sdrgsx.f3

!>
!> 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).
!>
!> 

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.
!> 

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