---
name: "dchkgg.f"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/dchkgg.f"
---

# dchkgg.f(3)

## NAME

TESTING/EIG/dchkgg.f

## SYNOPSIS

### Functions/Subroutines

subroutine **dchkgg** (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)\
 **DCHKGG**

## Function/Subroutine Documentation

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

**DCHKGG**

**Purpose:**

```
!>
!> DCHKGG  checks the nonsymmetric generalized eigenvalue problem
!> routines.
!>                                T          T        T
!> DGGHRD 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
!> DHGEQZ 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
!>
!> DTGEVC 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 DCHKGG is called, a number of matrix  () and a
!> number of matrix  are specified.  For each size ()
!> 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  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
!>       DTGEVC 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
!>       DTGEVC 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  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) 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,
!>          DCHKGG 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, DCHKGG
!>          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 DCHKGG 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.  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 DOUBLE PRECISION
!>          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 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, H, T, S1, P1, S2, and P2.
!>          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.
!>
```

*H*

```
!>          H is DOUBLE PRECISION array, dimension (LDA, max(NN))
!>          The upper Hessenberg matrix computed from A by DGGHRD.
!>
```

*T*

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

*S1*

```
!>          S1 is DOUBLE PRECISION array, dimension (LDA, max(NN))
!>          The Schur (block upper triangular) matrix computed from H by
!>          DHGEQZ when Q and Z are also computed.
!>
```

*S2*

```
!>          S2 is DOUBLE PRECISION array, dimension (LDA, max(NN))
!>          The Schur (block upper triangular) matrix computed from H by
!>          DHGEQZ when Q and Z are not computed.
!>
```

*P1*

```
!>          P1 is DOUBLE PRECISION array, dimension (LDA, max(NN))
!>          The upper triangular matrix computed from T by DHGEQZ
!>          when Q and Z are also computed.
!>
```

*P2*

```
!>          P2 is DOUBLE PRECISION array, dimension (LDA, max(NN))
!>          The upper triangular matrix computed from T by DHGEQZ
!>          when Q and Z are not computed.
!>
```

*U*

```
!>          U is DOUBLE PRECISION array, dimension (LDU, max(NN))
!>          The (left) orthogonal matrix computed by DGGHRD.
!>
```

*LDU*

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

*V*

```
!>          V is DOUBLE PRECISION array, dimension (LDU, max(NN))
!>          The (right) orthogonal matrix computed by DGGHRD.
!>
```

*Q*

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

*Z*

```
!>          Z is DOUBLE PRECISION array, dimension (LDU, max(NN))
!>          The (left) orthogonal matrix computed by DHGEQZ.
!>
```

*ALPHR1*

```
!>          ALPHR1 is DOUBLE PRECISION array, dimension (max(NN))
!>
```

*ALPHI1*

```
!>          ALPHI1 is DOUBLE PRECISION array, dimension (max(NN))
!>
```

*BETA1*

```
!>          BETA1 is DOUBLE PRECISION array, dimension (max(NN))
!>
!>          The generalized eigenvalues of (A,B) computed by DHGEQZ
!>          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 DOUBLE PRECISION array, dimension (max(NN))
!>
```

*ALPHI3*

```
!>          ALPHI3 is DOUBLE PRECISION array, dimension (max(NN))
!>
```

*BETA3*

```
!>          BETA3 is DOUBLE PRECISION array, dimension (max(NN))
!>
```

*EVECTL*

```
!>          EVECTL is DOUBLE PRECISION array, dimension (LDU, max(NN))
!>          The (block lower triangular) left eigenvector matrix for
!>          the matrices in S1 and P1.  (See DTGEVC for the format.)
!>
```

*EVECTR*

```
!>          EVECTR is DOUBLE PRECISION array, dimension (LDU, max(NN))
!>          The (block upper triangular) right eigenvector matrix for
!>          the matrices in S1 and P1.  (See DTGEVC for the format.)
!>
```

*WORK*

```
!>          WORK is DOUBLE PRECISION 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 DOUBLE PRECISION 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 **dchkgg.f**.

## Author

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