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282 lines
8.2 KiB
282 lines
8.2 KiB
*> \brief \b DTBT03
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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 DTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
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* SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
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* RESID )
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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, N, NRHS
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* DOUBLE PRECISION RESID, SCALE, TSCAL
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), CNORM( * ),
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* $ WORK( * ), X( LDX, * )
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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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*> DTBT03 computes the residual for the solution to a scaled triangular
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*> system of equations A*x = s*b or A'*x = s*b when A is a
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*> triangular band matrix. Here A' is the transpose of A, s is a scalar,
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*> and x and b are N by NRHS matrices. The test ratio is the maximum
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*> over the number of right hand sides of
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*> norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
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*> where op(A) denotes A or A' and EPS is the machine epsilon.
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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 operation applied to A.
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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 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 superdiagonals or subdiagonals of the
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*> triangular band matrix A. 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 right hand sides, i.e., the number of columns
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*> of the matrices X and B. 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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*> \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] SCALE
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*> \verbatim
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*> SCALE is DOUBLE PRECISION
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*> The scaling factor s used in solving the triangular system.
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*> \endverbatim
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*>
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*> \param[in] CNORM
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*> \verbatim
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*> CNORM is DOUBLE PRECISION array, dimension (N)
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*> The 1-norms of the columns of A, not counting the diagonal.
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*> \endverbatim
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*>
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*> \param[in] TSCAL
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*> \verbatim
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*> TSCAL is DOUBLE PRECISION
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*> The scaling factor used in computing the 1-norms in CNORM.
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*> CNORM actually contains the column norms of TSCAL*A.
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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 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] 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] 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[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension (N)
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*> \endverbatim
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*>
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*> \param[out] RESID
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*> \verbatim
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*> RESID is DOUBLE PRECISION
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*> The maximum over the number of right hand sides of
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*> norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * 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 DTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB,
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$ SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK,
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$ RESID )
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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, N, NRHS
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DOUBLE PRECISION RESID, SCALE, TSCAL
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), CNORM( * ),
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$ WORK( * ), X( LDX, * )
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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 ONE, ZERO
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PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
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* ..
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* .. Local Scalars ..
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INTEGER IX, J
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DOUBLE PRECISION BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
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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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* .. External Subroutines ..
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EXTERNAL DAXPY, DCOPY, DSCAL, DTBMV
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, MAX
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* ..
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* .. Executable Statements ..
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*
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* Quick exit if N = 0
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*
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IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
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RESID = ZERO
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RETURN
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END IF
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EPS = DLAMCH( 'Epsilon' )
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SMLNUM = DLAMCH( 'Safe minimum' )
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BIGNUM = ONE / SMLNUM
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*
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* Compute the norm of the triangular matrix A using the column
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* norms already computed by DLATBS.
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*
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TNORM = ZERO
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IF( LSAME( DIAG, 'N' ) ) THEN
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IF( LSAME( UPLO, 'U' ) ) THEN
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DO 10 J = 1, N
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TNORM = MAX( TNORM, TSCAL*ABS( AB( KD+1, J ) )+
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$ CNORM( J ) )
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10 CONTINUE
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ELSE
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DO 20 J = 1, N
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TNORM = MAX( TNORM, TSCAL*ABS( AB( 1, J ) )+CNORM( J ) )
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20 CONTINUE
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END IF
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ELSE
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DO 30 J = 1, N
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TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
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30 CONTINUE
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END IF
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*
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* Compute the maximum over the number of right hand sides of
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* norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
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*
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RESID = ZERO
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DO 40 J = 1, NRHS
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CALL DCOPY( N, X( 1, J ), 1, WORK, 1 )
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IX = IDAMAX( N, WORK, 1 )
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XNORM = MAX( ONE, ABS( X( IX, J ) ) )
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XSCAL = ( ONE / XNORM ) / DBLE( KD+1 )
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CALL DSCAL( N, XSCAL, WORK, 1 )
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CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
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CALL DAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 )
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IX = IDAMAX( N, WORK, 1 )
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ERR = TSCAL*ABS( WORK( IX ) )
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IX = IDAMAX( N, X( 1, J ), 1 )
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XNORM = ABS( X( IX, J ) )
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IF( ERR*SMLNUM.LE.XNORM ) THEN
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IF( XNORM.GT.ZERO )
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$ ERR = ERR / XNORM
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ELSE
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IF( ERR.GT.ZERO )
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$ ERR = ONE / EPS
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END IF
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IF( ERR*SMLNUM.LE.TNORM ) THEN
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IF( TNORM.GT.ZERO )
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$ ERR = ERR / TNORM
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ELSE
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IF( ERR.GT.ZERO )
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$ ERR = ONE / EPS
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END IF
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RESID = MAX( RESID, ERR )
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40 CONTINUE
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*
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RETURN
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*
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* End of DTBT03
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*
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END
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