226 lines
6.2 KiB
Fortran
226 lines
6.2 KiB
Fortran
*> \brief \b DLAHILB
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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 DLAHILB(N, NRHS, A, LDA, X, LDX, B, LDB, WORK, INFO)
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*
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* .. Scalar Arguments ..
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* INTEGER N, NRHS, LDA, LDX, LDB, INFO
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* .. Array Arguments ..
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* DOUBLE PRECISION A(LDA, N), X(LDX, NRHS), B(LDB, NRHS), WORK(N)
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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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*> DLAHILB generates an N by N scaled Hilbert matrix in A along with
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*> NRHS right-hand sides in B and solutions in X such that A*X=B.
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*>
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*> The Hilbert matrix is scaled by M = LCM(1, 2, ..., 2*N-1) so that all
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*> entries are integers. The right-hand sides are the first NRHS
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*> columns of M * the identity matrix, and the solutions are the
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*> first NRHS columns of the inverse Hilbert matrix.
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*>
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*> The condition number of the Hilbert matrix grows exponentially with
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*> its size, roughly as O(e ** (3.5*N)). Additionally, the inverse
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*> Hilbert matrices beyond a relatively small dimension cannot be
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*> generated exactly without extra precision. Precision is exhausted
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*> when the largest entry in the inverse Hilbert matrix is greater than
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*> 2 to the power of the number of bits in the fraction of the data type
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*> used plus one, which is 24 for single precision.
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*>
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*> In single, the generated solution is exact for N <= 6 and has
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*> small componentwise error for 7 <= N <= 11.
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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] N
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*> \verbatim
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*> N is INTEGER
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*> The dimension of the matrix A.
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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 NRHS
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*> The requested number of right-hand sides.
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*> \endverbatim
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*>
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*> \param[out] A
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*> \verbatim
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*> A is DOUBLE PRECISION array, dimension (LDA, N)
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*> The generated scaled Hilbert matrix.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= N.
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*> \endverbatim
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*>
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*> \param[out] X
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*> \verbatim
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*> X is DOUBLE PRECISION array, dimension (LDX, NRHS)
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*> The generated exact solutions. Currently, the first NRHS
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*> columns of the inverse Hilbert matrix.
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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 >= N.
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*> \endverbatim
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*>
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*> \param[out] B
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*> \verbatim
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*> B is DOUBLE PRECISION array, dimension (LDB, NRHS)
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*> The generated right-hand sides. Currently, the first NRHS
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*> columns of LCM(1, 2, ..., 2*N-1) * the identity matrix.
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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 >= 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] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> = 1: N is too large; the data is still generated but may not
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*> be not exact.
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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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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*> \date November 2011
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*
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*> \ingroup double_lin
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*
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* =====================================================================
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SUBROUTINE DLAHILB(N, NRHS, A, LDA, X, LDX, B, LDB, WORK, INFO)
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*
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* -- LAPACK test routine (version 3.4.0) --
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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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* November 2011
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*
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* .. Scalar Arguments ..
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INTEGER N, NRHS, LDA, LDX, LDB, INFO
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* .. Array Arguments ..
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DOUBLE PRECISION A(LDA, N), X(LDX, NRHS), B(LDB, NRHS), WORK(N)
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* ..
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*
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* =====================================================================
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* .. Local Scalars ..
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INTEGER TM, TI, R
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INTEGER M
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INTEGER I, J
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COMPLEX*16 TMP
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* ..
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* .. Parameters ..
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* NMAX_EXACT the largest dimension where the generated data is
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* exact.
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* NMAX_APPROX the largest dimension where the generated data has
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* a small componentwise relative error.
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INTEGER NMAX_EXACT, NMAX_APPROX
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PARAMETER (NMAX_EXACT = 6, NMAX_APPROX = 11)
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* ..
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* .. External Functions
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EXTERNAL DLASET
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INTRINSIC DBLE
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* ..
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* .. Executable Statements ..
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*
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* Test the input arguments
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*
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INFO = 0
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IF (N .LT. 0 .OR. N .GT. NMAX_APPROX) THEN
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INFO = -1
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ELSE IF (NRHS .LT. 0) THEN
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INFO = -2
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ELSE IF (LDA .LT. N) THEN
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INFO = -4
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ELSE IF (LDX .LT. N) THEN
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INFO = -6
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ELSE IF (LDB .LT. N) THEN
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INFO = -8
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END IF
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IF (INFO .LT. 0) THEN
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CALL XERBLA('DLAHILB', -INFO)
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RETURN
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END IF
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IF (N .GT. NMAX_EXACT) THEN
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INFO = 1
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END IF
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*
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* Compute M = the LCM of the integers [1, 2*N-1]. The largest
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* reasonable N is small enough that integers suffice (up to N = 11).
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M = 1
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DO I = 2, (2*N-1)
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TM = M
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TI = I
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R = MOD(TM, TI)
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DO WHILE (R .NE. 0)
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TM = TI
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TI = R
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R = MOD(TM, TI)
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END DO
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M = (M / TI) * I
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END DO
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*
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* Generate the scaled Hilbert matrix in A
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DO J = 1, N
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DO I = 1, N
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A(I, J) = DBLE(M) / (I + J - 1)
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END DO
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END DO
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*
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* Generate matrix B as simply the first NRHS columns of M * the
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* identity.
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TMP = DBLE(M)
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CALL DLASET('Full', N, NRHS, 0.0D+0, TMP, B, LDB)
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*
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* Generate the true solutions in X. Because B = the first NRHS
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* columns of M*I, the true solutions are just the first NRHS columns
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* of the inverse Hilbert matrix.
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WORK(1) = N
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DO J = 2, N
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WORK(J) = ( ( (WORK(J-1)/(J-1)) * (J-1 - N) ) /(J-1) )
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$ * (N +J -1)
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END DO
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*
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DO J = 1, NRHS
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DO I = 1, N
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X(I, J) = (WORK(I)*WORK(J)) / (I + J - 1)
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END DO
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END DO
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*
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END
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