.TH "SRC/ctgsna.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
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.SH NAME
SRC/ctgsna.f
.SH SYNOPSIS
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.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBctgsna\fP (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)"
.br
.RI "\fBCTGSNA\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine ctgsna (character job, character howmny, logical, dimension( * ) select, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) s, real, dimension( * ) dif, integer mm, integer m, complex, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)"

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\fBCTGSNA\fP  
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\fBPurpose:\fP
.RS 4

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.nf
!>
!> CTGSNA estimates reciprocal condition numbers for specified
!> eigenvalues and/or eigenvectors of a matrix pair (A, B)\&.
!>
!> (A, B) must be in generalized Schur canonical form, that is, A and
!> B are both upper triangular\&.
!> 
.fi
.PP
 
.RE
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\fBParameters\fP
.RS 4
\fIJOB\fP 
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.nf
!>          JOB is CHARACTER*1
!>          Specifies whether condition numbers are required for
!>          eigenvalues (S) or eigenvectors (DIF):
!>          = 'E': for eigenvalues only (S);
!>          = 'V': for eigenvectors only (DIF);
!>          = 'B': for both eigenvalues and eigenvectors (S and DIF)\&.
!> 
.fi
.PP
.br
\fIHOWMNY\fP 
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!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT\&.
!> 
.fi
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.br
\fISELECT\fP 
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.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 corresponding j-th eigenvalue and/or eigenvector,
!>          SELECT(j) must be set to \&.TRUE\&.\&.
!>          If HOWMNY = 'A', SELECT is not referenced\&.
!> 
.fi
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.br
\fIN\fP 
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!>          N is INTEGER
!>          The order of the square matrix pair (A, B)\&. N >= 0\&.
!> 
.fi
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\fIA\fP 
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!>          A is COMPLEX array, dimension (LDA,N)
!>          The upper triangular matrix A in the pair (A,B)\&.
!> 
.fi
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\fILDA\fP 
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!>          LDA is INTEGER
!>          The leading dimension of the array A\&. LDA >= max(1,N)\&.
!> 
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\fIB\fP 
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!>          B is COMPLEX array, dimension (LDB,N)
!>          The upper triangular matrix B in the pair (A, B)\&.
!> 
.fi
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\fILDB\fP 
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!>          LDB is INTEGER
!>          The leading dimension of the array B\&. LDB >= max(1,N)\&.
!> 
.fi
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\fIVL\fP 
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!>          VL is COMPLEX array, dimension (LDVL,M)
!>          IF JOB = 'E' or 'B', VL must contain left eigenvectors of
!>          (A, B), corresponding to the eigenpairs specified by HOWMNY
!>          and SELECT\&.  The eigenvectors must be stored in consecutive
!>          columns of VL, as returned by CTGEVC\&.
!>          If JOB = 'V', VL is not referenced\&.
!> 
.fi
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.br
\fILDVL\fP 
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!>          LDVL is INTEGER
!>          The leading dimension of the array VL\&. LDVL >= 1; and
!>          If JOB = 'E' or 'B', LDVL >= N\&.
!> 
.fi
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.br
\fIVR\fP 
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!>          VR is COMPLEX array, dimension (LDVR,M)
!>          IF JOB = 'E' or 'B', VR must contain right eigenvectors of
!>          (A, B), corresponding to the eigenpairs specified by HOWMNY
!>          and SELECT\&.  The eigenvectors must be stored in consecutive
!>          columns of VR, as returned by CTGEVC\&.
!>          If JOB = 'V', VR is not referenced\&.
!> 
.fi
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\fILDVR\fP 
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!>          LDVR is INTEGER
!>          The leading dimension of the array VR\&. LDVR >= 1;
!>          If JOB = 'E' or 'B', LDVR >= N\&.
!> 
.fi
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.br
\fIS\fP 
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!>          S is REAL array, dimension (MM)
!>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
!>          selected eigenvalues, stored in consecutive elements of the
!>          array\&.
!>          If JOB = 'V', S is not referenced\&.
!> 
.fi
.PP
.br
\fIDIF\fP 
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.nf
!>          DIF is REAL 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 the eigenvalues cannot be reordered to compute DIF(j),
!>          DIF(j) is set to 0; this can only occur when the true value
!>          would be very small anyway\&.
!>          For each eigenvalue/vector specified by SELECT, DIF stores
!>          a Frobenius norm-based estimate of Difl\&.
!>          If JOB = 'E', DIF is not referenced\&.
!> 
.fi
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\fIMM\fP 
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!>          MM is INTEGER
!>          The number of elements in the arrays S and DIF\&. MM >= M\&.
!> 
.fi
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\fIM\fP 
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!>          M is INTEGER
!>          The number of elements of the arrays S and DIF used to store
!>          the specified condition numbers; for each selected eigenvalue
!>          one element is used\&. If HOWMNY = 'A', M is set to N\&.
!> 
.fi
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.br
\fIWORK\fP 
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!>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
!> 
.fi
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\fILWORK\fP 
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!>          LWORK is INTEGER
!>          The dimension of the array WORK\&. LWORK >= max(1,N)\&.
!>          If JOB = 'V' or 'B', LWORK >= max(1,2*N*N)\&.
!> 
.fi
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\fIIWORK\fP 
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!>          IWORK is INTEGER array, dimension (N+2)
!>          If JOB = 'E', IWORK is not referenced\&.
!> 
.fi
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\fIINFO\fP 
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!>          INFO is INTEGER
!>          = 0: Successful exit
!>          < 0: If INFO = -i, the i-th argument had an illegal value
!> 
.fi
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.RE
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\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

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Univ\&. of California Berkeley 

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Univ\&. of Colorado Denver 

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NAG Ltd\&. 
.RE
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\fBFurther Details:\fP
.RS 4

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!>
!>  The reciprocal of the condition number of the i-th generalized
!>  eigenvalue w = (a, b) is defined as
!>
!>          S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))
!>
!>  where u and v are the right and left eigenvectors of (A, B)
!>  corresponding to w; |z| denotes the absolute value of the complex
!>  number, and norm(u) denotes the 2-norm of the vector u\&. The pair
!>  (a, b) corresponds to an eigenvalue w = a/b (= v**HAu/v**HBu) of the
!>  matrix pair (A, B)\&. If both a and b equal zero, then (A,B) is
!>  singular and S(I) = -1 is returned\&.
!>
!>  An approximate error bound on the chordal distance between the i-th
!>  computed generalized eigenvalue w and the corresponding exact
!>  eigenvalue lambda is
!>
!>          chord(w, lambda) <=   EPS * norm(A, B) / S(I),
!>
!>  where EPS is the machine precision\&.
!>
!>  The reciprocal of the condition number of the right eigenvector u
!>  and left eigenvector v corresponding to the generalized eigenvalue w
!>  is defined as follows\&. Suppose
!>
!>                   (A, B) = ( a   *  ) ( b  *  )  1
!>                            ( 0  A22 ),( 0 B22 )  n-1
!>                              1  n-1     1 n-1
!>
!>  Then the reciprocal condition number DIF(I) is
!>
!>          Difl[(a, b), (A22, B22)]  = sigma-min( Zl )
!>
!>  where sigma-min(Zl) denotes the smallest singular value of
!>
!>         Zl = [ kron(a, In-1) -kron(1, A22) ]
!>              [ kron(b, In-1) -kron(1, B22) ]\&.
!>
!>  Here In-1 is the identity matrix of size n-1 and X**H is the conjugate
!>  transpose of X\&. kron(X, Y) is the Kronecker product between the
!>  matrices X and Y\&.
!>
!>  We approximate the smallest singular value of Zl with an upper
!>  bound\&. This is done by CLATDF\&.
!>
!>  An approximate error bound for a computed eigenvector VL(i) or
!>  VR(i) is given by
!>
!>                      EPS * norm(A, B) / DIF(i)\&.
!>
!>  See ref\&. [2-3] for more details and further references\&.
!> 
.fi
.PP
 
.RE
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\fBContributors:\fP
.RS 4
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden\&. 
.RE
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\fBReferences:\fP
.RS 4

.PP
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!>
!>  [1] B\&. Kagstrom; A Direct Method for Reordering Eigenvalues in the
!>      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
!>      M\&.S\&. Moonen et al (eds), Linear Algebra for Large Scale and
!>      Real-Time Applications, Kluwer Academic Publ\&. 1993, pp 195-218\&.
!>
!>  [2] B\&. Kagstrom and P\&. Poromaa; Computing Eigenspaces with Specified
!>      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
!>      Estimation: Theory, Algorithms and Software, Report
!>      UMINF - 94\&.04, Department of Computing Science, Umea University,
!>      S-901 87 Umea, Sweden, 1994\&. Also as LAPACK Working Note 87\&.
!>      To appear in Numerical Algorithms, 1996\&.
!>
!>  [3] B\&. Kagstrom and P\&. Poromaa, LAPACK-Style Algorithms and Software
!>      for Solving the Generalized Sylvester Equation and Estimating the
!>      Separation between Regular Matrix Pairs, Report UMINF - 93\&.23,
!>      Department of Computing Science, Umea University, S-901 87 Umea,
!>      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
!>      Note 75\&.
!>      To appear in ACM Trans\&. on Math\&. Software, Vol 22, No 1, 1996\&.
!> 
.fi
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.RE
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Definition at line \fB308\fP of file \fBctgsna\&.f\fP\&.
.SH "Author"
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Generated automatically by Doxygen for LAPACK from the source code\&.