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ctrsna (3)
  • >> ctrsna (3) ( Solaris man: Библиотечные вызовы )
  • 
    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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