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NAME DTRSNA - 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 DTRSNA( 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( * ) DOUBLE PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * ) PURPOSE DTRSNA 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 DHSEQR), 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 con- dition 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DHSEIN or DTREVC. 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) DOUBLE PRECISION 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 DHSEIN or DTREVC. 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) DOUBLE PRECISION 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 eigenpair, unless all eigenpairs are selected). If JOB = 'V', S is not referenced. SEP (output) DOUBLE PRECISION 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. (LDWORK,N+1) WORK (workspace) DOUBLE PRECISION array, dimension 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)