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NAME

       ctgsna.f -

SYNOPSIS

   Functions/Subroutines
       subroutine ctgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM,
           M, WORK, LWORK, IWORK, INFO)
           CTGSNA

Function/Subroutine Documentation

   subroutine ctgsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN,
       complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB,
       complex, dimension( ldvl, * )VL, integerLDVL, complex, dimension( ldvr, * )VR,
       integerLDVR, real, dimension( * )S, real, dimension( * )DIF, integerMM, integerM, complex,
       dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)
       CTGSNA

       Purpose:

            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.

       Parameters:
           JOB

                     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).

           HOWMNY

                     HOWMNY is CHARACTER*1
                     = 'A': compute condition numbers for all eigenpairs;
                     = 'S': compute condition numbers for selected eigenpairs
                            specified by the array SELECT.

           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 corresponding j-th eigenvalue and/or eigenvector,
                     SELECT(j) must be set to .TRUE..
                     If HOWMNY = 'A', SELECT is not referenced.

           N

                     N is INTEGER
                     The order of the square matrix pair (A, B). N >= 0.

           A

                     A is COMPLEX array, dimension (LDA,N)
                     The upper triangular matrix A in the pair (A,B).

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,N).

           B

                     B is COMPLEX array, dimension (LDB,N)
                     The upper triangular matrix B in the pair (A, B).

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,N).

           VL

                     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.

           LDVL

                     LDVL is INTEGER
                     The leading dimension of the array VL. LDVL >= 1; and
                     If JOB = 'E' or 'B', LDVL >= N.

           VR

                     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.

           LDVR

                     LDVR is INTEGER
                     The leading dimension of the array VR. LDVR >= 1;
                     If JOB = 'E' or 'B', LDVR >= N.

           S

                     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.

           DIF

                     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.

           MM

                     MM is INTEGER
                     The number of elements in the arrays S and DIF. MM >= M.

           M

                     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.

           WORK

                     WORK is COMPLEX array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK >= max(1,N).
                     If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).

           IWORK

                     IWORK is INTEGER array, dimension (N+2)
                     If JOB = 'E', IWORK is not referenced.

           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.

       Date:
           November 2011

       Further Details:

             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.

       Contributors:
           Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
           87 Umea, Sweden.

       References:

             [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.

       Definition at line 310 of file ctgsna.f.

Author

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