ctrsna (l)  Linux Manuals
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)
NAME
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)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 unitary).ARGUMENTS
 JOB (input) CHARACTER*1

Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (SEP):
= aqEaq: for eigenvalues only (S);
= aqVaq: for eigenvectors only (SEP);
= aqBaq: for both eigenvalues and eigenvectors (S and SEP).  HOWMNY (input) CHARACTER*1

= aqAaq: compute condition numbers for all eigenpairs;
= aqSaq: compute condition numbers for selected eigenpairs specified by the array SELECT.  SELECT (input) LOGICAL array, dimension (N)
 If HOWMNY = aqSaq, 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 = aqAaq, 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 = aqEaq or aqBaq, 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 = aqVaq, VL is not referenced.
 LDVL (input) INTEGER
 The leading dimension of the array VL. LDVL >= 1; and if JOB = aqEaq or aqBaq, LDVL >= N.
 VR (input) COMPLEX array, dimension (LDVR,M)
 If JOB = aqEaq or aqBaq, 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 = aqVaq, VR is not referenced.
 LDVR (input) INTEGER
 The leading dimension of the array VR. LDVR >= 1; and if JOB = aqEaq or aqBaq, LDVR >= N.
 S (output) REAL array, dimension (MM)
 If JOB = aqEaq or aqBaq, 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 = aqVaq, S is not referenced.
 SEP (output) REAL array, dimension (MM)
 If JOB = aqVaq or aqBaq, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If JOB = aqEaq, SEP is not referenced.
 MM (input) INTEGER
 The number of elements in the arrays S (if JOB = aqEaq or aqBaq) and/or SEP (if JOB = aqVaq or aqBaq). MM >= M.
 M (output) INTEGER
 The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = aqAaq, M is set to N.
 WORK (workspace) COMPLEX array, dimension (LDWORK,N+6)
 If JOB = aqEaq, WORK is not referenced.
 LDWORK (input) INTEGER
 The leading dimension of the array WORK. LDWORK >= 1; and if JOB = aqVaq or aqBaq, LDWORK >= N.
 RWORK (workspace) REAL array, dimension (N)
 If JOB = aqEaq, 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 aswhere u and v are the right and left eigenvectors of T corresponding to lambda; vaq 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
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
Then the reciprocal condition number is
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