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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, * )



     #include <sunperf.h>

     void ctrsna(char job, char howmny, int *select, int n,  com-
               plex  *t,  int ldt, complex *vl, int ldvl, complex
               *vr, int ldvr, float *s, float *sep, int  mm,  int
               *m, int *info) ;

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  eigen-
               pairs;
               = 'S':  compute  condition  numbers  for  selected
               eigenpairs 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 condition 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  eigen-
               vectors  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
               CHSEIN  or CTREVC.  If JOB = 'V', VL is not refer-
               enced.

     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  eigen-
               vectors  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
               CHSEIN  or CTREVC.  If JOB = 'V', VR is not refer-
               enced.

     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
               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  gen-
               eral 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 con-
               dition   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 (if  JOB  =
               'E'  or  'B') and/or SEP (if JOB = 'V' or 'B'). MM
               >= M.

     M         (output) 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      (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  ille-
               gal 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
     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)

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