.TH STRSNA 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME STRSNA - 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) .SH SYNOPSIS .TP 19 SUBROUTINE STRSNA( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, INFO ) .TP 19 .ti +4 CHARACTER HOWMNY, JOB .TP 19 .ti +4 INTEGER INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N .TP 19 .ti +4 LOGICAL SELECT( * ) .TP 19 .ti +4 INTEGER IWORK( * ) .TP 19 .ti +4 REAL S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * ) .SH 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 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. .br .SH ARGUMENTS .TP 8 JOB (input) CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP): .br = \(aqE\(aq: for eigenvalues only (S); .br = \(aqV\(aq: for eigenvectors only (SEP); .br = \(aqB\(aq: for both eigenvalues and eigenvectors (S and SEP). .TP 8 HOWMNY (input) CHARACTER*1 .br = \(aqA\(aq: compute condition numbers for all eigenpairs; .br = \(aqS\(aq: compute condition numbers for selected eigenpairs specified by the array SELECT. .TP 8 SELECT (input) LOGICAL array, dimension (N) If HOWMNY = \(aqS\(aq, 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 = \(aqA\(aq, SELECT is not referenced. .TP 8 N (input) INTEGER The order of the matrix T. N >= 0. .TP 8 T (input) REAL array, dimension (LDT,N) The upper quasi-triangular matrix T, in Schur canonical form. .TP 8 LDT (input) INTEGER The leading dimension of the array T. LDT >= max(1,N). .TP 8 VL (input) REAL array, dimension (LDVL,M) If JOB = \(aqE\(aq or \(aqB\(aq, VL must contain left eigenvectors 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 consecutive columns of VL, as returned by SHSEIN or STREVC. If JOB = \(aqV\(aq, VL is not referenced. .TP 8 LDVL (input) INTEGER The leading dimension of the array VL. LDVL >= 1; and if JOB = \(aqE\(aq or \(aqB\(aq, LDVL >= N. .TP 8 VR (input) REAL array, dimension (LDVR,M) If JOB = \(aqE\(aq or \(aqB\(aq, VR must contain right eigenvectors 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 consecutive columns of VR, as returned by SHSEIN or STREVC. If JOB = \(aqV\(aq, VR is not referenced. .TP 8 LDVR (input) INTEGER The leading dimension of the array VR. LDVR >= 1; and if JOB = \(aqE\(aq or \(aqB\(aq, LDVR >= N. .TP 8 S (output) REAL array, dimension (MM) If JOB = \(aqE\(aq or \(aqB\(aq, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate 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 eigenpair, unless all eigenpairs are selected). If JOB = \(aqV\(aq, S is not referenced. .TP 8 SEP (output) REAL array, dimension (MM) If JOB = \(aqV\(aq or \(aqB\(aq, the estimated reciprocal condition 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 = \(aqE\(aq, SEP is not referenced. .TP 8 MM (input) INTEGER The number of elements in the arrays S (if JOB = \(aqE\(aq or \(aqB\(aq) and/or SEP (if JOB = \(aqV\(aq or \(aqB\(aq). MM >= M. .TP 8 M (output) INTEGER The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = \(aqA\(aq, M is set to N. .TP 8 WORK (workspace) REAL array, dimension (LDWORK,N+6) If JOB = \(aqE\(aq, WORK is not referenced. .TP 8 LDWORK (input) INTEGER The leading dimension of the array WORK. LDWORK >= 1; and if JOB = \(aqV\(aq or \(aqB\(aq, LDWORK >= N. .TP 8 IWORK (workspace) INTEGER array, dimension (2*(N-1)) If JOB = \(aqE\(aq, IWORK is not referenced. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .SH FURTHER DETAILS The reciprocal of the condition number of an eigenvalue lambda is defined as .br S(lambda) = |v\(aq*u| / (norm(u)*norm(v)) .br where u and v are the right and left eigenvectors of T corresponding to lambda; v\(aq 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. .br An approximate error bound for a computed eigenvalue W(i) is given by EPS * norm(T) / S(i) .br where EPS is the machine precision. .br The reciprocal of the condition number of the right eigenvector u corresponding to lambda is defined as follows. Suppose .br T = ( lambda c ) .br ( 0 T22 ) .br Then the reciprocal condition number is .br SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) .br 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)). .br An approximate error bound for a computed right eigenvector VR(i) is given by .br EPS * norm(T) / SEP(i) .br