schkgg.f - Man Page

TESTING/EIG/schkgg.f

Synopsis

Functions/Subroutines

subroutine schkgg (nsizes, nn, ntypes, dotype, iseed, thresh, tstdif, thrshn, nounit, a, lda, b, h, t, s1, s2, p1, p2, u, ldu, v, q, z, alphr1, alphi1, beta1, alphr3, alphi3, beta3, evectl, evectr, work, lwork, llwork, result, info)
SCHKGG

Function/Subroutine Documentation

subroutine schkgg (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, logical tstdif, real thrshn, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) h, real, dimension( lda, * ) t, real, dimension( lda, * ) s1, real, dimension( lda, * ) s2, real, dimension( lda, * ) p1, real, dimension( lda, * ) p2, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldu, * ) v, real, dimension( ldu, * ) q, real, dimension( ldu, * ) z, real, dimension( * ) alphr1, real, dimension( * ) alphi1, real, dimension( * ) beta1, real, dimension( * ) alphr3, real, dimension( * ) alphi3, real, dimension( * ) beta3, real, dimension( ldu, * ) evectl, real, dimension( ldu, * ) evectr, real, dimension( * ) work, integer lwork, logical, dimension( * ) llwork, real, dimension( 15 ) result, integer info)

SCHKGG

Purpose:

 SCHKGG  checks the nonsymmetric generalized eigenvalue problem
 routines.
                                T          T        T
 SGGHRD factors A and B as U H V  and U T V , where   means
 transpose, H is hessenberg, T is triangular and U and V are
 orthogonal.
                                 T          T
 SHGEQZ factors H and T as  Q S Z  and Q P Z , where P 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)),
 where alpha(j)=S(j,j) and beta(j)=P(j,j) -- thus,
 w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue
 problem

     det( A - w(j) B ) = 0

 and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent
 problem

     det( m(j) A - B ) = 0

 STGEVC computes the matrix L of left eigenvectors and the matrix R
 of right eigenvectors for the matrix pair ( S, P ).  In the
 description below,  l and r are left and right eigenvectors
 corresponding to the generalized eigenvalues (alpha,beta).

 When SCHKGG 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, one matrix will be generated and used
 to test the nonsymmetric eigenroutines.  For each matrix, 15
 tests will be performed.  The first twelve 'test ratios' should be
 small -- O(1).  They will be compared with the threshold THRESH:

                  T
 (1)   | A - U H V  | / ( |A| n ulp )

                  T
 (2)   | B - U T V  | / ( |B| n ulp )

               T
 (3)   | I - UU  | / ( n ulp )

               T
 (4)   | I - VV  | / ( n ulp )

                  T
 (5)   | H - Q S Z  | / ( |H| n ulp )

                  T
 (6)   | T - Q P Z  | / ( |T| n ulp )

               T
 (7)   | I - QQ  | / ( n ulp )

               T
 (8)   | I - ZZ  | / ( n ulp )

 (9)   max over all left eigenvalue/-vector pairs (beta/alpha,l) of

    | l**H * (beta S - alpha P) | / ( ulp max( |beta S|, |alpha P| ) )

 (10)  max over all left eigenvalue/-vector pairs (beta/alpha,l') of
                           T
   | l'**H * (beta H - alpha T) | / ( ulp max( |beta H|, |alpha T| ) )

       where the eigenvectors l' are the result of passing Q to
       STGEVC and back transforming (HOWMNY='B').

 (11)  max over all right eigenvalue/-vector pairs (beta/alpha,r) of

       | (beta S - alpha T) r | / ( ulp max( |beta S|, |alpha T| ) )

 (12)  max over all right eigenvalue/-vector pairs (beta/alpha,r') of

       | (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )

       where the eigenvectors r' are the result of passing Z to
       STGEVC and back transforming (HOWMNY='B').

 The last three test ratios will usually be small, but there is no
 mathematical requirement that they be so.  They are therefore
 compared with THRESH only if TSTDIF is .TRUE.

 (13)  | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )

 (14)  | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )

 (15)  max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
            |beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp

 In addition, the normalization of L and R are checked, and compared
 with the threshold THRSHN.

 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) U ( J , J ) V     where U and V are random orthogonal matrices.

 (17) U ( T1, T2 ) V    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) U ( T1, T2 ) V    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
                        diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
                        s = machine precision.

 (19) U ( T1, T2 ) V    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) U ( T1, T2 ) V    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )

 (21) U ( T1, T2 ) V    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) U ( big*T1, small*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )

 (23) U ( small*T1, big*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )

 (24) U ( small*T1, small*T2 ) V  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )

 (25) U ( big*T1, big*T2 ) V      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )

 (26) U ( T1, T2 ) V     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,
          SCHKGG does nothing.  It must be at least zero.

NN

          NN is INTEGER array, dimension (NSIZES)
          An array containing the sizes to be used for the matrices.
          Zero values will be skipped.  The values must be at least
          zero.

NTYPES

          NTYPES is INTEGER
          The number of elements in DOTYPE.   If it is zero, SCHKGG
          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.  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 SCHKGG to continue the same random number
          sequence.

THRESH

          THRESH is REAL
          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.  It must be at least zero.

TSTDIF

          TSTDIF is LOGICAL
          Specifies whether test ratios 13-15 will be computed and
          compared with THRESH.
          = .FALSE.: Only test ratios 1-12 will be computed and tested.
                     Ratios 13-15 will be set to zero.
          = .TRUE.:  All the test ratios 1-15 will be computed and
                     tested.

THRSHN

          THRSHN is REAL
          Threshold for reporting eigenvector normalization error.
          If the normalization of any eigenvector differs from 1 by
          more than THRSHN*ulp, then a special error message will be
          printed.  (This is handled separately from the other tests,
          since only a compiler or programming error should cause an
          error message, at least if THRSHN is at least 5--10.)

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 REAL 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, H, T, S1, P1, S2, and P2.
          It must be at least 1 and at least max( NN ).

B

          B is REAL 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.

H

          H is REAL array, dimension (LDA, max(NN))
          The upper Hessenberg matrix computed from A by SGGHRD.

T

          T is REAL array, dimension (LDA, max(NN))
          The upper triangular matrix computed from B by SGGHRD.

S1

          S1 is REAL array, dimension (LDA, max(NN))
          The Schur (block upper triangular) matrix computed from H by
          SHGEQZ when Q and Z are also computed.

S2

          S2 is REAL array, dimension (LDA, max(NN))
          The Schur (block upper triangular) matrix computed from H by
          SHGEQZ when Q and Z are not computed.

P1

          P1 is REAL array, dimension (LDA, max(NN))
          The upper triangular matrix computed from T by SHGEQZ
          when Q and Z are also computed.

P2

          P2 is REAL array, dimension (LDA, max(NN))
          The upper triangular matrix computed from T by SHGEQZ
          when Q and Z are not computed.

U

          U is REAL array, dimension (LDU, max(NN))
          The (left) orthogonal matrix computed by SGGHRD.

LDU

          LDU is INTEGER
          The leading dimension of U, V, Q, Z, EVECTL, and EVECTR.  It
          must be at least 1 and at least max( NN ).

V

          V is REAL array, dimension (LDU, max(NN))
          The (right) orthogonal matrix computed by SGGHRD.

Q

          Q is REAL array, dimension (LDU, max(NN))
          The (left) orthogonal matrix computed by SHGEQZ.

Z

          Z is REAL array, dimension (LDU, max(NN))
          The (left) orthogonal matrix computed by SHGEQZ.

ALPHR1

          ALPHR1 is REAL array, dimension (max(NN))

ALPHI1

          ALPHI1 is REAL array, dimension (max(NN))

BETA1

          BETA1 is REAL array, dimension (max(NN))

          The generalized eigenvalues of (A,B) computed by SHGEQZ
          when Q, Z, and the full Schur matrices are computed.
          On exit, ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the k-th
          generalized eigenvalue of the matrices in A and B.

ALPHR3

          ALPHR3 is REAL array, dimension (max(NN))

ALPHI3

          ALPHI3 is REAL array, dimension (max(NN))

BETA3

          BETA3 is REAL array, dimension (max(NN))

EVECTL

          EVECTL is REAL array, dimension (LDU, max(NN))
          The (block lower triangular) left eigenvector matrix for
          the matrices in S1 and P1.  (See STGEVC for the format.)

EVECTR

          EVECTR is REAL array, dimension (LDU, max(NN))
          The (block upper triangular) right eigenvector matrix for
          the matrices in S1 and P1.  (See STGEVC for the format.)

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The number of entries in WORK.  This must be at least
          max( 2 * N**2, 6*N, 1 ), for all N=NN(j).

LLWORK

          LLWORK is LOGICAL array, dimension (max(NN))

RESULT

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

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 506 of file schkgg.f.

Author

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Referenced By

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

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