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# strsna

NAME
STRSNA - estimate 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)

SYNOPSIS
SUBROUTINE STRSNA( 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( * )

REAL           S( * ), SEP( * ), T( LDT, * ), VL( LDVL,
* ), VR( LDVR, * ), WORK( LDWORK, * )

PURPOSE
STRSNA 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 SHSEQR),
that is, block upper triangular with 1-by-1 and 2-by-2 diag-
onal blocks; each 2-by-2 diagonal block has its diagonal
elements equal and its off-diagonal elements of opposite
sign.

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

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) INTEGER
The order of the matrix T. N >= 0.

T       (input) REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T, in Schur canon-
ical form.

LDT     (input) INTEGER
The leading dimension of the array T. LDT >=
max(1,N).

VL      (input) REAL array, dimension (LDVL,M)
If JOB = 'E' or 'B', VL must contain left eigenvec-
tors 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 con-
secutive columns of VL, as returned by SHSEIN or
STREVC.  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) REAL array, dimension (LDVR,M)
If JOB = 'E' or 'B', VR must contain right eigenvec-
tors 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 con-
secutive columns of VR, as returned by SHSEIN or
STREVC.  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 con-
secutive elements of the array. For a complex conju-
gate pair of eigenvalues two consecutive 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 eigen-
pair, unless all eigenpairs are selected).  If JOB =

'V', S is not referenced.

SEP     (output) REAL 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. 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 refer-
enced.

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; for each
selected real eigenvalue one element is used, and
for each selected complex conjugate pair of eigen-
values, two elements are used. If HOWMNY = 'A', M is
set to N.

WORK    (workspace) REAL 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 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)