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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, * )
#include <sunperf.h>
void dtrsna(char job, char howmny, int *select, int n, dou-
ble *t, int ldt, double *vl, int ldvl, double *vr,
int ldvr, double *s, double *sep, int mm, int *m,
int *info) ;
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 eigen-
pairs;
= '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.
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 eigen-
vectors of T (or of any Q*T*Q**T with Q orthogo-
nal), 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 refer-
enced.
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 eigen-
vectors of T (or of any Q*T*Q**T with Q orthogo-
nal), 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 refer-
enced.
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 ele-
ments 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 gen-
eral 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 con-
dition 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.
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 ille-
gal 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)
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