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*> \brief \b ZLAHILB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZLAHILB( N, NRHS, A, LDA, X, LDX, B, LDB, WORK,
* INFO, PATH)
*
* .. Scalar Arguments ..
* INTEGER N, NRHS, LDA, LDX, LDB, INFO
* .. Array Arguments ..
* DOUBLE PRECISION WORK(N)
* COMPLEX*16 A(LDA,N), X(LDX, NRHS), B(LDB, NRHS)
* CHARACTER*3 PATH
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLAHILB generates an N by N scaled Hilbert matrix in A along with
*> NRHS right-hand sides in B and solutions in X such that A*X=B.
*>
*> The Hilbert matrix is scaled by M = LCM(1, 2, ..., 2*N-1) so that all
*> entries are integers. The right-hand sides are the first NRHS
*> columns of M * the identity matrix, and the solutions are the
*> first NRHS columns of the inverse Hilbert matrix.
*>
*> The condition number of the Hilbert matrix grows exponentially with
*> its size, roughly as O(e ** (3.5*N)). Additionally, the inverse
*> Hilbert matrices beyond a relatively small dimension cannot be
*> generated exactly without extra precision. Precision is exhausted
*> when the largest entry in the inverse Hilbert matrix is greater than
*> 2 to the power of the number of bits in the fraction of the data type
*> used plus one, which is 24 for single precision.
*>
*> In single, the generated solution is exact for N <= 6 and has
*> small componentwise error for 7 <= N <= 11.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the matrix A.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The requested number of right-hand sides.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA, N)
*> The generated scaled Hilbert matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= N.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is COMPLEX array, dimension (LDX, NRHS)
*> The generated exact solutions. Currently, the first NRHS
*> columns of the inverse Hilbert matrix.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= N.
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*> B is REAL array, dimension (LDB, NRHS)
*> The generated right-hand sides. Currently, the first NRHS
*> columns of LCM(1, 2, ..., 2*N-1) * the identity matrix.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> = 1: N is too large; the data is still generated but may not
*> be not exact.
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*>
*> \param[in] PATH
*> \verbatim
*> PATH is CHARACTER*3
*> The LAPACK path name.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex16_matgen
*
* =====================================================================
SUBROUTINE ZLAHILB( N, NRHS, A, LDA, X, LDX, B, LDB, WORK,
$ INFO, PATH)
*
* -- LAPACK test routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER N, NRHS, LDA, LDX, LDB, INFO
* .. Array Arguments ..
DOUBLE PRECISION WORK(N)
COMPLEX*16 A(LDA,N), X(LDX, NRHS), B(LDB, NRHS)
CHARACTER*3 PATH
* ..
*
* =====================================================================
* .. Local Scalars ..
INTEGER TM, TI, R
INTEGER M
INTEGER I, J
COMPLEX*16 TMP
CHARACTER*2 C2
* .. Parameters ..
* NMAX_EXACT the largest dimension where the generated data is
* exact.
* NMAX_APPROX the largest dimension where the generated data has
* a small componentwise relative error.
* ??? complex uses how many bits ???
INTEGER NMAX_EXACT, NMAX_APPROX, SIZE_D
PARAMETER (NMAX_EXACT = 6, NMAX_APPROX = 11, SIZE_D = 8)
* d's are generated from random permuation of those eight elements.
COMPLEX*16 d1(8), d2(8), invd1(8), invd2(8)
DATA D1 /(-1,0),(0,1),(-1,-1),(0,-1),(1,0),(-1,1),(1,1),(1,-1)/
DATA D2 /(-1,0),(0,-1),(-1,1),(0,1),(1,0),(-1,-1),(1,-1),(1,1)/
DATA INVD1 /(-1,0),(0,-1),(-.5,.5),(0,1),(1,0),
$ (-.5,-.5),(.5,-.5),(.5,.5)/
DATA INVD2 /(-1,0),(0,1),(-.5,-.5),(0,-1),(1,0),
$ (-.5,.5),(.5,.5),(.5,-.5)/
* ..
* .. External Functions
EXTERNAL ZLASET, LSAMEN
INTRINSIC DBLE
LOGICAL LSAMEN
* ..
* .. Executable Statements ..
C2 = PATH( 2: 3 )
*
* Test the input arguments
*
INFO = 0
IF (N .LT. 0 .OR. N .GT. NMAX_APPROX) THEN
INFO = -1
ELSE IF (NRHS .LT. 0) THEN
INFO = -2
ELSE IF (LDA .LT. N) THEN
INFO = -4
ELSE IF (LDX .LT. N) THEN
INFO = -6
ELSE IF (LDB .LT. N) THEN
INFO = -8
END IF
IF (INFO .LT. 0) THEN
CALL XERBLA('ZLAHILB', -INFO)
RETURN
END IF
IF (N .GT. NMAX_EXACT) THEN
INFO = 1
END IF
* Compute M = the LCM of the integers [1, 2*N-1]. The largest
* reasonable N is small enough that integers suffice (up to N = 11).
M = 1
DO I = 2, (2*N-1)
TM = M
TI = I
R = MOD(TM, TI)
DO WHILE (R .NE. 0)
TM = TI
TI = R
R = MOD(TM, TI)
END DO
M = (M / TI) * I
END DO
* Generate the scaled Hilbert matrix in A
* If we are testing SY routines,
* take D1_i = D2_i, else, D1_i = D2_i*
IF ( LSAMEN( 2, C2, 'SY' ) ) THEN
DO J = 1, N
DO I = 1, N
A(I, J) = D1(MOD(J,SIZE_D)+1) * (DBLE(M) / (I + J - 1))
$ * D1(MOD(I,SIZE_D)+1)
END DO
END DO
ELSE
DO J = 1, N
DO I = 1, N
A(I, J) = D1(MOD(J,SIZE_D)+1) * (DBLE(M) / (I + J - 1))
$ * D2(MOD(I,SIZE_D)+1)
END DO
END DO
END IF
* Generate matrix B as simply the first NRHS columns of M * the
* identity.
TMP = DBLE(M)
CALL ZLASET('Full', N, NRHS, (0.0D+0,0.0D+0), TMP, B, LDB)
* Generate the true solutions in X. Because B = the first NRHS
* columns of M*I, the true solutions are just the first NRHS columns
* of the inverse Hilbert matrix.
WORK(1) = N
DO J = 2, N
WORK(J) = ( ( (WORK(J-1)/(J-1)) * (J-1 - N) ) /(J-1) )
$ * (N +J -1)
END DO
* If we are testing SY routines,
* take D1_i = D2_i, else, D1_i = D2_i*
IF ( LSAMEN( 2, C2, 'SY' ) ) THEN
DO J = 1, NRHS
DO I = 1, N
X(I, J) = INVD1(MOD(J,SIZE_D)+1) *
$ ((WORK(I)*WORK(J)) / (I + J - 1))
$ * INVD1(MOD(I,SIZE_D)+1)
END DO
END DO
ELSE
DO J = 1, NRHS
DO I = 1, N
X(I, J) = INVD2(MOD(J,SIZE_D)+1) *
$ ((WORK(I)*WORK(J)) / (I + J - 1))
$ * INVD1(MOD(I,SIZE_D)+1)
END DO
END DO
END IF
END
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