ztrsna()

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

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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

[in]	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). !>
[in]	HOWMNY	!> HOWMNY is CHARACTER*1 !> = 'A': compute condition numbers for all eigenpairs; !> = 'S': compute condition numbers for selected eigenpairs !> specified by the array SELECT. !>
[in]	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. !>
[in]	N	!> N is INTEGER !> The order of the matrix T. N >= 0. !>
[in]	T	!> T is COMPLEX*16 array, dimension (LDT,N) !> The upper triangular matrix T. !>
[in]	LDT	!> LDT is INTEGER !> The leading dimension of the array T. LDT >= max(1,N). !>
[in]	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. !>
[in]	LDVL	!> LDVL is INTEGER !> The leading dimension of the array VL. !> LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. !>
[in]	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. !>
[in]	LDVR	!> LDVR is INTEGER !> The leading dimension of the array VR. !> LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. !>
[out]	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. !>
[out]	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. !>
[in]	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. !>
[out]	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. !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension (LDWORK,N+6) !> If JOB = 'E', WORK is not referenced. !>
[in]	LDWORK	!> LDWORK is INTEGER !> The leading dimension of the array WORK. !> LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. !>
[out]	RWORK	!> RWORK is DOUBLE PRECISION array, dimension (N) !> If JOB = 'E', RWORK is not referenced. !>
[out]	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)
!>