ctrsna (l) - Linux Manuals

NAME

CTRSNA - estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q unitary)

SYNOPSIS

SUBROUTINE CTRSNA(
JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO )

CHARACTER HOWMNY, JOB

INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N

LOGICAL SELECT( * )

REAL RWORK( * ), S( * ), SEP( * )

COMPLEX T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * )

PURPOSE

CTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q unitary).

ARGUMENTS

JOB (input) CHARACTER*1
Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP):
= aqEaq: for eigenvalues only (S);
= aqVaq: for eigenvectors only (SEP);
= aqBaq: for both eigenvalues and eigenvectors (S and SEP).
HOWMNY (input) CHARACTER*1

= aqAaq: compute condition numbers for all eigenpairs;
= aqSaq: compute condition numbers for selected eigenpairs specified by the array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = aqSaq, SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the j-th eigenpair, SELECT(j) must be set to .TRUE.. If HOWMNY = aqAaq, SELECT is not referenced.
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input) COMPLEX array, dimension (LDT,N)
The upper triangular matrix T.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL (input) COMPLEX array, dimension (LDVL,M)
If JOB = aqEaq or aqBaq, VL must contain left eigenvectors of T (or of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by CHSEIN or CTREVC. If JOB = aqVaq, VL is not referenced.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; and if JOB = aqEaq or aqBaq, LDVL >= N.
VR (input) COMPLEX array, dimension (LDVR,M)
If JOB = aqEaq or aqBaq, VR must contain right eigenvectors of T (or of any Q*T*Q**H with Q unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by CHSEIN or CTREVC. If JOB = aqVaq, VR is not referenced.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1; and if JOB = aqEaq or aqBaq, LDVR >= N.
S (output) REAL array, dimension (MM)
If JOB = aqEaq or aqBaq, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. 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 = aqVaq, S is not referenced.
SEP (output) REAL array, dimension (MM)
If JOB = aqVaq or aqBaq, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If JOB = aqEaq, SEP is not referenced.
MM (input) INTEGER
The number of elements in the arrays S (if JOB = aqEaq or aqBaq) and/or SEP (if JOB = aqVaq or aqBaq). 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 = aqAaq, M is set to N.
WORK (workspace) COMPLEX array, dimension (LDWORK,N+6)
If JOB = aqEaq, WORK is not referenced.
LDWORK (input) INTEGER
The leading dimension of the array WORK. LDWORK >= 1; and if JOB = aqVaq or aqBaq, LDWORK >= N.
RWORK (workspace) REAL array, dimension (N)
If JOB = aqEaq, RWORK 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) |vaq*u| (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding to lambda; vaq 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

lambda   )

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)