
DTRSNA(l) ) DTRSNA(l)
NAME
DTRSNA  estimate reciprocal condition numbers for specified eigenvalues and/or right
eigenvectors of a real upper quasitriangular 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 quasitriangular 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 1by1 and 2by2 diagonal blocks; each 2by2 diagonal block has
its diagonal elements equal and its offdiagonal elements of opposite sign.
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 eigenpair corresponding to a real
eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select condition numbers cor
responding 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 quasitriangular 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 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) 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 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) 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 ei
genvalues two consecutive elements of S are set to the same value. Thus S(j),
SEP(j), and the jth columns of VL and VR all correspond to the same eigenpair
(but not in general the jth 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 condition numbers of the selected
eigenvectors, stored in consecutive elements of the array. For a complex eigenvec
tor 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 (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) 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 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 conjugatetranspose 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 DTRSNA(l) 
