dtrsna(3P) Sun Performance Library dtrsna(3P)NAMEdtrsna - estimate reciprocal condition numbers for specified eigenval‐
ues 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, WORK1, INFO)
CHARACTER * 1 JOB, HOWMNY
INTEGER N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO
INTEGER WORK1(*)
LOGICAL SELECT(*)
DOUBLE PRECISION T(LDT,*), VL(LDVL,*), VR(LDVR,*), S(*), SEP(*),
WORK(LDWORK,*)
SUBROUTINE DTRSNA_64(JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
LDVR, S, SEP, MM, M, WORK, LDWORK, WORK1, INFO)
CHARACTER * 1 JOB, HOWMNY
INTEGER*8 N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO
INTEGER*8 WORK1(*)
LOGICAL*8 SELECT(*)
DOUBLE PRECISION T(LDT,*), VL(LDVL,*), VR(LDVR,*), S(*), SEP(*),
WORK(LDWORK,*)
F95 INTERFACE
SUBROUTINE TRSNA(JOB, HOWMNY, SELECT, N, T, [LDT], VL, [LDVL], VR,
[LDVR], S, SEP, MM, M, [WORK], [LDWORK], [WORK1], [INFO])
CHARACTER(LEN=1) :: JOB, HOWMNY
INTEGER :: N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO
INTEGER, DIMENSION(:) :: WORK1
LOGICAL, DIMENSION(:) :: SELECT
REAL(8), DIMENSION(:) :: S, SEP
REAL(8), DIMENSION(:,:) :: T, VL, VR, WORK
SUBROUTINE TRSNA_64(JOB, HOWMNY, SELECT, N, T, [LDT], VL, [LDVL], VR,
[LDVR], S, SEP, MM, M, [WORK], [LDWORK], [WORK1], [INFO])
CHARACTER(LEN=1) :: JOB, HOWMNY
INTEGER(8) :: N, LDT, LDVL, LDVR, MM, M, LDWORK, INFO
INTEGER(8), DIMENSION(:) :: WORK1
LOGICAL(8), DIMENSION(:) :: SELECT
REAL(8), DIMENSION(:) :: S, SEP
REAL(8), DIMENSION(:,:) :: T, VL, VR, WORK
C INTERFACE
#include <sunperf.h>
void dtrsna(char job, char howmny, int *select, int n, double *t, int
ldt, double *vl, int ldvl, double *vr, int ldvr, double *s,
double *sep, int mm, int *m, int ldwork, int *info);
void dtrsna_64(char job, char howmny, long *select, long n, double *t,
long ldt, double *vl, long ldvl, double *vr, long ldvr, dou‐
ble *s, double *sep, long mm, long *m, long ldwork, long
*info);
PURPOSEdtrsna 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-diag‐
onal elements of opposite sign.
ARGUMENTS
JOB (input)
Specifies whether condition numbers are required for eigen‐
values (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)
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT (input)
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) The order of the matrix T. N >= 0.
T (input) The upper quasi-triangular matrix T, in Schur canonical form.
LDT (input)
The leading dimension of the array T. LDT >= max(1,N).
VL (input)
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)
The leading dimension of the array VL. LDVL >= 1; and if JOB
= 'E' or 'B', LDVL >= N.
VR (input)
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)
The leading dimension of the array VR. LDVR >= 1; and if JOB
= 'E' or 'B', LDVR >= N.
S (output)
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 con‐
secutive 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)
If JOB = 'V' or 'B', the estimated reciprocal condition num‐
bers of the selected eigenvectors, stored in consecutive ele‐
ments 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)
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)
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)
dimension(LDWORK,N+6) If JOB = 'E', WORK is not referenced.
LDWORK (input)
The leading dimension of the array WORK. LDWORK >= 1; and if
JOB = 'V' or 'B', LDWORK >= N.
WORK1 (workspace)
dimension(2*N) If JOB = 'E', WORK1 is not referenced.
INFO (output)
= 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 cor‐
responding 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)).
An approximate error bound for a computed right eigenvector VR(i) is
given by
EPS * norm(T) / SEP(i)
6 Mar 2009 dtrsna(3P)
