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NAME CTRSNA - 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 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 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 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 = '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 CHSEIN or CTREVC. 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 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 CHSEIN or CTREVC. 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. 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) 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. 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 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) REAL 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)