279 lines
7.9 KiB
Fortran
279 lines
7.9 KiB
Fortran
*> \brief \b CLAHILB
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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 CLAHILB( N, NRHS, A, LDA, X, LDX, B, LDB, WORK,
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* INFO, PATH)
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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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* REAL WORK(N)
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* COMPLEX A(LDA,N), X(LDX, NRHS), B(LDB, NRHS)
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* CHARACTER*3 PATH
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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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*> CLAHILB 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 INTEGER
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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 COMPLEX 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 COMPLEX 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 REAL 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 REAL 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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*> \param[in] PATH
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*> \verbatim
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*> PATH is CHARACTER*3
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*> The LAPACK path name.
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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 2015
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*
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*> \ingroup complex_matgen
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*
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* =====================================================================
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SUBROUTINE CLAHILB( N, NRHS, A, LDA, X, LDX, B, LDB, WORK,
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$ INFO, PATH)
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*
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* -- LAPACK test routine (version 3.6.0) --
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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 2015
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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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REAL WORK(N)
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COMPLEX A(LDA,N), X(LDX, NRHS), B(LDB, NRHS)
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CHARACTER*3 PATH
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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 TMP
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CHARACTER*2 C2
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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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* ??? complex uses how many bits ???
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INTEGER NMAX_EXACT, NMAX_APPROX, SIZE_D
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PARAMETER (NMAX_EXACT = 6, NMAX_APPROX = 11, SIZE_D = 8)
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* d's are generated from random permuation of those eight elements.
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COMPLEX D1(8), D2(8), INVD1(8), INVD2(8)
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DATA D1 /(-1,0),(0,1),(-1,-1),(0,-1),(1,0),(-1,1),(1,1),(1,-1)/
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DATA D2 /(-1,0),(0,-1),(-1,1),(0,1),(1,0),(-1,-1),(1,-1),(1,1)/
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DATA INVD1 /(-1,0),(0,-1),(-.5,.5),(0,1),(1,0),
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$ (-.5,-.5),(.5,-.5),(.5,.5)/
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DATA INVD2 /(-1,0),(0,1),(-.5,-.5),(0,-1),(1,0),
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$ (-.5,.5),(.5,.5),(.5,-.5)/
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* ..
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* .. External Functions
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EXTERNAL CLASET, LSAMEN
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INTRINSIC REAL
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LOGICAL LSAMEN
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* ..
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* .. Executable Statements ..
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C2 = PATH( 2: 3 )
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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('CLAHILB', -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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* 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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* Generate the scaled Hilbert matrix in A
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* If we are testing SY routines, take
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* D1_i = D2_i, else, D1_i = D2_i*
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IF ( LSAMEN( 2, C2, 'SY' ) ) THEN
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DO J = 1, N
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DO I = 1, N
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A(I, J) = D1(MOD(J,SIZE_D)+1) * (REAL(M) / (I + J - 1))
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$ * D1(MOD(I,SIZE_D)+1)
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END DO
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END DO
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ELSE
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DO J = 1, N
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DO I = 1, N
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A(I, J) = D1(MOD(J,SIZE_D)+1) * (REAL(M) / (I + J - 1))
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$ * D2(MOD(I,SIZE_D)+1)
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END DO
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END DO
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END IF
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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 = REAL(M)
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CALL CLASET('Full', N, NRHS, (0.0,0.0), TMP, B, LDB)
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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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* If we are testing SY routines,
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* take D1_i = D2_i, else, D1_i = D2_i*
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IF ( LSAMEN( 2, C2, 'SY' ) ) THEN
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DO J = 1, NRHS
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DO I = 1, N
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X(I, J) =
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$ INVD1(MOD(J,SIZE_D)+1) *
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$ ((WORK(I)*WORK(J)) / (I + J - 1))
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$ * INVD1(MOD(I,SIZE_D)+1)
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END DO
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END DO
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ELSE
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DO J = 1, NRHS
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DO I = 1, N
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X(I, J) =
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$ INVD2(MOD(J,SIZE_D)+1) *
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$ ((WORK(I)*WORK(J)) / (I + J - 1))
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$ * INVD1(MOD(I,SIZE_D)+1)
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END DO
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END DO
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END IF
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END
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