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# ztrsna

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NAME
ZTRSNA - 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 ZTRSNA( 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( * )

DOUBLE         PRECISION RWORK( * ), S( * ), SEP( * )

COMPLEX*16     T( LDT, * ), VL( LDVL, * ), VR( LDVR, *
), WORK( LDWORK, * )

PURPOSE
ZTRSNA 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 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 eigenpairs;
= 'S': compute condition numbers for selected eigen-
pairs 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 con-
dition numbers for the j-th 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*16 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*16 array, dimension (LDVL,M)
If JOB = 'E' or 'B', VL must contain left eigenvec-
tors 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 con-
secutive columns of VL, as returned by ZHSEIN or
ZTREVC.  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*16 array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvec-
tors 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 con-
secutive columns of VR, as returned by ZHSEIN or
ZTREVC.  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 con-
secutive 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 = 'V', S is not referenced.

SEP     (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal condi-
tion 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 and SEP. MM
>= M.

M       (output) INTEGER
The number of elements of the arrays S and SEP used

to store the specified condition numbers. If HOWMNY
= 'A', M is set to N.

WORK    (workspace) COMPLEX*16 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) DOUBLE PRECISION array, dimension (N)
If JOB = 'E', 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) = |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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