dtbt03.f

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*> \brief \b DTBT03
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE DTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
*                          SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
*                          RESID )
*
*       .. Scalar Arguments ..
*       CHARACTER          DIAG, TRANS, UPLO
*       INTEGER            KD, LDAB, LDB, LDX, N, NRHS
*       DOUBLE PRECISION   RESID, SCALE, TSCAL
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), CNORM( * ),
*      $                   WORK( * ), X( LDX, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DTBT03 computes the residual for the solution to a scaled triangular
*> system of equations  A*x = s*b  or  A'*x = s*b  when A is a
*> triangular band matrix. Here A' is the transpose of A, s is a scalar,
*> and x and b are N by NRHS matrices.  The test ratio is the maximum
*> over the number of right hand sides of
*>    norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
*> where op(A) denotes A or A' and EPS is the machine epsilon.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix A is upper or lower triangular.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the operation applied to A.
*>          = 'N':  A *x = b  (No transpose)
*>          = 'T':  A'*x = b  (Transpose)
*>          = 'C':  A'*x = b  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*>          DIAG is CHARACTER*1
*>          Specifies whether or not the matrix A is unit triangular.
*>          = 'N':  Non-unit triangular
*>          = 'U':  Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of superdiagonals or subdiagonals of the
*>          triangular band matrix A.  KD >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of the matrices X and B.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] AB
*> \verbatim
*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
*>          The upper or lower triangular band matrix A, stored in the
*>          first kd+1 rows of the array. The j-th column of A is stored
*>          in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \endverbatim
*>
*> \param[in] SCALE
*> \verbatim
*>          SCALE is DOUBLE PRECISION
*>          The scaling factor s used in solving the triangular system.
*> \endverbatim
*>
*> \param[in] CNORM
*> \verbatim
*>          CNORM is DOUBLE PRECISION array, dimension (N)
*>          The 1-norms of the columns of A, not counting the diagonal.
*> \endverbatim
*>
*> \param[in] TSCAL
*> \verbatim
*>          TSCAL is DOUBLE PRECISION
*>          The scaling factor used in computing the 1-norms in CNORM.
*>          CNORM actually contains the column norms of TSCAL*A.
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*>          The computed solution vectors for the system of linear
*>          equations.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*>          The right hand side vectors for the system of linear
*>          equations.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*>          RESID is DOUBLE PRECISION
*>          The maximum over the number of right hand sides of
*>          norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup double_lin
*
*  =====================================================================
      SUBROUTINE dtbt03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
     $                   SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
     $                   RESID )
*
*  -- 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 ..
      CHARACTER          DIAG, TRANS, UPLO
      INTEGER            KD, LDAB, LDB, LDX, N, NRHS
      DOUBLE PRECISION   RESID, SCALE, TSCAL
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), CNORM( * ),
     $                   WORK( * ), X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0d+0, zero = 0.0d+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            IX, J
      DOUBLE PRECISION   BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           lsame, idamax, dlamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           daxpy, dcopy, dscal, dtbmv
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, max
*     ..
*     .. Executable Statements ..
*
*     Quick exit if N = 0
*
      IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
         resid = zero
         RETURN
      END IF
      eps = dlamch( 'Epsilon' )
      smlnum = dlamch( 'Safe minimum' )
      bignum = one / smlnum
*
*     Compute the norm of the triangular matrix A using the column
*     norms already computed by DLATBS.
*
      tnorm = zero
      IF( lsame( diag, 'N' ) ) THEN
         IF( lsame( uplo, 'U' ) ) THEN
            DO 10 j = 1, n
               tnorm = max( tnorm, tscal*abs( ab( kd+1, j ) )+
     $                 cnorm( j ) )
   10       CONTINUE
         ELSE
            DO 20 j = 1, n
               tnorm = max( tnorm, tscal*abs( ab( 1, j ) )+cnorm( j ) )
   20       CONTINUE
         END IF
      ELSE
         DO 30 j = 1, n
            tnorm = max( tnorm, tscal+cnorm( j ) )
   30    CONTINUE
      END IF
*
*     Compute the maximum over the number of right hand sides of
*        norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
*
      resid = zero
      DO 40 j = 1, nrhs
         CALL dcopy( n, x( 1, j ), 1, work, 1 )
         ix = idamax( n, work, 1 )
         xnorm = max( one, abs( x( ix, j ) ) )
         xscal = ( one / xnorm ) / dble( kd+1 )
         CALL dscal( n, xscal, work, 1 )
         CALL dtbmv( uplo, trans, diag, n, kd, ab, ldab, work, 1 )
         CALL daxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
         ix = idamax( n, work, 1 )
         err = tscal*abs( work( ix ) )
         ix = idamax( n, x( 1, j ), 1 )
         xnorm = abs( x( ix, j ) )
         IF( err*smlnum.LE.xnorm ) THEN
            IF( xnorm.GT.zero )
     $         err = err / xnorm
         ELSE
            IF( err.GT.zero )
     $         err = one / eps
         END IF
         IF( err*smlnum.LE.tnorm ) THEN
            IF( tnorm.GT.zero )
     $         err = err / tnorm
         ELSE
            IF( err.GT.zero )
     $         err = one / eps
         END IF
         resid = max( resid, err )
   40 CONTINUE
*
      RETURN
*
*     End of DTBT03
*
      SUBROUTINE dtbt03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, …
      END