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332 lines
10 KiB
332 lines
10 KiB
2 years ago
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*> \brief \b DTBT05
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DTBT05( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
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* LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS )
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*
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* .. Scalar Arguments ..
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* CHARACTER DIAG, TRANS, UPLO
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* INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
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* $ FERR( * ), RESLTS( * ), X( LDX, * ),
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* $ XACT( LDXACT, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> DTBT05 tests the error bounds from iterative refinement for the
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*> computed solution to a system of equations A*X = B, where A is a
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*> triangular band matrix.
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*>
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*> RESLTS(1) = test of the error bound
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*> = norm(X - XACT) / ( norm(X) * FERR )
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*>
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*> A large value is returned if this ratio is not less than one.
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*>
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*> RESLTS(2) = residual from the iterative refinement routine
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*> = the maximum of BERR / ( NZ*EPS + (*) ), where
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*> (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
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*> and NZ = max. number of nonzeros in any row of A, plus 1
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> Specifies whether the matrix A is upper or lower triangular.
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*> = 'U': Upper triangular
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*> = 'L': Lower triangular
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*> \endverbatim
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*>
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*> \param[in] TRANS
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*> \verbatim
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*> TRANS is CHARACTER*1
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*> Specifies the form of the system of equations.
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*> = 'N': A * X = B (No transpose)
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*> = 'T': A'* X = B (Transpose)
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*> = 'C': A'* X = B (Conjugate transpose = Transpose)
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*> \endverbatim
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*>
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*> \param[in] DIAG
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*> \verbatim
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*> DIAG is CHARACTER*1
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*> Specifies whether or not the matrix A is unit triangular.
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*> = 'N': Non-unit triangular
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*> = 'U': Unit triangular
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of rows of the matrices X, B, and XACT, and the
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*> order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] KD
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*> \verbatim
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*> KD is INTEGER
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*> The number of super-diagonals of the matrix A if UPLO = 'U',
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*> or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
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*> \endverbatim
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*>
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*> \param[in] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of columns of the matrices X, B, and XACT.
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*> NRHS >= 0.
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*> \endverbatim
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*>
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*> \param[in] AB
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*> \verbatim
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*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
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*> The upper or lower triangular band matrix A, stored in the
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*> first kd+1 rows of the array. The j-th column of A is stored
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*> in the j-th column of the array AB as follows:
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*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
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*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
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*> If DIAG = 'U', the diagonal elements of A are not referenced
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*> and are assumed to be 1.
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*> \endverbatim
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*>
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*> \param[in] LDAB
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*> \verbatim
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*> LDAB is INTEGER
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*> The leading dimension of the array AB. LDAB >= KD+1.
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*> \endverbatim
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*>
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*> \param[in] B
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*> \verbatim
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*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
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*> The right hand side vectors for the system of linear
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*> equations.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the array B. LDB >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] X
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*> \verbatim
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*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
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*> The computed solution vectors. Each vector is stored as a
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*> column of the matrix X.
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*> \endverbatim
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*>
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*> \param[in] LDX
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*> \verbatim
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*> LDX is INTEGER
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*> The leading dimension of the array X. LDX >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] XACT
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*> \verbatim
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*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS)
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*> The exact solution vectors. Each vector is stored as a
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*> column of the matrix XACT.
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*> \endverbatim
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*>
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*> \param[in] LDXACT
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*> \verbatim
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*> LDXACT is INTEGER
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*> The leading dimension of the array XACT. LDXACT >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] FERR
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*> \verbatim
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*> FERR is DOUBLE PRECISION array, dimension (NRHS)
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*> The estimated forward error bounds for each solution vector
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*> X. If XTRUE is the true solution, FERR bounds the magnitude
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*> of the largest entry in (X - XTRUE) divided by the magnitude
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*> of the largest entry in X.
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*> \endverbatim
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*>
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*> \param[in] BERR
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*> \verbatim
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*> BERR is DOUBLE PRECISION array, dimension (NRHS)
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*> The componentwise relative backward error of each solution
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*> vector (i.e., the smallest relative change in any entry of A
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*> or B that makes X an exact solution).
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*> \endverbatim
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*>
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*> \param[out] RESLTS
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*> \verbatim
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*> RESLTS is DOUBLE PRECISION array, dimension (2)
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*> The maximum over the NRHS solution vectors of the ratios:
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*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
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*> RESLTS(2) = BERR / ( NZ*EPS + (*) )
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup double_lin
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*
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* =====================================================================
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SUBROUTINE DTBT05( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
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$ LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS )
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*
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* -- LAPACK test routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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CHARACTER DIAG, TRANS, UPLO
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INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
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$ FERR( * ), RESLTS( * ), X( LDX, * ),
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$ XACT( LDXACT, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOTRAN, UNIT, UPPER
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INTEGER I, IFU, IMAX, J, K, NZ
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DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER IDAMAX
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DOUBLE PRECISION DLAMCH
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EXTERNAL LSAME, IDAMAX, DLAMCH
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Quick exit if N = 0 or NRHS = 0.
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*
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IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
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RESLTS( 1 ) = ZERO
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RESLTS( 2 ) = ZERO
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RETURN
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END IF
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*
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EPS = DLAMCH( 'Epsilon' )
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UNFL = DLAMCH( 'Safe minimum' )
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OVFL = ONE / UNFL
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UPPER = LSAME( UPLO, 'U' )
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NOTRAN = LSAME( TRANS, 'N' )
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UNIT = LSAME( DIAG, 'U' )
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NZ = MIN( KD, N-1 ) + 1
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*
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* Test 1: Compute the maximum of
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* norm(X - XACT) / ( norm(X) * FERR )
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* over all the vectors X and XACT using the infinity-norm.
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*
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ERRBND = ZERO
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DO 30 J = 1, NRHS
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IMAX = IDAMAX( N, X( 1, J ), 1 )
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XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
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DIFF = ZERO
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DO 10 I = 1, N
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DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
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10 CONTINUE
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*
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IF( XNORM.GT.ONE ) THEN
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GO TO 20
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ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
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GO TO 20
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ELSE
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ERRBND = ONE / EPS
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GO TO 30
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END IF
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*
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20 CONTINUE
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IF( DIFF / XNORM.LE.FERR( J ) ) THEN
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ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
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ELSE
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ERRBND = ONE / EPS
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END IF
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30 CONTINUE
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RESLTS( 1 ) = ERRBND
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*
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* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
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* (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
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*
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IFU = 0
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IF( UNIT )
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$ IFU = 1
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DO 90 K = 1, NRHS
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DO 80 I = 1, N
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TMP = ABS( B( I, K ) )
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IF( UPPER ) THEN
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IF( .NOT.NOTRAN ) THEN
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DO 40 J = MAX( I-KD, 1 ), I - IFU
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TMP = TMP + ABS( AB( KD+1-I+J, I ) )*
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$ ABS( X( J, K ) )
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40 CONTINUE
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IF( UNIT )
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$ TMP = TMP + ABS( X( I, K ) )
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ELSE
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IF( UNIT )
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$ TMP = TMP + ABS( X( I, K ) )
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DO 50 J = I + IFU, MIN( I+KD, N )
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TMP = TMP + ABS( AB( KD+1+I-J, J ) )*
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$ ABS( X( J, K ) )
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50 CONTINUE
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END IF
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ELSE
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IF( NOTRAN ) THEN
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DO 60 J = MAX( I-KD, 1 ), I - IFU
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TMP = TMP + ABS( AB( 1+I-J, J ) )*ABS( X( J, K ) )
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60 CONTINUE
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IF( UNIT )
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$ TMP = TMP + ABS( X( I, K ) )
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ELSE
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IF( UNIT )
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$ TMP = TMP + ABS( X( I, K ) )
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DO 70 J = I + IFU, MIN( I+KD, N )
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TMP = TMP + ABS( AB( 1+J-I, I ) )*ABS( X( J, K ) )
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70 CONTINUE
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END IF
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END IF
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IF( I.EQ.1 ) THEN
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AXBI = TMP
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ELSE
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AXBI = MIN( AXBI, TMP )
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END IF
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80 CONTINUE
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TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
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IF( K.EQ.1 ) THEN
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RESLTS( 2 ) = TMP
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ELSE
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RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
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END IF
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90 CONTINUE
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*
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RETURN
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*
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* End of DTBT05
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*
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END
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