ddrgsx.f

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*> \brief \b DDRGSX
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE DDRGSX( NSIZE, NCMAX, THRESH, NIN, NOUT, A, LDA, B, AI,
*                          BI, Z, Q, ALPHAR, ALPHAI, BETA, C, LDC, S,
*                          WORK, LWORK, IWORK, LIWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDC, LIWORK, LWORK, NCMAX, NIN,
*      $                   NOUT, NSIZE
*       DOUBLE PRECISION   THRESH
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), AI( LDA, * ), ALPHAI( * ),
*      $                   ALPHAR( * ), B( LDA, * ), BETA( * ),
*      $                   BI( LDA, * ), C( LDC, * ), Q( LDA, * ), S( * ),
*      $                   WORK( * ), Z( LDA, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DDRGSX checks the nonsymmetric generalized eigenvalue (Schur form)
*> problem expert driver DGGESX.
*>
*> DGGESX 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 DDRGSX is called with NSIZE > 0, five (5) types of built-in
*> matrix pairs are used to test the routine DGGESX.
*>
*> When DDRGSX is called with NSIZE = 0, it reads in test matrix data
*> to test DGGESX.
*>
*> 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 DGGESX, the reordering "failed"
*>       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 DGESVD) 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 DLATM5).
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NSIZE
*> \verbatim
*>          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 DGGESX.
*> \endverbatim
*>
*> \param[in] NCMAX
*> \verbatim
*>          NCMAX is INTEGER
*>          Maximum allowable NMAX for generating Kroneker matrix
*>          in call to DLAKF2
*> \endverbatim
*>
*> \param[in] THRESH
*> \verbatim
*>          THRESH is DOUBLE PRECISION
*>          A test will count as "failed" if the "error", 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.
*> \endverbatim
*>
*> \param[in] NIN
*> \verbatim
*>          NIN is INTEGER
*>          The FORTRAN unit number for reading in the data file of
*>          problems to solve.
*> \endverbatim
*>
*> \param[in] NOUT
*> \verbatim
*>          NOUT is INTEGER
*>          The FORTRAN unit number for printing out error messages
*>          (e.g., if a routine returns IINFO not equal to 0.)
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, NSIZE)
*>          Used to store the matrix whose eigenvalues are to be
*>          computed.  On exit, A contains the last matrix actually used.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDA, NSIZE)
*>          Used to store the matrix whose eigenvalues are to be
*>          computed.  On exit, B contains the last matrix actually used.
*> \endverbatim
*>
*> \param[out] AI
*> \verbatim
*>          AI is DOUBLE PRECISION array, dimension (LDA, NSIZE)
*>          Copy of A, modified by DGGESX.
*> \endverbatim
*>
*> \param[out] BI
*> \verbatim
*>          BI is DOUBLE PRECISION array, dimension (LDA, NSIZE)
*>          Copy of B, modified by DGGESX.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is DOUBLE PRECISION array, dimension (LDA, NSIZE)
*>          Z holds the left Schur vectors computed by DGGESX.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is DOUBLE PRECISION array, dimension (LDA, NSIZE)
*>          Q holds the right Schur vectors computed by DGGESX.
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (NSIZE)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (NSIZE)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (NSIZE)
*>
*>          On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*>          C is DOUBLE PRECISION array, dimension (LDC, LDC)
*>          Store the matrix generated by subroutine DLAKF2, this is the
*>          matrix formed by Kronecker products used for estimating
*>          DIF.
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*>          LDC is INTEGER
*>          The leading dimension of C. LDC >= max(1, LDA*LDA/2 ).
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (LDC)
*>          Singular values of C
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          LWORK >= MAX( 5*NSIZE*NSIZE/2 - 2, 10*(NSIZE+1) )
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (LIWORK)
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK. LIWORK >= NSIZE + 6.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (LDA)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          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.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup double_eig
*
*  =====================================================================
      SUBROUTINE ddrgsx( NSIZE, NCMAX, THRESH, NIN, NOUT, A, LDA, B, AI,
     $                   BI, Z, Q, ALPHAR, ALPHAI, BETA, C, LDC, S,
     $                   WORK, LWORK, IWORK, LIWORK, BWORK, INFO )
*
*  -- LAPACK test routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDC, LIWORK, LWORK, NCMAX, NIN,
     $                   nout, nsize
      DOUBLE PRECISION   THRESH
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( lda, * ), AI( lda, * ), ALPHAI( * ),
     $                   alphar( * ), b( lda, * ), beta( * ),
     $                   bi( lda, * ), c( ldc, * ), q( lda, * ), s( * ),
     $                   work( * ), z( lda, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TEN
      parameter( zero = 0.0d+0, one = 1.0d+0, ten = 1.0d+1 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILABAD
      CHARACTER          SENSE
      INTEGER            BDSPAC, I, I1, IFUNC, IINFO, J, LINFO, MAXWRK,
     $                   minwrk, mm, mn2, nerrs, nptknt, ntest, ntestt,
     $                   prtype, qba, qbb
      DOUBLE PRECISION   ABNRM, BIGNUM, DIFTRU, PLTRU, SMLNUM, TEMP1,
     $                   temp2, thrsh2, ulp, ulpinv, weight
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DIFEST( 2 ), PL( 2 ), RESULT( 10 )
*     ..
*     .. External Functions ..
      LOGICAL            DLCTSX
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           dlctsx, ilaenv, dlamch, dlange
*     ..
*     .. External Subroutines ..
      EXTERNAL           alasvm, dgesvd, dget51, dget53, dggesx, dlabad,
     $                   dlacpy, dlakf2, dlaset, dlatm5, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, sqrt
*     ..
*     .. Scalars in Common ..
      LOGICAL            FS
      INTEGER            K, M, MPLUSN, N
*     ..
*     .. Common blocks ..
      COMMON             / mn / m, n, mplusn, k, fs
*     ..
*     .. Executable Statements ..
*
*     Check for errors
*
      IF( nsize.LT.0 ) THEN
         info = -1
      ELSE IF( thresh.LT.zero ) THEN
         info = -2
      ELSE IF( nin.LE.0 ) THEN
         info = -3
      ELSE IF( nout.LE.0 ) THEN
         info = -4
      ELSE IF( lda.LT.1 .OR. lda.LT.nsize ) THEN
         info = -6
      ELSE IF( ldc.LT.1 .OR. ldc.LT.nsize*nsize / 2 ) THEN
         info = -17
      ELSE IF( liwork.LT.nsize+6 ) THEN
         info = -21
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.)
*
      minwrk = 1
      IF( info.EQ.0 .AND. lwork.GE.1 ) THEN
         minwrk = max( 10*( nsize+1 ), 5*nsize*nsize / 2 )
*
*        workspace for sggesx
*
         maxwrk = 9*( nsize+1 ) + nsize*
     $            ilaenv( 1, 'DGEQRF', ' ', nsize, 1, nsize, 0 )
         maxwrk = max( maxwrk, 9*( nsize+1 )+nsize*
     $            ilaenv( 1, 'DORGQR', ' ', nsize, 1, nsize, -1 ) )
*
*        workspace for dgesvd
*
         bdspac = 5*nsize*nsize / 2
         maxwrk = max( maxwrk, 3*nsize*nsize / 2+nsize*nsize*
     $            ilaenv( 1, 'DGEBRD', ' ', nsize*nsize / 2,
     $            nsize*nsize / 2, -1, -1 ) )
         maxwrk = max( maxwrk, bdspac )
*
         maxwrk = max( maxwrk, minwrk )
*
         work( 1 ) = maxwrk
      END IF
*
      IF( lwork.LT.minwrk )
     $   info = -19
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DDRGSX', -info )
         RETURN
      END IF
*
*     Important constants
*
      ulp = dlamch( 'P' )
      ulpinv = one / ulp
      smlnum = dlamch( 'S' ) / ulp
      bignum = one / smlnum
      CALL dlabad( smlnum, bignum )
      thrsh2 = ten*thresh
      ntestt = 0
      nerrs = 0
*
*     Go to the tests for read-in matrix pairs
*
      ifunc = 0
      IF( nsize.EQ.0 )
     $   GO TO 70
*
*     Test the built-in matrix pairs.
*     Loop over different functions (IFUNC) of DGGESX, types (PRTYPE)
*     of test matrices, different size (M+N)
*
      prtype = 0
      qba = 3
      qbb = 4
      weight = sqrt( ulp )
*
      DO 60 ifunc = 0, 3
         DO 50 prtype = 1, 5
            DO 40 m = 1, nsize - 1
               DO 30 n = 1, nsize - m
*
                  weight = one / weight
                  mplusn = m + n
*
*                 Generate test matrices
*
                  fs = .true.
                  k = 0
*
                  CALL dlaset( 'Full', mplusn, mplusn, zero, zero, ai,
     $                         lda )
                  CALL dlaset( 'Full', mplusn, mplusn, zero, zero, bi,
     $                         lda )
*
                  CALL dlatm5( prtype, m, n, ai, lda, ai( m+1, m+1 ),
     $                         lda, ai( 1, m+1 ), lda, bi, lda,
     $                         bi( m+1, m+1 ), lda, bi( 1, m+1 ), lda,
     $                         q, lda, z, lda, weight, qba, qbb )
*
*                 Compute the Schur factorization and swapping the
*                 m-by-m (1,1)-blocks with n-by-n (2,2)-blocks.
*                 Swapping is accomplished via the function DLCTSX
*                 which is supplied below.
*
                  IF( ifunc.EQ.0 ) THEN
                     sense = 'N'
                  ELSE IF( ifunc.EQ.1 ) THEN
                     sense = 'E'
                  ELSE IF( ifunc.EQ.2 ) THEN
                     sense = 'V'
                  ELSE IF( ifunc.EQ.3 ) THEN
                     sense = 'B'
                  END IF
*
                  CALL dlacpy( 'Full', mplusn, mplusn, ai, lda, a, lda )
                  CALL dlacpy( 'Full', mplusn, mplusn, bi, lda, b, lda )
*
                  CALL dggesx( 'V', 'V', 'S', dlctsx, sense, mplusn, ai,
     $                         lda, bi, lda, mm, alphar, alphai, beta,
     $                         q, lda, z, lda, pl, difest, work, lwork,
     $                         iwork, liwork, bwork, linfo )
*
                  IF( linfo.NE.0 .AND. linfo.NE.mplusn+2 ) THEN
                     result( 1 ) = ulpinv
                     WRITE( nout, fmt = 9999 )'DGGESX', linfo, mplusn,
     $                  prtype
                     info = linfo
                     GO TO 30
                  END IF
*
*                 Compute the norm(A, B)
*
                  CALL dlacpy( 'Full', mplusn, mplusn, ai, lda, work,
     $                         mplusn )
                  CALL dlacpy( 'Full', mplusn, mplusn, bi, lda,
     $                         work( mplusn*mplusn+1 ), mplusn )
                  abnrm = dlange( 'Fro', mplusn, 2*mplusn, work, mplusn,
     $                    work )
*
*                 Do tests (1) to (4)
*
                  CALL dget51( 1, mplusn, a, lda, ai, lda, q, lda, z,
     $                         lda, work, result( 1 ) )
                  CALL dget51( 1, mplusn, b, lda, bi, lda, q, lda, z,
     $                         lda, work, result( 2 ) )
                  CALL dget51( 3, mplusn, b, lda, bi, lda, q, lda, q,
     $                         lda, work, result( 3 ) )
                  CALL dget51( 3, mplusn, b, lda, bi, lda, z, lda, z,
     $                         lda, work, result( 4 ) )
                  ntest = 4
*
*                 Do tests (5) and (6): check Schur form of A and
*                 compare eigenvalues with diagonals.
*
                  temp1 = zero
                  result( 5 ) = zero
                  result( 6 ) = zero
*
                  DO 10 j = 1, mplusn
                     ilabad = .false.
                     IF( alphai( j ).EQ.zero ) THEN
                        temp2 = ( abs( alphar( j )-ai( j, j ) ) /
     $                          max( smlnum, abs( alphar( j ) ),
     $                          abs( ai( j, j ) ) )+
     $                          abs( beta( j )-bi( j, j ) ) /
     $                          max( smlnum, abs( beta( j ) ),
     $                          abs( bi( j, j ) ) ) ) / ulp
                        IF( j.LT.mplusn ) THEN
                           IF( ai( j+1, j ).NE.zero ) THEN
                              ilabad = .true.
                              result( 5 ) = ulpinv
                           END IF
                        END IF
                        IF( j.GT.1 ) THEN
                           IF( ai( j, j-1 ).NE.zero ) THEN
                              ilabad = .true.
                              result( 5 ) = ulpinv
                           END IF
                        END IF
                     ELSE
                        IF( alphai( j ).GT.zero ) THEN
                           i1 = j
                        ELSE
                           i1 = j - 1
                        END IF
                        IF( i1.LE.0 .OR. i1.GE.mplusn ) THEN
                           ilabad = .true.
                        ELSE IF( i1.LT.mplusn-1 ) THEN
                           IF( ai( i1+2, i1+1 ).NE.zero ) THEN
                              ilabad = .true.
                              result( 5 ) = ulpinv
                           END IF
                        ELSE IF( i1.GT.1 ) THEN
                           IF( ai( i1, i1-1 ).NE.zero ) THEN
                              ilabad = .true.
                              result( 5 ) = ulpinv
                           END IF
                        END IF
                        IF( .NOT.ilabad ) THEN
                           CALL dget53( ai( i1, i1 ), lda, bi( i1, i1 ),
     $                                  lda, beta( j ), alphar( j ),
     $                                  alphai( j ), temp2, iinfo )
                           IF( iinfo.GE.3 ) THEN
                              WRITE( nout, fmt = 9997 )iinfo, j,
     $                           mplusn, prtype
                              info = abs( iinfo )
                           END IF
                        ELSE
                           temp2 = ulpinv
                        END IF
                     END IF
                     temp1 = max( temp1, temp2 )
                     IF( ilabad ) THEN
                        WRITE( nout, fmt = 9996 )j, mplusn, prtype
                     END IF
   10             CONTINUE
                  result( 6 ) = temp1
                  ntest = ntest + 2
*
*                 Test (7) (if sorting worked)
*
                  result( 7 ) = zero
                  IF( linfo.EQ.mplusn+3 ) THEN
                     result( 7 ) = ulpinv
                  ELSE IF( mm.NE.n ) THEN
                     result( 7 ) = ulpinv
                  END IF
                  ntest = ntest + 1
*
*                 Test (8): compare the estimated value DIF and its
*                 value. first, compute the exact DIF.
*
                  result( 8 ) = zero
                  mn2 = mm*( mplusn-mm )*2
                  IF( ifunc.GE.2 .AND. mn2.LE.ncmax*ncmax ) THEN
*
*                    Note: for either following two causes, there are
*                    almost same number of test cases fail the test.
*
                     CALL dlakf2( mm, mplusn-mm, ai, lda,
     $                            ai( mm+1, mm+1 ), bi,
     $                            bi( mm+1, mm+1 ), c, ldc )
*
                     CALL dgesvd( 'N', 'N', mn2, mn2, c, ldc, s, work,
     $                            1, work( 2 ), 1, work( 3 ), lwork-2,
     $                            info )
                     diftru = s( mn2 )
*
                     IF( difest( 2 ).EQ.zero ) THEN
                        IF( diftru.GT.abnrm*ulp )
     $                     result( 8 ) = ulpinv
                     ELSE IF( diftru.EQ.zero ) THEN
                        IF( difest( 2 ).GT.abnrm*ulp )
     $                     result( 8 ) = ulpinv
                     ELSE IF( ( diftru.GT.thrsh2*difest( 2 ) ) .OR.
     $                        ( diftru*thrsh2.LT.difest( 2 ) ) ) THEN
                        result( 8 ) = max( diftru / difest( 2 ),
     $                                difest( 2 ) / diftru )
                     END IF
                     ntest = ntest + 1
                  END IF
*
*                 Test (9)
*
                  result( 9 ) = zero
                  IF( linfo.EQ.( mplusn+2 ) ) THEN
                     IF( diftru.GT.abnrm*ulp )
     $                  result( 9 ) = ulpinv
                     IF( ( ifunc.GT.1 ) .AND. ( difest( 2 ).NE.zero ) )
     $                  result( 9 ) = ulpinv
                     IF( ( ifunc.EQ.1 ) .AND. ( pl( 1 ).NE.zero ) )
     $                  result( 9 ) = ulpinv
                     ntest = ntest + 1
                  END IF
*
                  ntestt = ntestt + ntest
*
*                 Print out tests which fail.
*
                  DO 20 j = 1, 9
                     IF( result( j ).GE.thresh ) THEN
*
*                       If this is the first test to fail,
*                       print a header to the data file.
*
                        IF( nerrs.EQ.0 ) THEN
                           WRITE( nout, fmt = 9995 )'DGX'
*
*                          Matrix types
*
                           WRITE( nout, fmt = 9993 )
*
*                          Tests performed
*
                           WRITE( nout, fmt = 9992 )'orthogonal', '''',
     $                        'transpose', ( '''', i = 1, 4 )
*
                        END IF
                        nerrs = nerrs + 1
                        IF( result( j ).LT.10000.0d0 ) THEN
                           WRITE( nout, fmt = 9991 )mplusn, prtype,
     $                        weight, m, j, result( j )
                        ELSE
                           WRITE( nout, fmt = 9990 )mplusn, prtype,
     $                        weight, m, j, result( j )
                        END IF
                     END IF
   20             CONTINUE
*
   30          CONTINUE
   40       CONTINUE
   50    CONTINUE
   60 CONTINUE
*
      GO TO 150
*
   70 CONTINUE
*
*     Read in data from file to check accuracy of condition estimation
*     Read input data until N=0
*
      nptknt = 0
*
   80 CONTINUE
      READ( nin, fmt = *, end = 140 )mplusn
      IF( mplusn.EQ.0 )
     $   GO TO 140
      READ( nin, fmt = *, end = 140 )n
      DO 90 i = 1, mplusn
         READ( nin, fmt = * )( ai( i, j ), j = 1, mplusn )
   90 CONTINUE
      DO 100 i = 1, mplusn
         READ( nin, fmt = * )( bi( i, j ), j = 1, mplusn )
  100 CONTINUE
      READ( nin, fmt = * )pltru, diftru
*
      nptknt = nptknt + 1
      fs = .true.
      k = 0
      m = mplusn - n
*
      CALL dlacpy( 'Full', mplusn, mplusn, ai, lda, a, lda )
      CALL dlacpy( 'Full', mplusn, mplusn, bi, lda, b, lda )
*
*     Compute the Schur factorization while swaping the
*     m-by-m (1,1)-blocks with n-by-n (2,2)-blocks.
*
      CALL dggesx( 'V', 'V', 'S', dlctsx, 'B', mplusn, ai, lda, bi, lda,
     $             mm, alphar, alphai, beta, q, lda, z, lda, pl, difest,
     $             work, lwork, iwork, liwork, bwork, linfo )
*
      IF( linfo.NE.0 .AND. linfo.NE.mplusn+2 ) THEN
         result( 1 ) = ulpinv
         WRITE( nout, fmt = 9998 )'DGGESX', linfo, mplusn, nptknt
         GO TO 130
      END IF
*
*     Compute the norm(A, B)
*        (should this be norm of (A,B) or (AI,BI)?)
*
      CALL dlacpy( 'Full', mplusn, mplusn, ai, lda, work, mplusn )
      CALL dlacpy( 'Full', mplusn, mplusn, bi, lda,
     $             work( mplusn*mplusn+1 ), mplusn )
      abnrm = dlange( 'Fro', mplusn, 2*mplusn, work, mplusn, work )
*
*     Do tests (1) to (4)
*
      CALL dget51( 1, mplusn, a, lda, ai, lda, q, lda, z, lda, work,
     $             result( 1 ) )
      CALL dget51( 1, mplusn, b, lda, bi, lda, q, lda, z, lda, work,
     $             result( 2 ) )
      CALL dget51( 3, mplusn, b, lda, bi, lda, q, lda, q, lda, work,
     $             result( 3 ) )
      CALL dget51( 3, mplusn, b, lda, bi, lda, z, lda, z, lda, work,
     $             result( 4 ) )
*
*     Do tests (5) and (6): check Schur form of A and compare
*     eigenvalues with diagonals.
*
      ntest = 6
      temp1 = zero
      result( 5 ) = zero
      result( 6 ) = zero
*
      DO 110 j = 1, mplusn
         ilabad = .false.
         IF( alphai( j ).EQ.zero ) THEN
            temp2 = ( abs( alphar( j )-ai( j, j ) ) /
     $              max( smlnum, abs( alphar( j ) ), abs( ai( j,
     $              j ) ) )+abs( beta( j )-bi( j, j ) ) /
     $              max( smlnum, abs( beta( j ) ), abs( bi( j, j ) ) ) )
     $               / ulp
            IF( j.LT.mplusn ) THEN
               IF( ai( j+1, j ).NE.zero ) THEN
                  ilabad = .true.
                  result( 5 ) = ulpinv
               END IF
            END IF
            IF( j.GT.1 ) THEN
               IF( ai( j, j-1 ).NE.zero ) THEN
                  ilabad = .true.
                  result( 5 ) = ulpinv
               END IF
            END IF
         ELSE
            IF( alphai( j ).GT.zero ) THEN
               i1 = j
            ELSE
               i1 = j - 1
            END IF
            IF( i1.LE.0 .OR. i1.GE.mplusn ) THEN
               ilabad = .true.
            ELSE IF( i1.LT.mplusn-1 ) THEN
               IF( ai( i1+2, i1+1 ).NE.zero ) THEN
                  ilabad = .true.
                  result( 5 ) = ulpinv
               END IF
            ELSE IF( i1.GT.1 ) THEN
               IF( ai( i1, i1-1 ).NE.zero ) THEN
                  ilabad = .true.
                  result( 5 ) = ulpinv
               END IF
            END IF
            IF( .NOT.ilabad ) THEN
               CALL dget53( ai( i1, i1 ), lda, bi( i1, i1 ), lda,
     $                      beta( j ), alphar( j ), alphai( j ), temp2,
     $                      iinfo )
               IF( iinfo.GE.3 ) THEN
                  WRITE( nout, fmt = 9997 )iinfo, j, mplusn, nptknt
                  info = abs( iinfo )
               END IF
            ELSE
               temp2 = ulpinv
            END IF
         END IF
         temp1 = max( temp1, temp2 )
         IF( ilabad ) THEN
            WRITE( nout, fmt = 9996 )j, mplusn, nptknt
         END IF
  110 CONTINUE
      result( 6 ) = temp1
*
*     Test (7) (if sorting worked)  <--------- need to be checked.
*
      ntest = 7
      result( 7 ) = zero
      IF( linfo.EQ.mplusn+3 )
     $   result( 7 ) = ulpinv
*
*     Test (8): compare the estimated value of DIF and its true value.
*
      ntest = 8
      result( 8 ) = zero
      IF( difest( 2 ).EQ.zero ) THEN
         IF( diftru.GT.abnrm*ulp )
     $      result( 8 ) = ulpinv
      ELSE IF( diftru.EQ.zero ) THEN
         IF( difest( 2 ).GT.abnrm*ulp )
     $      result( 8 ) = ulpinv
      ELSE IF( ( diftru.GT.thrsh2*difest( 2 ) ) .OR.
     $         ( diftru*thrsh2.LT.difest( 2 ) ) ) THEN
         result( 8 ) = max( diftru / difest( 2 ), difest( 2 ) / diftru )
      END IF
*
*     Test (9)
*
      ntest = 9
      result( 9 ) = zero
      IF( linfo.EQ.( mplusn+2 ) ) THEN
         IF( diftru.GT.abnrm*ulp )
     $      result( 9 ) = ulpinv
         IF( ( ifunc.GT.1 ) .AND. ( difest( 2 ).NE.zero ) )
     $      result( 9 ) = ulpinv
         IF( ( ifunc.EQ.1 ) .AND. ( pl( 1 ).NE.zero ) )
     $      result( 9 ) = ulpinv
      END IF
*
*     Test (10): compare the estimated value of PL and it true value.
*
      ntest = 10
      result( 10 ) = zero
      IF( pl( 1 ).EQ.zero ) THEN
         IF( pltru.GT.abnrm*ulp )
     $      result( 10 ) = ulpinv
      ELSE IF( pltru.EQ.zero ) THEN
         IF( pl( 1 ).GT.abnrm*ulp )
     $      result( 10 ) = ulpinv
      ELSE IF( ( pltru.GT.thresh*pl( 1 ) ) .OR.
     $         ( pltru*thresh.LT.pl( 1 ) ) ) THEN
         result( 10 ) = ulpinv
      END IF
*
      ntestt = ntestt + ntest
*
*     Print out tests which fail.
*
      DO 120 j = 1, ntest
         IF( result( j ).GE.thresh ) THEN
*
*           If this is the first test to fail,
*           print a header to the data file.
*
            IF( nerrs.EQ.0 ) THEN
               WRITE( nout, fmt = 9995 )'DGX'
*
*              Matrix types
*
               WRITE( nout, fmt = 9994 )
*
*              Tests performed
*
               WRITE( nout, fmt = 9992 )'orthogonal', '''',
     $            'transpose', ( '''', i = 1, 4 )
*
            END IF
            nerrs = nerrs + 1
            IF( result( j ).LT.10000.0d0 ) THEN
               WRITE( nout, fmt = 9989 )nptknt, mplusn, j, result( j )
            ELSE
               WRITE( nout, fmt = 9988 )nptknt, mplusn, j, result( j )
            END IF
         END IF
*
  120 CONTINUE
*
  130 CONTINUE
      GO TO 80
  140 CONTINUE
*
  150 CONTINUE
*
*     Summary
*
      CALL alasvm( 'DGX', nout, nerrs, ntestt, 0 )
*
      work( 1 ) = maxwrk
*
      RETURN
*
 9999 FORMAT( ' DDRGSX: ', a, ' returned INFO=', i6, '.', / 9x, 'N=',
     $      i6, ', JTYPE=', i6, ')' )
*
 9998 FORMAT( ' DDRGSX: ', a, ' returned INFO=', i6, '.', / 9x, 'N=',
     $      i6, ', Input Example #', i2, ')' )
*
 9997 FORMAT( ' DDRGSX: DGET53 returned INFO=', i1, ' for eigenvalue ',
     $      i6, '.', / 9x, 'N=', i6, ', JTYPE=', i6, ')' )
*
 9996 FORMAT( ' DDRGSX: S not in Schur form at eigenvalue ', i6, '.',
     $      / 9x, 'N=', i6, ', JTYPE=', i6, ')' )
*
 9995 FORMAT( / 1x, a3, ' -- Real Expert Generalized Schur form',
     $      ' problem driver' )
*
 9994 FORMAT( 'Input Example' )
*
 9993 FORMAT( ' Matrix types: ', /
     $      '  1:  A is a block diagonal matrix of Jordan blocks ',
     $      'and B is the identity ', / '      matrix, ',
     $      / '  2:  A and B are upper triangular matrices, ',
     $      / '  3:  A and B are as type 2, but each second diagonal ',
     $      'block in A_11 and ', /
     $      '      each third diaongal block in A_22 are 2x2 blocks,',
     $      / '  4:  A and B are block diagonal matrices, ',
     $      / '  5:  (A,B) has potentially close or common ',
     $      'eigenvalues.', / )
*
 9992 FORMAT( / ' Tests performed:  (S is Schur, T is triangular, ',
     $      'Q and Z are ', a, ',', / 19x,
     $      ' a is alpha, b is beta, and ', a, ' means ', a, '.)',
     $      / '  1 = | A - Q S Z', a,
     $      ' | / ( |A| n ulp )      2 = | B - Q T Z', a,
     $      ' | / ( |B| n ulp )', / '  3 = | I - QQ', a,
     $      ' | / ( n ulp )             4 = | I - ZZ', a,
     $      ' | / ( n ulp )', / '  5 = 1/ULP  if A is not in ',
     $      'Schur form S', / '  6 = difference between (alpha,beta)',
     $      ' and diagonals of (S,T)', /
     $      '  7 = 1/ULP  if SDIM is not the correct number of ',
     $      'selected eigenvalues', /
     $      '  8 = 1/ULP  if DIFEST/DIFTRU > 10*THRESH or ',
     $      'DIFTRU/DIFEST > 10*THRESH',
     $      / '  9 = 1/ULP  if DIFEST <> 0 or DIFTRU > ULP*norm(A,B) ',
     $      'when reordering fails', /
     $      ' 10 = 1/ULP  if PLEST/PLTRU > THRESH or ',
     $      'PLTRU/PLEST > THRESH', /
     $      '    ( Test 10 is only for input examples )', / )
 9991 FORMAT( ' Matrix order=', i2, ', type=', i2, ', a=', d10.3,
     $      ', order(A_11)=', i2, ', result ', i2, ' is ', 0p, f8.2 )
 9990 FORMAT( ' Matrix order=', i2, ', type=', i2, ', a=', d10.3,
     $      ', order(A_11)=', i2, ', result ', i2, ' is ', 0p, d10.3 )
 9989 FORMAT( ' Input example #', i2, ', matrix order=', i4, ',',
     $      ' result ', i2, ' is', 0p, f8.2 )
 9988 FORMAT( ' Input example #', i2, ', matrix order=', i4, ',',
     $      ' result ', i2, ' is', 1p, d10.3 )
*
*     End of DDRGSX
*
      END