
ctrsna.f(3) LAPACK ctrsna.f(3)
NAME
ctrsna.f 
SYNOPSIS
Functions/Subroutines
subroutine ctrsna (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M,
WORK, LDWORK, RWORK, INFO)
CTRSNA
Function/Subroutine Documentation
subroutine ctrsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN,
complex, dimension( ldt, * )T, integerLDT, complex, dimension( ldvl, * )VL, integerLDVL,
complex, dimension( ldvr, * )VR, integerLDVR, real, dimension( * )S, real, dimension( *
)SEP, integerMM, integerM, complex, dimension( ldwork, * )WORK, integerLDWORK, real,
dimension( * )RWORK, integerINFO)
CTRSNA
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).
Parameters:
JOB
JOB is 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
HOWMNY is CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT
SELECT is 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
N is INTEGER
The order of the matrix T. N >= 0.
T
T is COMPLEX array, dimension (LDT,N)
The upper triangular matrix T.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
VL
VL is 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
LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
VR
VR is 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
LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
S
S is 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 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
SEP is 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
MM is 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
M is INTEGER
The number of elements of the arrays S and/or SEP actually
used to store the estimated condition numbers.
If HOWMNY = 'A', M is set to N.
WORK
WORK is COMPLEX array, dimension (LDWORK,N+6)
If JOB = 'E', WORK is not referenced.
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
RWORK
RWORK is REAL array, dimension (N)
If JOB = 'E', RWORK is not referenced.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Further Details:
The reciprocal of the condition number of an eigenvalue lambda is
defined as
S(lambda) = v**H*u / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of T corresponding
to lambda; v**H 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
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)
Definition at line 248 of file ctrsna.f.
Author
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Version 3.4.2 Tue Sep 25 2012 ctrsna.f(3) 
