SRC/ztrsna.f(3) Library Functions Manual SRC/ztrsna.f(3) NAME SRC/ztrsna.f SYNOPSIS Functions/Subroutines subroutine ztrsna (job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info) ZTRSNA Function/Subroutine Documentation subroutine ztrsna (character job, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, complex*16, dimension( ldwork, * ) work, integer ldwork, double precision, dimension( * ) rwork, integer info) ZTRSNA Purpose: !> !> ZTRSNA 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 j-th 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*16 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*16 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 !> ZHSEIN or ZTREVC. !> 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*16 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 !> ZHSEIN or ZTREVC. !> 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 DOUBLE PRECISION 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 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 !> SEP is DOUBLE PRECISION 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*16 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 DOUBLE PRECISION array, dimension (N) !> If JOB = 'E', RWORK is not referenced. !> INFO !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. 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 ) = 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) !> Definition at line 246 of file ztrsna.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 SRC/ztrsna.f(3)