zdrges.f - Man Page

TESTING/EIG/zdrges.f

Synopsis

Functions/Subroutines

subroutine zdrges (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, b, s, t, q, ldq, z, alpha, beta, work, lwork, rwork, result, bwork, info)
ZDRGES

Function/Subroutine Documentation

subroutine zdrges (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, double precision thresh, integer nounit, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( lda, * ) b, complex*16, dimension( lda, * ) s, complex*16, dimension( lda, * ) t, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldq, * ) z, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, double precision, dimension( 13 ) result, logical, dimension( * ) bwork, integer info)

ZDRGES

Purpose:

 ZDRGES checks the nonsymmetric generalized eigenvalue (Schur form)
 problem driver ZGGES.

 ZGGES factors A and B as Q*S*Z'  and Q*T*Z' , where ' means conjugate
 transpose, S and T are  upper triangular (i.e., in generalized Schur
 form), and Q and Z are unitary. 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 ZDRGES is called, a number of matrix 'sizes' ('N's') and a
 number of matrix 'TYPES' are specified.  For each size ('N')
 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. triangular form) (no sorting of
       eigenvalues)

 (6)   if eigenvalues = diagonal elements of the Schur form (S, T),
       i.e., test the maximum over j of D(j)  where:

                     |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
           D(j) = ------------------------ + -----------------------
                  max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

       (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 elements of the Schur form (S, T),
       i.e. test the maximum over j of D(j)  where:

                     |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
           D(j) = ------------------------ + -----------------------
                  max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

       (with sorting of eigenvalues).

 (12)  if sorting worked and SDIM is the number of eigenvalues
       which were CELECTed.

 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 'types' are specified by a logical array DOTYPE( 1:NTYPES ); if
 DOTYPE(j) is .TRUE., then matrix type 'j' 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 'big' 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.
Parameters

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 '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.

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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDA, max(NN))
          The Schur form matrix computed from A by ZGGES.  On exit, S
          contains the Schur form matrix corresponding to the matrix
          in A.

T

          T is COMPLEX*16 array, dimension (LDA, max(NN))
          The upper triangular matrix computed from B by ZGGES.

Q

          Q is COMPLEX*16 array, dimension (LDQ, max(NN))
          The (left) orthogonal matrix computed by ZGGES.

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 COMPLEX*16 array, dimension( LDQ, max(NN) )
          The (right) orthogonal matrix computed by ZGGES.

ALPHA

          ALPHA is COMPLEX*16 array, dimension (max(NN))

BETA

          BETA is COMPLEX*16 array, dimension (max(NN))

          The generalized eigenvalues of (A,B) computed by ZGGES.
          ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A
          and B.

WORK

          WORK is COMPLEX*16 array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.  LWORK >= 3*N*N.

RWORK

          RWORK is DOUBLE PRECISION array, dimension ( 8*N )
          Real workspace.

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.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 378 of file zdrges.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

The man page zdrges(3) is an alias of zdrges.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK