schksb.f

Go to the documentation of this file.

*> \brief \b SCHKSB
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE SCHKSB( NSIZES, NN, NWDTHS, KK, NTYPES, DOTYPE, ISEED,
*                          THRESH, NOUNIT, A, LDA, SD, SE, U, LDU, WORK,
*                          LWORK, RESULT, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDU, LWORK, NOUNIT, NSIZES, NTYPES,
*      $                   NWDTHS
*       REAL               THRESH
*       ..
*       .. Array Arguments ..
*       LOGICAL            DOTYPE( * )
*       INTEGER            ISEED( 4 ), KK( * ), NN( * )
*       REAL               A( LDA, * ), RESULT( * ), SD( * ), SE( * ),
*      $                   U( LDU, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SCHKSB tests the reduction of a symmetric band matrix to tridiagonal
*> form, used with the symmetric eigenvalue problem.
*>
*> SSBTRD factors a symmetric band matrix A as  U S U' , where ' means
*> transpose, S is symmetric tridiagonal, and U is orthogonal.
*> SSBTRD can use either just the lower or just the upper triangle
*> of A; SCHKSB checks both cases.
*>
*> When SCHKSB is called, a number of matrix "sizes" ("n's"), a number
*> of bandwidths ("k's"), and a number of matrix "types" are
*> specified.  For each size ("n"), each bandwidth ("k") less than or
*> equal to "n", and each type of matrix, one matrix will be generated
*> and used to test the symmetric banded reduction routine.  For each
*> matrix, a number of tests will be performed:
*>
*> (1)     | A - V S V' | / ( |A| n ulp )  computed by SSBTRD with
*>                                         UPLO='U'
*>
*> (2)     | I - UU' | / ( n ulp )
*>
*> (3)     | A - V S V' | / ( |A| n ulp )  computed by SSBTRD with
*>                                         UPLO='L'
*>
*> (4)     | I - UU' | / ( n ulp )
*>
*> The "sizes" 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)  The zero matrix.
*> (2)  The identity matrix.
*>
*> (3)  A diagonal matrix with evenly spaced entries
*>      1, ..., ULP  and random signs.
*>      (ULP = (first number larger than 1) - 1 )
*> (4)  A diagonal matrix with geometrically spaced entries
*>      1, ..., ULP  and random signs.
*> (5)  A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
*>      and random signs.
*>
*> (6)  Same as (4), but multiplied by SQRT( overflow threshold )
*> (7)  Same as (4), but multiplied by SQRT( underflow threshold )
*>
*> (8)  A matrix of the form  U' D U, where U is orthogonal and
*>      D has evenly spaced entries 1, ..., ULP with random signs
*>      on the diagonal.
*>
*> (9)  A matrix of the form  U' D U, where U is orthogonal and
*>      D has geometrically spaced entries 1, ..., ULP with random
*>      signs on the diagonal.
*>
*> (10) A matrix of the form  U' D U, where U is orthogonal and
*>      D has "clustered" entries 1, ULP,..., ULP with random
*>      signs on the diagonal.
*>
*> (11) Same as (8), but multiplied by SQRT( overflow threshold )
*> (12) Same as (8), but multiplied by SQRT( underflow threshold )
*>
*> (13) Symmetric matrix with random entries chosen from (-1,1).
*> (14) Same as (13), but multiplied by SQRT( overflow threshold )
*> (15) Same as (13), but multiplied by SQRT( underflow threshold )
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NSIZES
*> \verbatim
*>          NSIZES is INTEGER
*>          The number of sizes of matrices to use.  If it is zero,
*>          SCHKSB does nothing.  It must be at least zero.
*> \endverbatim
*>
*> \param[in] NN
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] NWDTHS
*> \verbatim
*>          NWDTHS is INTEGER
*>          The number of bandwidths to use.  If it is zero,
*>          SCHKSB does nothing.  It must be at least zero.
*> \endverbatim
*>
*> \param[in] KK
*> \verbatim
*>          KK is INTEGER array, dimension (NWDTHS)
*>          An array containing the bandwidths to be used for the band
*>          matrices.  The values must be at least zero.
*> \endverbatim
*>
*> \param[in] NTYPES
*> \verbatim
*>          NTYPES is INTEGER
*>          The number of elements in DOTYPE.   If it is zero, SCHKSB
*>          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. .
*> \endverbatim
*>
*> \param[in] DOTYPE
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in,out] ISEED
*> \verbatim
*>          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 SCHKSB to continue the same random number
*>          sequence.
*> \endverbatim
*>
*> \param[in] THRESH
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] NOUNIT
*> \verbatim
*>          NOUNIT is INTEGER
*>          The FORTRAN unit number for printing out error messages
*>          (e.g., if a routine returns IINFO not equal to 0.)
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is REAL array, dimension
*>                            (LDA, max(NN))
*>          Used to hold the matrix whose eigenvalues are to be
*>          computed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  It must be at least 2 (not 1!)
*>          and at least max( KK )+1.
*> \endverbatim
*>
*> \param[out] SD
*> \verbatim
*>          SD is REAL array, dimension (max(NN))
*>          Used to hold the diagonal of the tridiagonal matrix computed
*>          by SSBTRD.
*> \endverbatim
*>
*> \param[out] SE
*> \verbatim
*>          SE is REAL array, dimension (max(NN))
*>          Used to hold the off-diagonal of the tridiagonal matrix
*>          computed by SSBTRD.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is REAL array, dimension (LDU, max(NN))
*>          Used to hold the orthogonal matrix computed by SSBTRD.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of U.  It must be at least 1
*>          and at least max( NN ).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The number of entries in WORK.  This must be at least
*>          max( LDA+1, max(NN)+1 )*max(NN).
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*>          RESULT is REAL array, dimension (4)
*>          The values computed by the tests described above.
*>          The values are currently limited to 1/ulp, to avoid
*>          overflow.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          If 0, then everything ran OK.
*>
*>-----------------------------------------------------------------------
*>
*>       Some Local Variables and Parameters:
*>       ---- ----- --------- --- ----------
*>       ZERO, ONE       Real 0 and 1.
*>       MAXTYP          The number of types defined.
*>       NTEST           The number of tests performed, or which can
*>                       be performed so far, for the current matrix.
*>       NTESTT          The total number of tests performed so far.
*>       NMAX            Largest value in NN.
*>       NMATS           The number of matrices generated so far.
*>       NERRS           The number of tests which have exceeded THRESH
*>                       so far.
*>       COND, IMODE     Values to be passed to the matrix generators.
*>       ANORM           Norm of A; passed to matrix generators.
*>
*>       OVFL, UNFL      Overflow and underflow thresholds.
*>       ULP, ULPINV     Finest relative precision and its inverse.
*>       RTOVFL, RTUNFL  Square roots of the previous 2 values.
*>               The following four arrays decode JTYPE:
*>       KTYPE(j)        The general type (1-10) for type "j".
*>       KMODE(j)        The MODE value to be passed to the matrix
*>                       generator for type "j".
*>       KMAGN(j)        The order of magnitude ( O(1),
*>                       O(overflow^(1/2) ), O(underflow^(1/2) )
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup single_eig
*
*  =====================================================================
      SUBROUTINE schksb( NSIZES, NN, NWDTHS, KK, NTYPES, DOTYPE, ISEED,
     $                   THRESH, NOUNIT, A, LDA, SD, SE, U, LDU, WORK,
     $                   LWORK, RESULT, INFO )
*
*  -- 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 ..
      INTEGER            INFO, LDA, LDU, LWORK, NOUNIT, NSIZES, NTYPES,
     $                   NWDTHS
      REAL               THRESH
*     ..
*     .. Array Arguments ..
      LOGICAL            DOTYPE( * )
      INTEGER            ISEED( 4 ), KK( * ), NN( * )
      REAL               A( LDA, * ), RESULT( * ), SD( * ), SE( * ),
     $                   u( ldu, * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE, TWO, TEN
      PARAMETER          ( ZERO = 0.0e0, one = 1.0e0, two = 2.0e0,
     $                   ten = 10.0e0 )
      REAL               HALF
      PARAMETER          ( HALF = one / two )
      INTEGER            MAXTYP
      parameter( maxtyp = 15 )
*     ..
*     .. Local Scalars ..
      LOGICAL            BADNN, BADNNB
      INTEGER            I, IINFO, IMODE, ITYPE, J, JC, JCOL, JR, JSIZE,
     $                   jtype, jwidth, k, kmax, mtypes, n, nerrs,
     $                   nmats, nmax, ntest, ntestt
      REAL               ANINV, ANORM, COND, OVFL, RTOVFL, RTUNFL,
     $                   TEMP1, ULP, ULPINV, UNFL
*     ..
*     .. Local Arrays ..
      INTEGER            IDUMMA( 1 ), IOLDSD( 4 ), KMAGN( MAXTYP ),
     $                   KMODE( MAXTYP ), KTYPE( MAXTYP )
*     ..
*     .. External Functions ..
      REAL               SLAMCH
      EXTERNAL           SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           slacpy, slasum, slatmr, slatms, slaset, ssbt21,
     $                   ssbtrd, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, real, sqrt
*     ..
*     .. Data statements ..
      DATA               ktype / 1, 2, 5*4, 5*5, 3*8 /
      DATA               kmagn / 2*1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1,
     $                   2, 3 /
      DATA               kmode / 2*0, 4, 3, 1, 4, 4, 4, 3, 1, 4, 4, 0,
     $                   0, 0 /
*     ..
*     .. Executable Statements ..
*
*     Check for errors
*
      ntestt = 0
      info = 0
*
*     Important constants
*
      badnn = .false.
      nmax = 1
      DO 10 j = 1, nsizes
         nmax = max( nmax, nn( j ) )
         IF( nn( j ).LT.0 )
     $      badnn = .true.
   10 CONTINUE
*
      badnnb = .false.
      kmax = 0
      DO 20 j = 1, nsizes
         kmax = max( kmax, kk( j ) )
         IF( kk( j ).LT.0 )
     $      badnnb = .true.
   20 CONTINUE
      kmax = min( nmax-1, kmax )
*
*     Check for errors
*
      IF( nsizes.LT.0 ) THEN
         info = -1
      ELSE IF( badnn ) THEN
         info = -2
      ELSE IF( nwdths.LT.0 ) THEN
         info = -3
      ELSE IF( badnnb ) THEN
         info = -4
      ELSE IF( ntypes.LT.0 ) THEN
         info = -5
      ELSE IF( lda.LT.kmax+1 ) THEN
         info = -11
      ELSE IF( ldu.LT.nmax ) THEN
         info = -15
      ELSE IF( ( max( lda, nmax )+1 )*nmax.GT.lwork ) THEN
         info = -17
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SCHKSB', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( nsizes.EQ.0 .OR. ntypes.EQ.0 .OR. nwdths.EQ.0 )
     $   RETURN
*
*     More Important constants
*
      unfl = slamch( 'Safe minimum' )
      ovfl = one / unfl
      ulp = slamch( 'Epsilon' )*slamch( 'Base' )
      ulpinv = one / ulp
      rtunfl = sqrt( unfl )
      rtovfl = sqrt( ovfl )
*
*     Loop over sizes, types
*
      nerrs = 0
      nmats = 0
*
      DO 190 jsize = 1, nsizes
         n = nn( jsize )
         aninv = one / real( max( 1, n ) )
*
         DO 180 jwidth = 1, nwdths
            k = kk( jwidth )
            IF( k.GT.n )
     $         GO TO 180
            k = max( 0, min( n-1, k ) )
*
            IF( nsizes.NE.1 ) THEN
               mtypes = min( maxtyp, ntypes )
            ELSE
               mtypes = min( maxtyp+1, ntypes )
            END IF
*
            DO 170 jtype = 1, mtypes
               IF( .NOT.dotype( jtype ) )
     $            GO TO 170
               nmats = nmats + 1
               ntest = 0
*
               DO 30 j = 1, 4
                  ioldsd( j ) = iseed( j )
   30          CONTINUE
*
*              Compute "A".
*              Store as "Upper"; later, we will copy to other format.
*
*              Control parameters:
*
*                  KMAGN  KMODE        KTYPE
*              =1  O(1)   clustered 1  zero
*              =2  large  clustered 2  identity
*              =3  small  exponential  (none)
*              =4         arithmetic   diagonal, (w/ eigenvalues)
*              =5         random log   symmetric, w/ eigenvalues
*              =6         random       (none)
*              =7                      random diagonal
*              =8                      random symmetric
*              =9                      positive definite
*              =10                     diagonally dominant tridiagonal
*
               IF( mtypes.GT.maxtyp )
     $            GO TO 100
*
               itype = ktype( jtype )
               imode = kmode( jtype )
*
*              Compute norm
*
               GO TO ( 40, 50, 60 )kmagn( jtype )
*
   40          CONTINUE
               anorm = one
               GO TO 70
*
   50          CONTINUE
               anorm = ( rtovfl*ulp )*aninv
               GO TO 70
*
   60          CONTINUE
               anorm = rtunfl*n*ulpinv
               GO TO 70
*
   70          CONTINUE
*
               CALL slaset( 'Full', lda, n, zero, zero, a, lda )
               iinfo = 0
               IF( jtype.LE.15 ) THEN
                  cond = ulpinv
               ELSE
                  cond = ulpinv*aninv / ten
               END IF
*
*              Special Matrices -- Identity & Jordan block
*
*                 Zero
*
               IF( itype.EQ.1 ) THEN
                  iinfo = 0
*
               ELSE IF( itype.EQ.2 ) THEN
*
*                 Identity
*
                  DO 80 jcol = 1, n
                     a( k+1, jcol ) = anorm
   80             CONTINUE
*
               ELSE IF( itype.EQ.4 ) THEN
*
*                 Diagonal Matrix, [Eigen]values Specified
*
                  CALL slatms( n, n, 'S', iseed, 'S', work, imode, cond,
     $                         anorm, 0, 0, 'Q', a( k+1, 1 ), lda,
     $                         work( n+1 ), iinfo )
*
               ELSE IF( itype.EQ.5 ) THEN
*
*                 Symmetric, eigenvalues specified
*
                  CALL slatms( n, n, 'S', iseed, 'S', work, imode, cond,
     $                         anorm, k, k, 'Q', a, lda, work( n+1 ),
     $                         iinfo )
*
               ELSE IF( itype.EQ.7 ) THEN
*
*                 Diagonal, random eigenvalues
*
                  CALL slatmr( n, n, 'S', iseed, 'S', work, 6, one, one,
     $                         'T', 'N', work( n+1 ), 1, one,
     $                         work( 2*n+1 ), 1, one, 'N', idumma, 0, 0,
     $                         zero, anorm, 'Q', a( k+1, 1 ), lda,
     $                         idumma, iinfo )
*
               ELSE IF( itype.EQ.8 ) THEN
*
*                 Symmetric, random eigenvalues
*
                  CALL slatmr( n, n, 'S', iseed, 'S', work, 6, one, one,
     $                         'T', 'N', work( n+1 ), 1, one,
     $                         work( 2*n+1 ), 1, one, 'N', idumma, k, k,
     $                         zero, anorm, 'Q', a, lda, idumma, iinfo )
*
               ELSE IF( itype.EQ.9 ) THEN
*
*                 Positive definite, eigenvalues specified.
*
                  CALL slatms( n, n, 'S', iseed, 'P', work, imode, cond,
     $                         anorm, k, k, 'Q', a, lda, work( n+1 ),
     $                         iinfo )
*
               ELSE IF( itype.EQ.10 ) THEN
*
*                 Positive definite tridiagonal, eigenvalues specified.
*
                  IF( n.GT.1 )
     $               k = max( 1, k )
                  CALL slatms( n, n, 'S', iseed, 'P', work, imode, cond,
     $                         anorm, 1, 1, 'Q', a( k, 1 ), lda,
     $                         work( n+1 ), iinfo )
                  DO 90 i = 2, n
                     temp1 = abs( a( k, i ) ) /
     $                       sqrt( abs( a( k+1, i-1 )*a( k+1, i ) ) )
                     IF( temp1.GT.half ) THEN
                        a( k, i ) = half*sqrt( abs( a( k+1,
     $                              i-1 )*a( k+1, i ) ) )
                     END IF
   90             CONTINUE
*
               ELSE
*
                  iinfo = 1
               END IF
*
               IF( iinfo.NE.0 ) THEN
                  WRITE( nounit, fmt = 9999 )'Generator', iinfo, n,
     $               jtype, ioldsd
                  info = abs( iinfo )
                  RETURN
               END IF
*
  100          CONTINUE
*
*              Call SSBTRD to compute S and U from upper triangle.
*
               CALL slacpy( ' ', k+1, n, a, lda, work, lda )
*
               ntest = 1
               CALL ssbtrd( 'V', 'U', n, k, work, lda, sd, se, u, ldu,
     $                      work( lda*n+1 ), iinfo )
*
               IF( iinfo.NE.0 ) THEN
                  WRITE( nounit, fmt = 9999 )'SSBTRD(U)', iinfo, n,
     $               jtype, ioldsd
                  info = abs( iinfo )
                  IF( iinfo.LT.0 ) THEN
                     RETURN
                  ELSE
                     result( 1 ) = ulpinv
                     GO TO 150
                  END IF
               END IF
*
*              Do tests 1 and 2
*
               CALL ssbt21( 'Upper', n, k, 1, a, lda, sd, se, u, ldu,
     $                      work, result( 1 ) )
*
*              Convert A from Upper-Triangle-Only storage to
*              Lower-Triangle-Only storage.
*
               DO 120 jc = 1, n
                  DO 110 jr = 0, min( k, n-jc )
                     a( jr+1, jc ) = a( k+1-jr, jc+jr )
  110             CONTINUE
  120          CONTINUE
               DO 140 jc = n + 1 - k, n
                  DO 130 jr = min( k, n-jc ) + 1, k
                     a( jr+1, jc ) = zero
  130             CONTINUE
  140          CONTINUE
*
*              Call SSBTRD to compute S and U from lower triangle
*
               CALL slacpy( ' ', k+1, n, a, lda, work, lda )
*
               ntest = 3
               CALL ssbtrd( 'V', 'L', n, k, work, lda, sd, se, u, ldu,
     $                      work( lda*n+1 ), iinfo )
*
               IF( iinfo.NE.0 ) THEN
                  WRITE( nounit, fmt = 9999 )'SSBTRD(L)', iinfo, n,
     $               jtype, ioldsd
                  info = abs( iinfo )
                  IF( iinfo.LT.0 ) THEN
                     RETURN
                  ELSE
                     result( 3 ) = ulpinv
                     GO TO 150
                  END IF
               END IF
               ntest = 4
*
*              Do tests 3 and 4
*
               CALL ssbt21( 'Lower', n, k, 1, a, lda, sd, se, u, ldu,
     $                      work, result( 3 ) )
*
*              End of Loop -- Check for RESULT(j) > THRESH
*
  150          CONTINUE
               ntestt = ntestt + ntest
*
*              Print out tests which fail.
*
               DO 160 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 = 9998 )'SSB'
                        WRITE( nounit, fmt = 9997 )
                        WRITE( nounit, fmt = 9996 )
                        WRITE( nounit, fmt = 9995 )'Symmetric'
                        WRITE( nounit, fmt = 9994 )'orthogonal', '''',
     $                     'transpose', ( '''', j = 1, 4 )
                     END IF
                     nerrs = nerrs + 1
                     WRITE( nounit, fmt = 9993 )n, k, ioldsd, jtype,
     $                  jr, result( jr )
                  END IF
  160          CONTINUE
*
  170       CONTINUE
  180    CONTINUE
  190 CONTINUE
*
*     Summary
*
      CALL slasum( 'SSB', nounit, nerrs, ntestt )
      RETURN
*
 9999 FORMAT( ' SCHKSB: ', a, ' returned INFO=', i6, '.', / 9x, 'N=',
     $      i6, ', JTYPE=', i6, ', ISEED=(', 3( i5, ',' ), i5, ')' )
*
 9998 FORMAT( / 1x, a3,
     $      ' -- Real Symmetric Banded Tridiagonal Reduction Routines' )
 9997 FORMAT( ' Matrix types (see SCHKSB for details): ' )
*
 9996 FORMAT( / ' Special Matrices:',
     $      / '  1=Zero matrix.                        ',
     $      '  5=Diagonal: clustered entries.',
     $      / '  2=Identity matrix.                    ',
     $      '  6=Diagonal: large, evenly spaced.',
     $      / '  3=Diagonal: evenly spaced entries.    ',
     $      '  7=Diagonal: small, evenly spaced.',
     $      / '  4=Diagonal: geometr. spaced entries.' )
 9995 FORMAT( ' Dense ', a, ' Banded Matrices:',
     $      / '  8=Evenly spaced eigenvals.            ',
     $      ' 12=Small, evenly spaced eigenvals.',
     $      / '  9=Geometrically spaced eigenvals.     ',
     $      ' 13=Matrix with random O(1) entries.',
     $      / ' 10=Clustered eigenvalues.              ',
     $      ' 14=Matrix with large random entries.',
     $      / ' 11=Large, evenly spaced eigenvals.     ',
     $      ' 15=Matrix with small random entries.' )
*
 9994 FORMAT( / ' Tests performed:   (S is Tridiag,  U is ', a, ',',
     $      / 20x, a, ' means ', a, '.', / ' UPLO=''U'':',
     $      / '  1= | A - U S U', a1, ' | / ( |A| n ulp )     ',
     $      '  2= | I - U U', a1, ' | / ( n ulp )', / ' UPLO=''L'':',
     $      / '  3= | A - U S U', a1, ' | / ( |A| n ulp )     ',
     $      '  4= | I - U U', a1, ' | / ( n ulp )' )
 9993 FORMAT( ' N=', i5, ', K=', i4, ', seed=', 4( i4, ',' ), ' type ',
     $      i2, ', test(', i2, ')=', g10.3 )
*
*     End of SCHKSB
*
      END