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NAME
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)
SYNOPSIS
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, * )
PURPOSE
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 diag-
onal blocks; each 2-by-2 diagonal block has its diagonal
elements equal and its off-diagonal elements of opposite
sign.
ARGUMENTS
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 eigen-
pairs 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 con-
dition 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.
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 canon-
ical 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 eigenvec-
tors 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 con-
secutive 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 eigenvec-
tors 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 con-
secutive 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 con-
secutive elements of the array. For a complex conju-
gate 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 condi-
tion 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 refer-
enced.
MM (input) INTEGER
The number of elements in the arrays S and SEP. MM
>= M.
M (output) INTEGER
The number of elements of the arrays S and SEP used
to store the specified condition numbers; for each
selected real eigenvalue one element is used, and
for each selected complex conjugate pair of eigen-
values, two elements are used. If HOWMNY = 'A', M is
set to N.
(LDWORK,N+1)
WORK (workspace) DOUBLE PRECISION array, dimension
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 recipro-
cal 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 eigen-
vector u corresponding to lambda is defined as follows. Sup-
pose
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)).
An approximate error bound for a computed right eigenvector
VR(i) is given by
EPS * norm(T) / SEP(i)