ssbt21.f

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*> \brief \b SSBT21
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE SSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
*                          RESULT )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            KA, KS, LDA, LDU, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
*      $                   U( LDU, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SSBT21  generally checks a decomposition of the form
*>
*>         A = U S U**T
*>
*> where **T means transpose, A is symmetric banded, U is
*> orthogonal, and S is diagonal (if KS=0) or symmetric
*> tridiagonal (if KS=1).
*>
*> Specifically:
*>
*>         RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and
*>         RESULT(2) = | I - U U**T | / ( n ulp )
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER
*>          If UPLO='U', the upper triangle of A and V will be used and
*>          the (strictly) lower triangle will not be referenced.
*>          If UPLO='L', the lower triangle of A and V will be used and
*>          the (strictly) upper triangle will not be referenced.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The size of the matrix.  If it is zero, SSBT21 does nothing.
*>          It must be at least zero.
*> \endverbatim
*>
*> \param[in] KA
*> \verbatim
*>          KA is INTEGER
*>          The bandwidth of the matrix A.  It must be at least zero.  If
*>          it is larger than N-1, then max( 0, N-1 ) will be used.
*> \endverbatim
*>
*> \param[in] KS
*> \verbatim
*>          KS is INTEGER
*>          The bandwidth of the matrix S.  It may only be zero or one.
*>          If zero, then S is diagonal, and E is not referenced.  If
*>          one, then S is symmetric tri-diagonal.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA, N)
*>          The original (unfactored) matrix.  It is assumed to be
*>          symmetric, and only the upper (UPLO='U') or only the lower
*>          (UPLO='L') will be referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  It must be at least 1
*>          and at least min( KA, N-1 ).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is REAL array, dimension (N)
*>          The diagonal of the (symmetric tri-) diagonal matrix S.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is REAL array, dimension (N-1)
*>          The off-diagonal of the (symmetric tri-) diagonal matrix S.
*>          E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
*>          (3,2) element, etc.
*>          Not referenced if KS=0.
*> \endverbatim
*>
*> \param[in] U
*> \verbatim
*>          U is REAL array, dimension (LDU, N)
*>          The orthogonal matrix in the decomposition, expressed as a
*>          dense matrix (i.e., not as a product of Householder
*>          transformations, Givens transformations, etc.)
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of U.  LDU must be at least N and
*>          at least 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (N**2+N)
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*>          RESULT is REAL array, dimension (2)
*>          The values computed by the two tests described above.  The
*>          values are currently limited to 1/ulp, to avoid overflow.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup single_eig
*
*  =====================================================================
      SUBROUTINE ssbt21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
     $                   RESULT )
*
*  -- LAPACK test routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            KA, KS, LDA, LDU, N
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
     $                   u( ldu, * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LOWER
      CHARACTER          CUPLO
      INTEGER            IKA, J, JC, JR, LW
      REAL               ANORM, ULP, UNFL, WNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SLAMCH, SLANGE, SLANSB, SLANSP
      EXTERNAL           lsame, slamch, slange, slansb, slansp
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgemm, sspr, sspr2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real
*     ..
*     .. Executable Statements ..
*
*     Constants
*
      result( 1 ) = zero
      result( 2 ) = zero
      IF( n.LE.0 )
     $   RETURN
*
      ika = max( 0, min( n-1, ka ) )
      lw = ( n*( n+1 ) ) / 2
*
      IF( lsame( uplo, 'U' ) ) THEN
         lower = .false.
         cuplo = 'U'
      ELSE
         lower = .true.
         cuplo = 'L'
      END IF
*
      unfl = slamch( 'Safe minimum' )
      ulp = slamch( 'Epsilon' )*slamch( 'Base' )
*
*     Some Error Checks
*
*     Do Test 1
*
*     Norm of A:
*
      anorm = max( slansb( '1', cuplo, n, ika, a, lda, work ), unfl )
*
*     Compute error matrix:    Error = A - U S U**T
*
*     Copy A from SB to SP storage format.
*
      j = 0
      DO 50 jc = 1, n
         IF( lower ) THEN
            DO 10 jr = 1, min( ika+1, n+1-jc )
               j = j + 1
               work( j ) = a( jr, jc )
   10       CONTINUE
            DO 20 jr = ika + 2, n + 1 - jc
               j = j + 1
               work( j ) = zero
   20       CONTINUE
         ELSE
            DO 30 jr = ika + 2, jc
               j = j + 1
               work( j ) = zero
   30       CONTINUE
            DO 40 jr = min( ika, jc-1 ), 0, -1
               j = j + 1
               work( j ) = a( ika+1-jr, jc )
   40       CONTINUE
         END IF
   50 CONTINUE
*
      DO 60 j = 1, n
         CALL sspr( cuplo, n, -d( j ), u( 1, j ), 1, work )
   60 CONTINUE
*
      IF( n.GT.1 .AND. ks.EQ.1 ) THEN
         DO 70 j = 1, n - 1
            CALL sspr2( cuplo, n, -e( j ), u( 1, j ), 1, u( 1, j+1 ), 1,
     $                  work )
   70    CONTINUE
      END IF
      wnorm = slansp( '1', cuplo, n, work, work( lw+1 ) )
*
      IF( anorm.GT.wnorm ) THEN
         result( 1 ) = ( wnorm / anorm ) / ( n*ulp )
      ELSE
         IF( anorm.LT.one ) THEN
            result( 1 ) = ( min( wnorm, n*anorm ) / anorm ) / ( n*ulp )
         ELSE
            result( 1 ) = min( wnorm / anorm, real( n ) ) / ( n*ulp )
         END IF
      END IF
*
*     Do Test 2
*
*     Compute  U U**T - I
*
      CALL sgemm( 'N', 'C', n, n, n, one, u, ldu, u, ldu, zero, work,
     $            n )
*
      DO 80 j = 1, n
         work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - one
   80 CONTINUE
*
      result( 2 ) = min( slange( '1', n, n, work, n, work( n**2+1 ) ),
     $              real( n ) ) / ( n*ulp )
*
      RETURN
*
*     End of SSBT21
*
      END