schkgg()

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  () 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
!>       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  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

[in]	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. !>
[in]	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. !>
[in]	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. . !>
[in]	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. !>
[in,out]	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. !>
[in]	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. It must be at least zero. !>
[in]	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. !>
[in]	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.) !>
[in]	NOUNIT	!> NOUNIT is INTEGER !> The FORTRAN unit number for printing out error messages !> (e.g., if a routine returns IINFO not equal to 0.) !>
[in,out]	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. !>
[in]	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 ). !>
[in,out]	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. !>
[out]	H	!> H is REAL array, dimension (LDA, max(NN)) !> The upper Hessenberg matrix computed from A by SGGHRD. !>
[out]	T	!> T is REAL array, dimension (LDA, max(NN)) !> The upper triangular matrix computed from B by SGGHRD. !>
[out]	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. !>
[out]	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. !>
[out]	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. !>
[out]	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. !>
[out]	U	!> U is REAL array, dimension (LDU, max(NN)) !> The (left) orthogonal matrix computed by SGGHRD. !>
[in]	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 ). !>
[out]	V	!> V is REAL array, dimension (LDU, max(NN)) !> The (right) orthogonal matrix computed by SGGHRD. !>
[out]	Q	!> Q is REAL array, dimension (LDU, max(NN)) !> The (left) orthogonal matrix computed by SHGEQZ. !>
[out]	Z	!> Z is REAL array, dimension (LDU, max(NN)) !> The (left) orthogonal matrix computed by SHGEQZ. !>
[out]	ALPHR1	!> ALPHR1 is REAL array, dimension (max(NN)) !>
[out]	ALPHI1	!> ALPHI1 is REAL array, dimension (max(NN)) !>
[out]	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. !>
[out]	ALPHR3	!> ALPHR3 is REAL array, dimension (max(NN)) !>
[out]	ALPHI3	!> ALPHI3 is REAL array, dimension (max(NN)) !>
[out]	BETA3	!> BETA3 is REAL array, dimension (max(NN)) !>
[out]	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.) !>
[out]	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.) !>
[out]	WORK	!> WORK is REAL array, dimension (LWORK) !>
[in]	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). !>
[out]	LLWORK	!> LLWORK is LOGICAL array, dimension (max(NN)) !>
[out]	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. !>
[out]	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.

*
*  -- LAPACK test routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            TSTDIF
      INTEGER            INFO, LDA, LDU, LWORK, NOUNIT, NSIZES, NTYPES
      REAL               THRESH, THRSHN
*     ..
*     .. Array Arguments ..
      LOGICAL            DOTYPE( * ), LLWORK( * )
      INTEGER            ISEED( 4 ), NN( * )
      REAL               A( LDA, * ), ALPHI1( * ), ALPHI3( * ),
     $                   ALPHR1( * ), ALPHR3( * ), B( LDA, * ),
     $                   BETA1( * ), BETA3( * ), EVECTL( LDU, * ),
     $                   EVECTR( LDU, * ), H( LDA, * ), P1( LDA, * ),
     $                   P2( LDA, * ), Q( LDU, * ), RESULT( 15 ),
     $                   S1( LDA, * ), S2( LDA, * ), T( LDA, * ),
     $                   U( LDU, * ), V( LDU, * ), WORK( * ),
     $                   Z( LDU, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0, one = 1.0 )
      INTEGER            MAXTYP
      parameter( maxtyp = 26 )
*     ..
*     .. Local Scalars ..
      LOGICAL            BADNN
      INTEGER            I1, IADD, IINFO, IN, J, JC, JR, JSIZE, JTYPE,
     $                   LWKOPT, MTYPES, N, N1, NERRS, NMATS, NMAX,
     $                   NTEST, NTESTT
      REAL               ANORM, BNORM, SAFMAX, SAFMIN, TEMP1, TEMP2,
     $                   ULP, ULPINV
*     ..
*     .. Local Arrays ..
      INTEGER            IASIGN( MAXTYP ), IBSIGN( MAXTYP ),
     $                   IOLDSD( 4 ), KADD( 6 ), KAMAGN( MAXTYP ),
     $                   KATYPE( MAXTYP ), KAZERO( MAXTYP ),
     $                   KBMAGN( MAXTYP ), KBTYPE( MAXTYP ),
     $                   KBZERO( MAXTYP ), KCLASS( MAXTYP ),
     $                   KTRIAN( MAXTYP ), KZ1( 6 ), KZ2( 6 )
      REAL               DUMMA( 4 ), RMAGN( 0: 3 )
*     ..
*     .. External Functions ..
      REAL               SLAMCH, SLANGE, SLARND
      EXTERNAL           slamch, slange, slarnd
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgeqr2, sget51, sget52, sgghrd, shgeqz, slacpy,
     $                   slarfg, slaset, slasum, slatm4, sorm2r, stgevc,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, real, sign
*     ..
*     .. Data statements ..
      DATA               kclass / 15*1, 10*2, 1*3 /
      DATA               kz1 / 0, 1, 2, 1, 3, 3 /
      DATA               kz2 / 0, 0, 1, 2, 1, 1 /
      DATA               kadd / 0, 0, 0, 0, 3, 2 /
      DATA               katype / 0, 1, 0, 1, 2, 3, 4, 1, 4, 4, 1, 1, 4,
     $                   4, 4, 2, 4, 5, 8, 7, 9, 4*4, 0 /
      DATA               kbtype / 0, 0, 1, 1, 2, -3, 1, 4, 1, 1, 4, 4,
     $                   1, 1, -4, 2, -4, 8*8, 0 /
      DATA               kazero / 6*1, 2, 1, 2*2, 2*1, 2*2, 3, 1, 3,
     $                   4*5, 4*3, 1 /
      DATA               kbzero / 6*1, 1, 2, 2*1, 2*2, 2*1, 4, 1, 4,
     $                   4*6, 4*4, 1 /
      DATA               kamagn / 8*1, 2, 3, 2, 3, 2, 3, 7*1, 2, 3, 3,
     $                   2, 1 /
      DATA               kbmagn / 8*1, 3, 2, 3, 2, 2, 3, 7*1, 3, 2, 3,
     $                   2, 1 /
      DATA               ktrian / 16*0, 10*1 /
      DATA               iasign / 6*0, 2, 0, 2*2, 2*0, 3*2, 0, 2, 3*0,
     $                   5*2, 0 /
      DATA               ibsign / 7*0, 2, 2*0, 2*2, 2*0, 2, 0, 2, 9*0 /
*     ..
*     .. Executable Statements ..
*
*     Check for errors
*
      info = 0
*
      badnn = .false.
      nmax = 1
      DO 10 j = 1, nsizes
         nmax = max( nmax, nn( j ) )
         IF( nn( j ).LT.0 )
     $      badnn = .true.
   10 CONTINUE
*
*     Maximum blocksize and shift -- we assume that blocksize and number
*     of shifts are monotone increasing functions of N.
*
      lwkopt = max( 6*nmax, 2*nmax*nmax, 1 )
*
*     Check for errors
*
      IF( nsizes.LT.0 ) THEN
         info = -1
      ELSE IF( badnn ) THEN
         info = -2
      ELSE IF( ntypes.LT.0 ) THEN
         info = -3
      ELSE IF( thresh.LT.zero ) THEN
         info = -6
      ELSE IF( lda.LE.1 .OR. lda.LT.nmax ) THEN
         info = -10
      ELSE IF( ldu.LE.1 .OR. ldu.LT.nmax ) THEN
         info = -19
      ELSE IF( lwkopt.GT.lwork ) THEN
         info = -30
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SCHKGG', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( nsizes.EQ.0 .OR. ntypes.EQ.0 )
     $   RETURN
*
      safmin = slamch( 'Safe minimum' )
      ulp = slamch( 'Epsilon' )*slamch( 'Base' )
      safmin = safmin / ulp
      safmax = one / safmin
      ulpinv = one / ulp
*
*     The values RMAGN(2:3) depend on N, see below.
*
      rmagn( 0 ) = zero
      rmagn( 1 ) = one
*
*     Loop over sizes, types
*
      ntestt = 0
      nerrs = 0
      nmats = 0
*
      DO 240 jsize = 1, nsizes
         n = nn( jsize )
         n1 = max( 1, n )
         rmagn( 2 ) = safmax*ulp / real( n1 )
         rmagn( 3 ) = safmin*ulpinv*n1
*
         IF( nsizes.NE.1 ) THEN
            mtypes = min( maxtyp, ntypes )
         ELSE
            mtypes = min( maxtyp+1, ntypes )
         END IF
*
         DO 230 jtype = 1, mtypes
            IF( .NOT.dotype( jtype ) )
     $         GO TO 230
            nmats = nmats + 1
            ntest = 0
*
*           Save ISEED in case of an error.
*
            DO 20 j = 1, 4
               ioldsd( j ) = iseed( j )
   20       CONTINUE
*
*           Initialize RESULT
*
            DO 30 j = 1, 15
               result( j ) = zero
   30       CONTINUE
*
*           Compute A and B
*
*           Description of control parameters:
*
*           KCLASS: =1 means w/o rotation, =2 means w/ rotation,
*                   =3 means random.
*           KATYPE: the "type" to be passed to SLATM4 for computing A.
*           KAZERO: the pattern of zeros on the diagonal for A:
*                   =1: ( xxx ), =2: (0, xxx ) =3: ( 0, 0, xxx, 0 ),
*                   =4: ( 0, xxx, 0, 0 ), =5: ( 0, 0, 1, xxx, 0 ),
*                   =6: ( 0, 1, 0, xxx, 0 ).  (xxx means a string of
*                   non-zero entries.)
*           KAMAGN: the magnitude of the matrix: =0: zero, =1: O(1),
*                   =2: large, =3: small.
*           IASIGN: 1 if the diagonal elements of A are to be
*                   multiplied by a random magnitude 1 number, =2 if
*                   randomly chosen diagonal blocks are to be rotated
*                   to form 2x2 blocks.
*           KBTYPE, KBZERO, KBMAGN, IBSIGN: the same, but for B.
*           KTRIAN: =0: don't fill in the upper triangle, =1: do.
*           KZ1, KZ2, KADD: used to implement KAZERO and KBZERO.
*           RMAGN: used to implement KAMAGN and KBMAGN.
*
            IF( mtypes.GT.maxtyp )
     $         GO TO 110
            iinfo = 0
            IF( kclass( jtype ).LT.3 ) THEN
*
*              Generate A (w/o rotation)
*
               IF( abs( katype( jtype ) ).EQ.3 ) THEN
                  in = 2*( ( n-1 ) / 2 ) + 1
                  IF( in.NE.n )
     $               CALL slaset( 'Full', n, n, zero, zero, a, lda )
               ELSE
                  in = n
               END IF
               CALL slatm4( katype( jtype ), in, kz1( kazero( jtype ) ),
     $                      kz2( kazero( jtype ) ), iasign( jtype ),
     $                      rmagn( kamagn( jtype ) ), ulp,
     $                      rmagn( ktrian( jtype )*kamagn( jtype ) ), 2,
     $                      iseed, a, lda )
               iadd = kadd( kazero( jtype ) )
               IF( iadd.GT.0 .AND. iadd.LE.n )
     $            a( iadd, iadd ) = rmagn( kamagn( jtype ) )
*
*              Generate B (w/o rotation)
*
               IF( abs( kbtype( jtype ) ).EQ.3 ) THEN
                  in = 2*( ( n-1 ) / 2 ) + 1
                  IF( in.NE.n )
     $               CALL slaset( 'Full', n, n, zero, zero, b, lda )
               ELSE
                  in = n
               END IF
               CALL slatm4( kbtype( jtype ), in, kz1( kbzero( jtype ) ),
     $                      kz2( kbzero( jtype ) ), ibsign( jtype ),
     $                      rmagn( kbmagn( jtype ) ), one,
     $                      rmagn( ktrian( jtype )*kbmagn( jtype ) ), 2,
     $                      iseed, b, lda )
               iadd = kadd( kbzero( jtype ) )
               IF( iadd.NE.0 .AND. iadd.LE.n )
     $            b( iadd, iadd ) = rmagn( kbmagn( jtype ) )
*
               IF( kclass( jtype ).EQ.2 .AND. n.GT.0 ) THEN
*
*                 Include rotations
*
*                 Generate U, V as Householder transformations times
*                 a diagonal matrix.
*
                  DO 50 jc = 1, n - 1
                     DO 40 jr = jc, n
                        u( jr, jc ) = slarnd( 3, iseed )
                        v( jr, jc ) = slarnd( 3, iseed )
   40                CONTINUE
                     CALL slarfg( n+1-jc, u( jc, jc ), u( jc+1, jc ), 1,
     $                            work( jc ) )
                     work( 2*n+jc ) = sign( one, u( jc, jc ) )
                     u( jc, jc ) = one
                     CALL slarfg( n+1-jc, v( jc, jc ), v( jc+1, jc ), 1,
     $                            work( n+jc ) )
                     work( 3*n+jc ) = sign( one, v( jc, jc ) )
                     v( jc, jc ) = one
   50             CONTINUE
                  u( n, n ) = one
                  work( n ) = zero
                  work( 3*n ) = sign( one, slarnd( 2, iseed ) )
                  v( n, n ) = one
                  work( 2*n ) = zero
                  work( 4*n ) = sign( one, slarnd( 2, iseed ) )
*
*                 Apply the diagonal matrices
*
                  DO 70 jc = 1, n
                     DO 60 jr = 1, n
                        a( jr, jc ) = work( 2*n+jr )*work( 3*n+jc )*
     $                                a( jr, jc )
                        b( jr, jc ) = work( 2*n+jr )*work( 3*n+jc )*
     $                                b( jr, jc )
   60                CONTINUE
   70             CONTINUE
                  CALL sorm2r( 'L', 'N', n, n, n-1, u, ldu, work, a,
     $                         lda, work( 2*n+1 ), iinfo )
                  IF( iinfo.NE.0 )
     $               GO TO 100
                  CALL sorm2r( 'R', 'T', n, n, n-1, v, ldu, work( n+1 ),
     $                         a, lda, work( 2*n+1 ), iinfo )
                  IF( iinfo.NE.0 )
     $               GO TO 100
                  CALL sorm2r( 'L', 'N', n, n, n-1, u, ldu, work, b,
     $                         lda, work( 2*n+1 ), iinfo )
                  IF( iinfo.NE.0 )
     $               GO TO 100
                  CALL sorm2r( 'R', 'T', n, n, n-1, v, ldu, work( n+1 ),
     $                         b, lda, work( 2*n+1 ), iinfo )
                  IF( iinfo.NE.0 )
     $               GO TO 100
               END IF
            ELSE
*
*              Random matrices
*
               DO 90 jc = 1, n
                  DO 80 jr = 1, n
                     a( jr, jc ) = rmagn( kamagn( jtype ) )*
     $                             slarnd( 2, iseed )
                     b( jr, jc ) = rmagn( kbmagn( jtype ) )*
     $                             slarnd( 2, iseed )
   80             CONTINUE
   90          CONTINUE
            END IF
*
            anorm = slange( '1', n, n, a, lda, work )
            bnorm = slange( '1', n, n, b, lda, work )
*
  100       CONTINUE
*
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'Generator', iinfo, n, jtype,
     $            ioldsd
               info = abs( iinfo )
               RETURN
            END IF
*
  110       CONTINUE
*
*           Call SGEQR2, SORM2R, and SGGHRD to compute H, T, U, and V
*
            CALL slacpy( ' ', n, n, a, lda, h, lda )
            CALL slacpy( ' ', n, n, b, lda, t, lda )
            ntest = 1
            result( 1 ) = ulpinv
*
            CALL sgeqr2( n, n, t, lda, work, work( n+1 ), iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'SGEQR2', iinfo, n, jtype,
     $            ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            CALL sorm2r( 'L', 'T', n, n, n, t, lda, work, h, lda,
     $                   work( n+1 ), iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'SORM2R', iinfo, n, jtype,
     $            ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            CALL slaset( 'Full', n, n, zero, one, u, ldu )
            CALL sorm2r( 'R', 'N', n, n, n, t, lda, work, u, ldu,
     $                   work( n+1 ), iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'SORM2R', iinfo, n, jtype,
     $            ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            CALL sgghrd( 'V', 'I', n, 1, n, h, lda, t, lda, u, ldu, v,
     $                   ldu, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'SGGHRD', iinfo, n, jtype,
     $            ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
            ntest = 4
*
*           Do tests 1--4
*
            CALL sget51( 1, n, a, lda, h, lda, u, ldu, v, ldu, work,
     $                   result( 1 ) )
            CALL sget51( 1, n, b, lda, t, lda, u, ldu, v, ldu, work,
     $                   result( 2 ) )
            CALL sget51( 3, n, b, lda, t, lda, u, ldu, u, ldu, work,
     $                   result( 3 ) )
            CALL sget51( 3, n, b, lda, t, lda, v, ldu, v, ldu, work,
     $                   result( 4 ) )
*
*           Call SHGEQZ to compute S1, P1, S2, P2, Q, and Z, do tests.
*
*           Compute T1 and UZ
*
*           Eigenvalues only
*
            CALL slacpy( ' ', n, n, h, lda, s2, lda )
            CALL slacpy( ' ', n, n, t, lda, p2, lda )
            ntest = 5
            result( 5 ) = ulpinv
*
            CALL shgeqz( 'E', 'N', 'N', n, 1, n, s2, lda, p2, lda,
     $                   alphr3, alphi3, beta3, q, ldu, z, ldu, work,
     $                   lwork, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'SHGEQZ(E)', iinfo, n, jtype,
     $            ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
*           Eigenvalues and Full Schur Form
*
            CALL slacpy( ' ', n, n, h, lda, s2, lda )
            CALL slacpy( ' ', n, n, t, lda, p2, lda )
*
            CALL shgeqz( 'S', 'N', 'N', n, 1, n, s2, lda, p2, lda,
     $                   alphr1, alphi1, beta1, q, ldu, z, ldu, work,
     $                   lwork, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'SHGEQZ(S)', iinfo, n, jtype,
     $            ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
*           Eigenvalues, Schur Form, and Schur Vectors
*
            CALL slacpy( ' ', n, n, h, lda, s1, lda )
            CALL slacpy( ' ', n, n, t, lda, p1, lda )
*
            CALL shgeqz( 'S', 'I', 'I', n, 1, n, s1, lda, p1, lda,
     $                   alphr1, alphi1, beta1, q, ldu, z, ldu, work,
     $                   lwork, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'SHGEQZ(V)', iinfo, n, jtype,
     $            ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            ntest = 8
*
*           Do Tests 5--8
*
            CALL sget51( 1, n, h, lda, s1, lda, q, ldu, z, ldu, work,
     $                   result( 5 ) )
            CALL sget51( 1, n, t, lda, p1, lda, q, ldu, z, ldu, work,
     $                   result( 6 ) )
            CALL sget51( 3, n, t, lda, p1, lda, q, ldu, q, ldu, work,
     $                   result( 7 ) )
            CALL sget51( 3, n, t, lda, p1, lda, z, ldu, z, ldu, work,
     $                   result( 8 ) )
*
*           Compute the Left and Right Eigenvectors of (S1,P1)
*
*           9: Compute the left eigenvector Matrix without
*              back transforming:
*
            ntest = 9
            result( 9 ) = ulpinv
*
*           To test "SELECT" option, compute half of the eigenvectors
*           in one call, and half in another
*
            i1 = n / 2
            DO 120 j = 1, i1
               llwork( j ) = .true.
  120       CONTINUE
            DO 130 j = i1 + 1, n
               llwork( j ) = .false.
  130       CONTINUE
*
            CALL stgevc( 'L', 'S', llwork, n, s1, lda, p1, lda, evectl,
     $                   ldu, dumma, ldu, n, in, work, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'STGEVC(L,S1)', iinfo, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            i1 = in
            DO 140 j = 1, i1
               llwork( j ) = .false.
  140       CONTINUE
            DO 150 j = i1 + 1, n
               llwork( j ) = .true.
  150       CONTINUE
*
            CALL stgevc( 'L', 'S', llwork, n, s1, lda, p1, lda,
     $                   evectl( 1, i1+1 ), ldu, dumma, ldu, n, in,
     $                   work, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'STGEVC(L,S2)', iinfo, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            CALL sget52( .true., n, s1, lda, p1, lda, evectl, ldu,
     $                   alphr1, alphi1, beta1, work, dumma( 1 ) )
            result( 9 ) = dumma( 1 )
            IF( dumma( 2 ).GT.thrshn ) THEN
               WRITE( nounit, fmt = 9998 )'Left', 'STGEVC(HOWMNY=S)',
     $            dumma( 2 ), n, jtype, ioldsd
            END IF
*
*           10: Compute the left eigenvector Matrix with
*               back transforming:
*
            ntest = 10
            result( 10 ) = ulpinv
            CALL slacpy( 'F', n, n, q, ldu, evectl, ldu )
            CALL stgevc( 'L', 'B', llwork, n, s1, lda, p1, lda, evectl,
     $                   ldu, dumma, ldu, n, in, work, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'STGEVC(L,B)', iinfo, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            CALL sget52( .true., n, h, lda, t, lda, evectl, ldu, alphr1,
     $                   alphi1, beta1, work, dumma( 1 ) )
            result( 10 ) = dumma( 1 )
            IF( dumma( 2 ).GT.thrshn ) THEN
               WRITE( nounit, fmt = 9998 )'Left', 'STGEVC(HOWMNY=B)',
     $            dumma( 2 ), n, jtype, ioldsd
            END IF
*
*           11: Compute the right eigenvector Matrix without
*               back transforming:
*
            ntest = 11
            result( 11 ) = ulpinv
*
*           To test "SELECT" option, compute half of the eigenvectors
*           in one call, and half in another
*
            i1 = n / 2
            DO 160 j = 1, i1
               llwork( j ) = .true.
  160       CONTINUE
            DO 170 j = i1 + 1, n
               llwork( j ) = .false.
  170       CONTINUE
*
            CALL stgevc( 'R', 'S', llwork, n, s1, lda, p1, lda, dumma,
     $                   ldu, evectr, ldu, n, in, work, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'STGEVC(R,S1)', iinfo, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            i1 = in
            DO 180 j = 1, i1
               llwork( j ) = .false.
  180       CONTINUE
            DO 190 j = i1 + 1, n
               llwork( j ) = .true.
  190       CONTINUE
*
            CALL stgevc( 'R', 'S', llwork, n, s1, lda, p1, lda, dumma,
     $                   ldu, evectr( 1, i1+1 ), ldu, n, in, work,
     $                   iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'STGEVC(R,S2)', iinfo, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            CALL sget52( .false., n, s1, lda, p1, lda, evectr, ldu,
     $                   alphr1, alphi1, beta1, work, dumma( 1 ) )
            result( 11 ) = dumma( 1 )
            IF( dumma( 2 ).GT.thresh ) THEN
               WRITE( nounit, fmt = 9998 )'Right', 'STGEVC(HOWMNY=S)',
     $            dumma( 2 ), n, jtype, ioldsd
            END IF
*
*           12: Compute the right eigenvector Matrix with
*               back transforming:
*
            ntest = 12
            result( 12 ) = ulpinv
            CALL slacpy( 'F', n, n, z, ldu, evectr, ldu )
            CALL stgevc( 'R', 'B', llwork, n, s1, lda, p1, lda, dumma,
     $                   ldu, evectr, ldu, n, in, work, iinfo )
            IF( iinfo.NE.0 ) THEN
               WRITE( nounit, fmt = 9999 )'STGEVC(R,B)', iinfo, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               GO TO 210
            END IF
*
            CALL sget52( .false., n, h, lda, t, lda, evectr, ldu,
     $                   alphr1, alphi1, beta1, work, dumma( 1 ) )
            result( 12 ) = dumma( 1 )
            IF( dumma( 2 ).GT.thresh ) THEN
               WRITE( nounit, fmt = 9998 )'Right', 'STGEVC(HOWMNY=B)',
     $            dumma( 2 ), n, jtype, ioldsd
            END IF
*
*           Tests 13--15 are done only on request
*
            IF( tstdif ) THEN
*
*              Do Tests 13--14
*
               CALL sget51( 2, n, s1, lda, s2, lda, q, ldu, z, ldu,
     $                      work, result( 13 ) )
               CALL sget51( 2, n, p1, lda, p2, lda, q, ldu, z, ldu,
     $                      work, result( 14 ) )
*
*              Do Test 15
*
               temp1 = zero
               temp2 = zero
               DO 200 j = 1, n
                  temp1 = max( temp1, abs( alphr1( j )-alphr3( j ) )+
     $                    abs( alphi1( j )-alphi3( j ) ) )
                  temp2 = max( temp2, abs( beta1( j )-beta3( j ) ) )
  200          CONTINUE
*
               temp1 = temp1 / max( safmin, ulp*max( temp1, anorm ) )
               temp2 = temp2 / max( safmin, ulp*max( temp2, bnorm ) )
               result( 15 ) = max( temp1, temp2 )
               ntest = 15
            ELSE
               result( 13 ) = zero
               result( 14 ) = zero
               result( 15 ) = zero
               ntest = 12
            END IF
*
*           End of Loop -- Check for RESULT(j) > THRESH
*
  210       CONTINUE
*
            ntestt = ntestt + ntest
*
*           Print out tests which fail.
*
            DO 220 jr = 1, ntest
               IF( result( jr ).GE.thresh ) THEN
*
*                 If this is the first test to fail,
*                 print a header to the data file.
*
                  IF( nerrs.EQ.0 ) THEN
                     WRITE( nounit, fmt = 9997 )'SGG'
*
*                    Matrix types
*
                     WRITE( nounit, fmt = 9996 )
                     WRITE( nounit, fmt = 9995 )
                     WRITE( nounit, fmt = 9994 )'Orthogonal'
*
*                    Tests performed
*
                     WRITE( nounit, fmt = 9993 )'orthogonal', '''',
     $                  'transpose', ( '''', j = 1, 10 )
*
                  END IF
                  nerrs = nerrs + 1
                  IF( result( jr ).LT.10000.0 ) THEN
                     WRITE( nounit, fmt = 9992 )n, jtype, ioldsd, jr,
     $                  result( jr )
                  ELSE
                     WRITE( nounit, fmt = 9991 )n, jtype, ioldsd, jr,
     $                  result( jr )
                  END IF
               END IF
  220       CONTINUE
*
  230    CONTINUE
  240 CONTINUE
*
*     Summary
*
      CALL slasum( 'SGG', nounit, nerrs, ntestt )
      RETURN
*
 9999 FORMAT( ' SCHKGG: ', a, ' returned INFO=', i6, '.', / 9x, 'N=',
     $      i6, ', JTYPE=', i6, ', ISEED=(', 3( i5, ',' ), i5, ')' )
*
 9998 FORMAT( ' SCHKGG: ', a, ' Eigenvectors from ', a, ' incorrectly ',
     $      'normalized.', / ' Bits of error=', 0p, g10.3, ',', 9x,
     $      'N=', i6, ', JTYPE=', i6, ', ISEED=(', 3( i5, ',' ), i5,
     $      ')' )
*
 9997 FORMAT( / 1x, a3, ' -- Real Generalized eigenvalue problem' )
*
 9996 FORMAT( ' Matrix types (see SCHKGG for details): ' )
*
 9995 FORMAT( ' Special Matrices:', 23x,
     $      '(J''=transposed Jordan block)',
     $      / '   1=(0,0)  2=(I,0)  3=(0,I)  4=(I,I)  5=(J'',J'')  ',
     $      '6=(diag(J'',I), diag(I,J''))', / ' Diagonal Matrices:  ( ',
     $      'D=diag(0,1,2,...) )', / '   7=(D,I)   9=(large*D, small*I',
     $      ')  11=(large*I, small*D)  13=(large*D, large*I)', /
     $      '   8=(I,D)  10=(small*D, large*I)  12=(small*I, large*D) ',
     $      ' 14=(small*D, small*I)', / '  15=(D, reversed D)' )
 9994 FORMAT( ' Matrices Rotated by Random ', a, ' Matrices U, V:',
     $      / '  16=Transposed Jordan Blocks             19=geometric ',
     $      'alpha, beta=0,1', / '  17=arithm. alpha&beta             ',
     $      '      20=arithmetic alpha, beta=0,1', / '  18=clustered ',
     $      'alpha, beta=0,1            21=random alpha, beta=0,1',
     $      / ' Large & Small Matrices:', / '  22=(large, small)   ',
     $      '23=(small,large)    24=(small,small)    25=(large,large)',
     $      / '  26=random O(1) matrices.' )
*
 9993 FORMAT( / ' Tests performed:   (H is Hessenberg, S is Schur, B, ',
     $      'T, P are triangular,', / 20x, 'U, V, Q, and Z are ', a,
     $      ', l and r are the', / 20x,
     $      'appropriate left and right eigenvectors, resp., a is',
     $      / 20x, 'alpha, b is beta, and ', a, ' means ', a, '.)',
     $      / ' 1 = | A - U H V', a,
     $      ' | / ( |A| n ulp )      2 = | B - U T V', a,
     $      ' | / ( |B| n ulp )', / ' 3 = | I - UU', a,
     $      ' | / ( n ulp )             4 = | I - VV', a,
     $      ' | / ( n ulp )', / ' 5 = | H - Q S Z', a,
     $      ' | / ( |H| n ulp )', 6x, '6 = | T - Q P Z', a,
     $      ' | / ( |T| n ulp )', / ' 7 = | I - QQ', a,
     $      ' | / ( n ulp )             8 = | I - ZZ', a,
     $      ' | / ( n ulp )', / ' 9 = max | ( b S - a P )', a,
     $      ' l | / const.  10 = max | ( b H - a T )', a,
     $      ' l | / const.', /
     $      ' 11= max | ( b S - a P ) r | / const.   12 = max | ( b H',
     $      ' - a T ) r | / const.', / 1x )
*
 9992 FORMAT( ' Matrix order=', i5, ', type=', i2, ', seed=',
     $      4( i4, ',' ), ' result ', i2, ' is', 0p, f8.2 )
 9991 FORMAT( ' Matrix order=', i5, ', type=', i2, ', seed=',
     $      4( i4, ',' ), ' result ', i2, ' is', 1p, e10.3 )
*
*     End of SCHKGG
*

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