.TH "ztrsna.f" 3 "Tue Nov 14 2017" "Version 3.8.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
ztrsna.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBztrsna\fP (JOB, HOWMNY, SELECT, \fBN\fP, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO)"
.br
.RI "\fBZTRSNA\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "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)"

.PP
\fBZTRSNA\fP  
.PP
\fBPurpose: \fP
.RS 4

.PP
.nf
 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).
.fi
.PP
 
.RE
.PP
\fBParameters:\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          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).
.fi
.PP
.br
\fIHOWMNY\fP 
.PP
.nf
          HOWMNY is CHARACTER*1
          = 'A': compute condition numbers for all eigenpairs;
          = 'S': compute condition numbers for selected eigenpairs
                 specified by the array SELECT.
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix T. N >= 0.
.fi
.PP
.br
\fIT\fP 
.PP
.nf
          T is COMPLEX*16 array, dimension (LDT,N)
          The upper triangular matrix T.
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fILDVL\fP 
.PP
.nf
          LDVL is INTEGER
          The leading dimension of the array VL.
          LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.
.fi
.PP
.br
\fIVR\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fILDVR\fP 
.PP
.nf
          LDVR is INTEGER
          The leading dimension of the array VR.
          LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.
.fi
.PP
.br
\fIS\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fISEP\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fIMM\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fIM\fP 
.PP
.nf
          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.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is COMPLEX*16 array, dimension (LDWORK,N+6)
          If JOB = 'E', WORK is not referenced.
.fi
.PP
.br
\fILDWORK\fP 
.PP
.nf
          LDWORK is INTEGER
          The leading dimension of the array WORK.
          LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.
.fi
.PP
.br
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (N)
          If JOB = 'E', RWORK is not referenced.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
.fi
.PP
 
.RE
.PP
\fBAuthor:\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBDate:\fP
.RS 4
November 2017 
.RE
.PP
\fBFurther Details: \fP
.RS 4

.PP
.nf
  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)
.fi
.PP
 
.RE
.PP

.PP
Definition at line 251 of file ztrsna\&.f\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.
