
CTRSNA(l) ) CTRSNA(l)
NAME
CTRSNA  estimate 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 uni
tary)
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 uni
tary).
ARGUMENTS
JOB (input) CHARACTER*1
Specifies whether condition numbers are required for eigenvalues (S) or eigenvec
tors (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 jth eigenpair, SELECT(j) 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) 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 = 'E' or 'B', 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 = '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) COMPLEX array, dimension (LDVR,M)
If JOB = 'E' or 'B', 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 = '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) REAL array, dimension (MM)
If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues,
stored in consecutive elements of the array. Thus S(j), SEP(j), and the jth col
umns 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 refer
enced.
SEP (output) REAL array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected
eigenvectors, stored in consecutive elements of the array. 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 esti
mated condition numbers. If HOWMNY = 'A', M is set to N.
WORK (workspace) COMPLEX 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.
RWORK (workspace) REAL array, dimension (N)
If JOB = 'E', RWORK is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith 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 con
ditioned). 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 ) = sigmamin( T22  lambda*I )
where sigmamin denotes the smallest singular value. We approximate the smallest singular
value by the reciprocal of an estimate of the onenorm 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)
LAPACK version 3.0 15 June 2000 CTRSNA(l) 
