ddrges.f3

!>
!> DDRGES checks the nonsymmetric generalized eigenvalue (Schur form)
!> problem driver DGGES.
!>
!> DGGES 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(j),beta(j)), j=1,...,n,
!> Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic
!> equation
!>                 det( A - w(j) B ) = 0
!> Optionally it also reorder the eigenvalues so that a selected
!> cluster of eigenvalues appears in the leading diagonal block of the
!> Schur forms.
!>
!> When DDRGES is called, a number of matrix  () and a
!> number of matrix  are specified.  For each size ()
!> and each TYPE of matrix, a pair of matrices (A, B) will be generated
!> and used for testing. For each matrix pair, the following 13 tests
!> will be performed and compared with the threshold THRESH except
!> the tests (5), (11) and (13).
!>
!>
!> (1)   | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
!>
!>
!> (2)   | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
!>
!>
!> (3)   | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
!>
!>
!> (4)   | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
!>
!> (5)   if A is in Schur form (i.e. quasi-triangular form)
!>       (no sorting of eigenvalues)
!>
!> (6)   if eigenvalues = diagonal blocks of the Schur form (S, T),
!>       i.e., test the 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.
!>       (no sorting of eigenvalues)
!>
!> (7)   | (A,B) - Q (S,T) Z' | / ( | (A,B) | n ulp )
!>            (with sorting of eigenvalues).
!>
!> (8)   | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).
!>
!> (9)   | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).
!>
!> (10)  if A is in Schur form (i.e. quasi-triangular form)
!>       (with sorting of eigenvalues).
!>
!> (11)  if eigenvalues = diagonal blocks of the Schur form (S, T),
!>       i.e. test the 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.
!>       (with sorting of eigenvalues).
!>
!> (12)  if sorting worked and SDIM is the number of eigenvalues
!>       which were SELECTed.
!>
!> Test Matrices
!> =============
!>
!> The sizes of the test matrices are specified by an array
!> NN(1:NSIZES); the value of each element NN(j) specifies one size.
!> The  are specified by a logical array DOTYPE( 1:NTYPES ); if
!> DOTYPE(j) is .TRUE., then matrix type  will be generated.
!> Currently, the list of possible types is:
!>
!> (1)  ( 0, 0 )         (a pair of zero matrices)
!>
!> (2)  ( I, 0 )         (an identity and a zero matrix)
!>
!> (3)  ( 0, I )         (an identity and a zero matrix)
!>
!> (4)  ( I, I )         (a pair of identity matrices)
!>
!>         t   t
!> (5)  ( J , J  )       (a pair of transposed Jordan blocks)
!>
!>                                     t                ( I   0  )
!> (6)  ( X, Y )         where  X = ( J   0  )  and Y = (      t )
!>                                  ( 0   I  )          ( 0   J  )
!>                       and I is a k x k identity and J a (k+1)x(k+1)
!>                       Jordan block; k=(N-1)/2
!>
!> (7)  ( D, I )         where D is diag( 0, 1,..., N-1 ) (a diagonal
!>                       matrix with those diagonal entries.)
!> (8)  ( I, D )
!>
!> (9)  ( big*D, small*I ) where  is near overflow and small=1/big
!>
!> (10) ( small*D, big*I )
!>
!> (11) ( big*I, small*D )
!>
!> (12) ( small*I, big*D )
!>
!> (13) ( big*D, big*I )
!>
!> (14) ( small*D, small*I )
!>
!> (15) ( D1, D2 )        where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
!>                        D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
!>           t   t
!> (16) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.
!>
!> (17) Q ( T1, T2 ) Z    where T1 and T2 are upper triangular matrices
!>                        with random O(1) entries above the diagonal
!>                        and diagonal entries diag(T1) =
!>                        ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
!>                        ( 0, N-3, N-4,..., 1, 0, 0 )
!>
!> (18) Q ( T1, T2 ) Z    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
!>                        diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
!>                        s = machine precision.
!>
!> (19) Q ( T1, T2 ) Z    diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
!>                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
!>
!>                                                        N-5
!> (20) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
!>                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
!>
!> (21) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
!>                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
!>                        where r1,..., r(N-4) are random.
!>
!> (22) Q ( big*T1, small*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
!>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
!>
!> (23) Q ( small*T1, big*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
!>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
!>
!> (24) Q ( small*T1, small*T2 ) Z  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
!>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
!>
!> (25) Q ( big*T1, big*T2 ) Z      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
!>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
!>
!> (26) Q ( T1, T2 ) Z     where T1 and T2 are random upper-triangular
!>                         matrices.
!>
!> 

NSIZES

!>          NSIZES is INTEGER
!>          The number of sizes of matrices to use.  If it is zero,
!>          DDRGES does nothing.  NSIZES >= 0.
!> 

NN

!>          NN is INTEGER array, dimension (NSIZES)
!>          An array containing the sizes to be used for the matrices.
!>          Zero values will be skipped.  NN >= 0.
!> 

NTYPES

!>          NTYPES is INTEGER
!>          The number of elements in DOTYPE.   If it is zero, DDRGES
!>          does nothing.  It must be at least zero.  If it is MAXTYP+1
!>          and NSIZES is 1, then an additional type, MAXTYP+1 is
!>          defined, which is to use whatever matrix is in A on input.
!>          This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
!>          DOTYPE(MAXTYP+1) is .TRUE. .
!> 

DOTYPE

!>          DOTYPE is LOGICAL array, dimension (NTYPES)
!>          If DOTYPE(j) is .TRUE., then for each size in NN a
!>          matrix of that size and of type j will be generated.
!>          If NTYPES is smaller than the maximum number of types
!>          defined (PARAMETER MAXTYP), then types NTYPES+1 through
!>          MAXTYP will not be generated. If NTYPES is larger
!>          than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
!>          will be ignored.
!> 

ISEED

!>          ISEED is INTEGER array, dimension (4)
!>          On entry ISEED specifies the seed of the random number
!>          generator. The array elements should be between 0 and 4095;
!>          if not they will be reduced mod 4096. Also, ISEED(4) must
!>          be odd.  The random number generator uses a linear
!>          congruential sequence limited to small integers, and so
!>          should produce machine independent random numbers. The
!>          values of ISEED are changed on exit, and can be used in the
!>          next call to DDRGES to continue the same random number
!>          sequence.
!> 

THRESH

!>          THRESH is DOUBLE PRECISION
!>          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.
!> 

NOUNIT

!>          NOUNIT 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 DOUBLE PRECISION array,
!>                                       dimension(LDA, max(NN))
!>          Used to hold the original A matrix.  Used as input only
!>          if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
!>          DOTYPE(MAXTYP+1)=.TRUE.
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of A, B, S, and T.
!>          It must be at least 1 and at least max( NN ).
!> 

B

!>          B is DOUBLE PRECISION array,
!>                                       dimension(LDA, max(NN))
!>          Used to hold the original B matrix.  Used as input only
!>          if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
!>          DOTYPE(MAXTYP+1)=.TRUE.
!> 

S

!>          S is DOUBLE PRECISION array, dimension (LDA, max(NN))
!>          The Schur form matrix computed from A by DGGES.  On exit, S
!>          contains the Schur form matrix corresponding to the matrix
!>          in A.
!> 

T

!>          T is DOUBLE PRECISION array, dimension (LDA, max(NN))
!>          The upper triangular matrix computed from B by DGGES.
!> 

Q

!>          Q is DOUBLE PRECISION array, dimension (LDQ, max(NN))
!>          The (left) orthogonal matrix computed by DGGES.
!> 

LDQ

!>          LDQ is INTEGER
!>          The leading dimension of Q and Z. It must
!>          be at least 1 and at least max( NN ).
!> 

Z

!>          Z is DOUBLE PRECISION array, dimension( LDQ, max(NN) )
!>          The (right) orthogonal matrix computed by DGGES.
!> 

ALPHAR

!>          ALPHAR is DOUBLE PRECISION array, dimension (max(NN))
!> 

ALPHAI

!>          ALPHAI is DOUBLE PRECISION array, dimension (max(NN))
!> 

BETA

!>          BETA is DOUBLE PRECISION array, dimension (max(NN))
!>
!>          The generalized eigenvalues of (A,B) computed by DGGES.
!>          ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th
!>          generalized eigenvalue of A and B.
!> 

WORK

!>          WORK is DOUBLE PRECISION array, dimension (LWORK)
!> 

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK.
!>          LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest
!>          matrix dimension.
!> 

RESULT

!>          RESULT is DOUBLE PRECISION array, dimension (15)
!>          The values computed by the tests described above.
!>          The values are currently limited to 1/ulp, to avoid overflow.
!> 

BWORK

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

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.  INFO is the
!>                absolute value of the INFO value returned.
!> 

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