LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ dlahilb()

subroutine dlahilb ( integer  N,
integer  NRHS,
double precision, dimension(lda, n)  A,
integer  LDA,
double precision, dimension(ldx, nrhs)  X,
integer  LDX,
double precision, dimension(ldb, nrhs)  B,
integer  LDB,
double precision, dimension(n)  WORK,
integer  INFO 
)

DLAHILB

Purpose:
 DLAHILB 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.
Parameters
[in]N
          N is INTEGER
          The dimension of the matrix A.
[in]NRHS
          NRHS is INTEGER
          The requested number of right-hand sides.
[out]A
          A is DOUBLE PRECISION array, dimension (LDA, N)
          The generated scaled Hilbert matrix.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= N.
[out]X
          X is DOUBLE PRECISION array, dimension (LDX, NRHS)
          The generated exact solutions.  Currently, the first NRHS
          columns of the inverse Hilbert matrix.
[in]LDX
          LDX is INTEGER
          The leading dimension of the array X.  LDX >= N.
[out]B
          B is DOUBLE PRECISION array, dimension (LDB, NRHS)
          The generated right-hand sides.  Currently, the first NRHS
          columns of LCM(1, 2, ..., 2*N-1) * the identity matrix.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= N.
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (N)
[out]INFO
          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
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 123 of file dlahilb.f.

124 *
125 * -- LAPACK test routine --
126 * -- LAPACK is a software package provided by Univ. of Tennessee, --
127 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
128 *
129 * .. Scalar Arguments ..
130  INTEGER N, NRHS, LDA, LDX, LDB, INFO
131 * .. Array Arguments ..
132  DOUBLE PRECISION A(LDA, N), X(LDX, NRHS), B(LDB, NRHS), WORK(N)
133 * ..
134 *
135 * =====================================================================
136 * .. Local Scalars ..
137  INTEGER TM, TI, R
138  INTEGER M
139  INTEGER I, J
140 * ..
141 * .. Parameters ..
142 * NMAX_EXACT the largest dimension where the generated data is
143 * exact.
144 * NMAX_APPROX the largest dimension where the generated data has
145 * a small componentwise relative error.
146  INTEGER NMAX_EXACT, NMAX_APPROX
147  parameter(nmax_exact = 6, nmax_approx = 11)
148 
149 * ..
150 * .. External Functions
151  EXTERNAL dlaset
152  INTRINSIC dble
153 * ..
154 * .. Executable Statements ..
155 *
156 * Test the input arguments
157 *
158  info = 0
159  IF (n .LT. 0 .OR. n .GT. nmax_approx) THEN
160  info = -1
161  ELSE IF (nrhs .LT. 0) THEN
162  info = -2
163  ELSE IF (lda .LT. n) THEN
164  info = -4
165  ELSE IF (ldx .LT. n) THEN
166  info = -6
167  ELSE IF (ldb .LT. n) THEN
168  info = -8
169  END IF
170  IF (info .LT. 0) THEN
171  CALL xerbla('DLAHILB', -info)
172  RETURN
173  END IF
174  IF (n .GT. nmax_exact) THEN
175  info = 1
176  END IF
177 *
178 * Compute M = the LCM of the integers [1, 2*N-1]. The largest
179 * reasonable N is small enough that integers suffice (up to N = 11).
180  m = 1
181  DO i = 2, (2*n-1)
182  tm = m
183  ti = i
184  r = mod(tm, ti)
185  DO WHILE (r .NE. 0)
186  tm = ti
187  ti = r
188  r = mod(tm, ti)
189  END DO
190  m = (m / ti) * i
191  END DO
192 *
193 * Generate the scaled Hilbert matrix in A
194  DO j = 1, n
195  DO i = 1, n
196  a(i, j) = dble(m) / (i + j - 1)
197  END DO
198  END DO
199 *
200 * Generate matrix B as simply the first NRHS columns of M * the
201 * identity.
202  CALL dlaset('Full', n, nrhs, 0.0d+0, dble(m), b, ldb)
203 
204 * Generate the true solutions in X. Because B = the first NRHS
205 * columns of M*I, the true solutions are just the first NRHS columns
206 * of the inverse Hilbert matrix.
207  work(1) = n
208  DO j = 2, n
209  work(j) = ( ( (work(j-1)/(j-1)) * (j-1 - n) ) /(j-1) )
210  $ * (n +j -1)
211  END DO
212 *
213  DO j = 1, nrhs
214  DO i = 1, n
215  x(i, j) = (work(i)*work(j)) / (i + j - 1)
216  END DO
217  END DO
218 *
subroutine dlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
DLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: dlaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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