ZTGSNA(3)      LAPACK routine of NEC Numeric Library Collection      ZTGSNA(3)



NAME
       ZTGSNA

SYNOPSIS
       SUBROUTINE ZTGSNA (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL,
           VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)



PURPOSE
            ZTGSNA 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.




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

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

           A         (input)
                     A is COMPLEX*16 array, dimension (LDA,N)
                     The upper triangular matrix A in the pair (A,B).

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

           B         (input)
                     B is COMPLEX*16 array, dimension (LDB,N)
                     The upper triangular matrix B in the pair (A, B).

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

           VL        (input)
                     VL is COMPLEX*16 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 ZTGEVC.
                     If JOB = 'V', VL is not referenced.

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

           VR        (input)
                     VR is COMPLEX*16 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 ZTGEVC.
                     If JOB = 'V', VR is not referenced.

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

           S         (output)
                     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.
                     If JOB = 'V', S is not referenced.

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

           M         (output)
                     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      (output)
                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

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

           INFO      (output)
                     INFO is INTEGER
                     = 0: Successful exit
                     < 0: If INFO = -i, the i-th argument had an illegal value






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 ZLATDF.

             An approximate error bound for a computed eigenvector VL(i) or
             VR(i) is given by

                                 EPS * norm(A, B) / DIF(i).




LAPACK routine                  31 October 2017                      ZTGSNA(3)