Q*T*Q**T with Q orthogonal).
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.
JOB (input) CHARACTER*1
Specifies whether condition numbers are required for eigenvalues
(S) or eigenvectors (SEP):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (SEP);
= 'B': for both eigenvalues and eigenvectors (S and SEP).
HOWMNY (input) CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers for
the eigenpair corresponding to a real eigenvalue w(j), SELECT(j)
must be set to .TRUE.. To select condition numbers corresponding
to a complex conjugate pair of eigenvalues w(j) and w(j+1),
either SELECT(j) or SELECT(j+1) or both, must be set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
Page 1
DTRSNA(3F) DTRSNA(3F)
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) DOUBLE PRECISION array, dimension (LDVL,M)
If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or
of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors must
be stored in consecutive columns of VL, as returned by DHSEIN or
DTREVC. If JOB = 'V', VL is not referenced.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; and if JOB =
'E' or 'B', LDVL >= N.
VR (input) DOUBLE PRECISION array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or
of any Q*T*Q**T with Q orthogonal), corresponding to the
eigenpairs specified by HOWMNY and SELECT. The eigenvectors must
be stored in consecutive columns of VR, as returned by DHSEIN or
DTREVC. If JOB = 'V', VR is not referenced.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1; and if JOB =
'E' or 'B', LDVR >= N.
S (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'E' or 'B', the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of S are set to the same value. Thus S(j),
SEP(j), and the j-th columns of VL and VR all correspond to the
same eigenpair (but not in general the j-th eigenpair, unless all
eigenpairs are selected). If JOB = 'V', S is not referenced.
SEP (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal condition numbers
of the selected eigenvectors, stored in consecutive elements of
the array. For a complex eigenvector two consecutive elements of
SEP are set to the same value. If the eigenvalues cannot be
reordered to compute SEP(j), SEP(j) is set to 0; this can only
occur when the true value would be very small anyway. If JOB =
'E', SEP is not referenced.
MM (input) INTEGER
The number of elements in the arrays S (if JOB = 'E' or 'B')
and/or SEP (if JOB = 'V' or 'B'). MM >= M.
Page 2
DTRSNA(3F) DTRSNA(3F)
M (output) INTEGER
The number of elements of the arrays S and/or SEP actually used
to store the estimated condition numbers. If HOWMNY = 'A', M is
set to N.
WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+1)
If JOB = 'E', WORK is not referenced.
LDWORK (input) INTEGER
The leading dimension of the array WORK. LDWORK >= 1; and if JOB
= 'V' or 'B', LDWORK >= N.
IWORK (workspace) INTEGER array, dimension (N)
If JOB = 'E', IWORK is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
FURTHER DETAILS
The reciprocal of the condition number of an eigenvalue lambda is defined
as
S(lambda) = |v'*u| / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding to
lambda; v' denotes the conjugate-transpose of v, and norm(u) denotes the
Euclidean norm. These reciprocal condition numbers always lie between
zero (very badly conditioned) and one (very well conditioned). If n = 1,
S(lambda) is defined to be 1.
An approximate error bound for a computed eigenvalue W(i) is given by
EPS * norm(T) / S(i)
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u
corresponding to lambda is defined as follows. Suppose
T = ( lambda c )
( 0 T22 )
Then the reciprocal condition number is
SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )
where sigma-min denotes the smallest singular value. We approximate the
smallest singular value by the reciprocal of an estimate of the one-norm
of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to be
abs(T(1,1)).
Page 3
DTRSNA(3F) DTRSNA(3F)
An approximate error bound for a computed right eigenvector VR(i) is
given by
EPS * norm(T) / SEP(i)
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