TESTING/EIG/sdrgsx.f(3)

NAME
TESTING/EIG/sdrgsx.f

SYNOPSIS

  Functions/Subroutines
subroutine sdrgsx (nsize, ncmax, thresh, nin, nout, a, lda, b, ai, bi, z, q, alphar, alphai, beta, c, ldc, s, work, lwork, iwork, liwork, bwork, info)
SDRGSX

Function/Subroutine Documentation

  subroutine sdrgsx (integer nsize, integer ncmax, real thresh, integer nin, integer nout, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) ai, real, dimension( lda, * ) bi, real, dimension( lda, * ) z, real, dimension( lda, * ) q, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) s, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

SDRGSX

Purpose:

    !>
    !> SDRGSX checks the nonsymmetric generalized eigenvalue (Schur form)
    !> problem expert driver SGGESX.
    !>
    !> SGGESX factors A and B as Q S Z' and Q T Z', where ' means
    !> transpose, T is upper triangular, S is in generalized Schur form
    !> (block upper triangular, with 1x1 and 2x2 blocks on the diagonal,
    !> the 2x2 blocks corresponding to complex conjugate pairs of
    !> generalized eigenvalues), and Q and Z are orthogonal.  It also
    !> computes the generalized eigenvalues (alpha(1),beta(1)), ...,
    !> (alpha(n),beta(n)). Thus, w(j) = alpha(j)/beta(j) is a root of the
    !> characteristic equation
    !>
    !>     det( A - w(j) B ) = 0
    !>
    !> Optionally it also reorders the eigenvalues so that a selected
    !> cluster of eigenvalues appears in the leading diagonal block of the
    !> Schur forms; computes a reciprocal condition number for the average
    !> of the selected eigenvalues; and computes a reciprocal condition
    !> number for the right and left deflating subspaces corresponding to
    !> the selected eigenvalues.
    !>
    !> When SDRGSX is called with NSIZE > 0, five (5) types of built-in
    !> matrix pairs are used to test the routine SGGESX.
    !>
    !> When SDRGSX is called with NSIZE = 0, it reads in test matrix data
    !> to test SGGESX.
    !>
    !> For each matrix pair, the following tests will be performed and
    !> compared with the threshold THRESH except for the tests (7) and (9):
    !>
    !> (1)   | A - Q S Z' | / ( |A| n ulp )
    !>
    !> (2)   | B - Q T Z' | / ( |B| n ulp )
    !>
    !> (3)   | I - QQ' | / ( n ulp )
    !>
    !> (4)   | I - ZZ' | / ( n ulp )
    !>
    !> (5)   if A is in Schur form (i.e. quasi-triangular form)
    !>
    !> (6)   maximum over j of D(j)  where:
    !>
    !>       if alpha(j) is real:
    !>                     |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
    !>           D(j) = ------------------------ + -----------------------
    !>                  max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)
    !>
    !>       if alpha(j) is complex:
    !>                                 | det( s S - w T ) |
    !>           D(j) = ---------------------------------------------------
    !>                  ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )
    !>
    !>           and S and T are here the 2 x 2 diagonal blocks of S and T
    !>           corresponding to the j-th and j+1-th eigenvalues.
    !>
    !> (7)   if sorting worked and SDIM is the number of eigenvalues
    !>       which were selected.
    !>
    !> (8)   the estimated value DIF does not differ from the true values of
    !>       Difu and Difl more than a factor 10*THRESH. If the estimate DIF
    !>       equals zero the corresponding true values of Difu and Difl
    !>       should be less than EPS*norm(A, B). If the true value of Difu
    !>       and Difl equal zero, the estimate DIF should be less than
    !>       EPS*norm(A, B).
    !>
    !> (9)   If INFO = N+3 is returned by SGGESX, the reordering
    !>       and we check that DIF = PL = PR = 0 and that the true value of
    !>       Difu and Difl is < EPS*norm(A, B). We count the events when
    !>       INFO=N+3.
    !>
    !> For read-in test matrices, the above tests are run except that the
    !> exact value for DIF (and PL) is input data.  Additionally, there is
    !> one more test run for read-in test matrices:
    !>
    !> (10)  the estimated value PL does not differ from the true value of
    !>       PLTRU more than a factor THRESH. If the estimate PL equals
    !>       zero the corresponding true value of PLTRU should be less than
    !>       EPS*norm(A, B). If the true value of PLTRU equal zero, the
    !>       estimate PL should be less than EPS*norm(A, B).
    !>
    !> Note that for the built-in tests, a total of 10*NSIZE*(NSIZE-1)
    !> matrix pairs are generated and tested. NSIZE should be kept small.
    !>
    !> SVD (routine SGESVD) is used for computing the true value of DIF_u
    !> and DIF_l when testing the built-in test problems.
    !>
    !> Built-in Test Matrices
    !> ======================
    !>
    !> All built-in test matrices are the 2 by 2 block of triangular
    !> matrices
    !>
    !>          A = [ A11 A12 ]    and      B = [ B11 B12 ]
    !>              [     A22 ]                 [     B22 ]
    !>
    !> where for different type of A11 and A22 are given as the following.
    !> A12 and B12 are chosen so that the generalized Sylvester equation
    !>
    !>          A11*R - L*A22 = -A12
    !>          B11*R - L*B22 = -B12
    !>
    !> have prescribed solution R and L.
    !>
    !> Type 1:  A11 = J_m(1,-1) and A_22 = J_k(1-a,1).
    !>          B11 = I_m, B22 = I_k
    !>          where J_k(a,b) is the k-by-k Jordan block with ``a'' on
    !>          diagonal and ``b'' on superdiagonal.
    !>
    !> Type 2:  A11 = (a_ij) = ( 2(.5-sin(i)) ) and
    !>          B11 = (b_ij) = ( 2(.5-sin(ij)) ) for i=1,...,m, j=i,...,m
    !>          A22 = (a_ij) = ( 2(.5-sin(i+j)) ) and
    !>          B22 = (b_ij) = ( 2(.5-sin(ij)) ) for i=m+1,...,k, j=i,...,k
    !>
    !> Type 3:  A11, A22 and B11, B22 are chosen as for Type 2, but each
    !>          second diagonal block in A_11 and each third diagonal block
    !>          in A_22 are made as 2 by 2 blocks.
    !>
    !> Type 4:  A11 = ( 20(.5 - sin(ij)) ) and B22 = ( 2(.5 - sin(i+j)) )
    !>             for i=1,...,m,  j=1,...,m and
    !>          A22 = ( 20(.5 - sin(i+j)) ) and B22 = ( 2(.5 - sin(ij)) )
    !>             for i=m+1,...,k,  j=m+1,...,k
    !>
    !> Type 5:  (A,B) and have potentially close or common eigenvalues and
    !>          very large departure from block diagonality A_11 is chosen
    !>          as the m x m leading submatrix of A_1:
    !>                  |  1  b                            |
    !>                  | -b  1                            |
    !>                  |        1+d  b                    |
    !>                  |         -b 1+d                   |
    !>           A_1 =  |                  d  1            |
    !>                  |                 -1  d            |
    !>                  |                        -d  1     |
    !>                  |                        -1 -d     |
    !>                  |                               1  |
    !>          and A_22 is chosen as the k x k leading submatrix of A_2:
    !>                  | -1  b                            |
    !>                  | -b -1                            |
    !>                  |       1-d  b                     |
    !>                  |       -b  1-d                    |
    !>           A_2 =  |                 d 1+b            |
    !>                  |               -1-b d             |
    !>                  |                       -d  1+b    |
    !>                  |                      -1+b  -d    |
    !>                  |                              1-d |
    !>          and matrix B are chosen as identity matrices (see SLATM5).
    !>
    !>

Parameters
    NSIZE

    !>          NSIZE is INTEGER
    !>          The maximum size of the matrices to use. NSIZE >= 0.
    !>          If NSIZE = 0, no built-in tests matrices are used, but
    !>          read-in test matrices are used to test SGGESX.
    !>

    NCMAX

    !>          NCMAX is INTEGER
    !>          Maximum allowable NMAX for generating Kroneker matrix
    !>          in call to SLAKF2
    !>

    THRESH

    !>          THRESH is REAL
    !>          A test will count as  if the , computed as
    !>          described above, exceeds THRESH.  Note that the error
    !>          is scaled to be O(1), so THRESH should be a reasonably
    !>          small multiple of 1, e.g., 10 or 100.  In particular,
    !>          it should not depend on the precision (single vs. double)
    !>          or the size of the matrix.  THRESH >= 0.
    !>

    NIN

    !>          NIN is INTEGER
    !>          The FORTRAN unit number for reading in the data file of
    !>          problems to solve.
    !>

    NOUT

    !>          NOUT is INTEGER
    !>          The FORTRAN unit number for printing out error messages
    !>          (e.g., if a routine returns IINFO not equal to 0.)
    !>

    A

    !>          A is REAL array, dimension (LDA, NSIZE)
    !>          Used to store the matrix whose eigenvalues are to be
    !>          computed.  On exit, A contains the last matrix actually used.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of A, B, AI, BI, Z and Q,
    !>          LDA >= max( 1, NSIZE ). For the read-in test,
    !>          LDA >= max( 1, N ), N is the size of the test matrices.
    !>

    B

    !>          B is REAL array, dimension (LDA, NSIZE)
    !>          Used to store the matrix whose eigenvalues are to be
    !>          computed.  On exit, B contains the last matrix actually used.
    !>

    AI

    !>          AI is REAL array, dimension (LDA, NSIZE)
    !>          Copy of A, modified by SGGESX.
    !>

    BI

    !>          BI is REAL array, dimension (LDA, NSIZE)
    !>          Copy of B, modified by SGGESX.
    !>

    Z

    !>          Z is REAL array, dimension (LDA, NSIZE)
    !>          Z holds the left Schur vectors computed by SGGESX.
    !>

    Q

    !>          Q is REAL array, dimension (LDA, NSIZE)
    !>          Q holds the right Schur vectors computed by SGGESX.
    !>

    ALPHAR

    !>          ALPHAR is REAL array, dimension (NSIZE)
    !>

    ALPHAI

    !>          ALPHAI is REAL array, dimension (NSIZE)
    !>

    BETA

    !>          BETA is REAL array, dimension (NSIZE)
    !> \verbatim
    !>          On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues.
    !>

    C

    !>          C is REAL array, dimension (LDC, LDC)
    !>          Store the matrix generated by subroutine SLAKF2, this is the
    !>          matrix formed by Kronecker products used for estimating
    !>          DIF.
    !>

    LDC

    !>          LDC is INTEGER
    !>          The leading dimension of C. LDC >= max(1, LDA*LDA/2 ).
    !>

    S

    !>          S is REAL array, dimension (LDC)
    !>          Singular values of C
    !>

    WORK

    !>          WORK is REAL array, dimension (LWORK)
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= MAX( 5*NSIZE*NSIZE/2 - 2, 10*(NSIZE+1) )
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (LIWORK)
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK. LIWORK >= NSIZE + 6.
    !>

    BWORK

    !>          BWORK is LOGICAL array, dimension (LDA)
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  A routine returned an error code.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 356 of file sdrgsx.f.

Author

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