LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ slahilb()

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

SLAHILB

Purpose:
``` SLAHILB 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 REAL 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 REAL 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 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.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= N.``` [out] WORK ` WORK is REAL 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```
Date
June 2017

Definition at line 126 of file slahilb.f.

126 *
127 * -- LAPACK test routine (version 3.7.1) --
128 * -- LAPACK is a software package provided by Univ. of Tennessee, --
129 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
130 * June 2017
131 *
132 * .. Scalar Arguments ..
133  INTEGER n, nrhs, lda, ldx, ldb, info
134 * .. Array Arguments ..
135  REAL a(lda, n), x(ldx, nrhs), b(ldb, nrhs), work(n)
136 * ..
137 *
138 * =====================================================================
139 * .. Local Scalars ..
140  INTEGER tm, ti, r
141  INTEGER m
142  INTEGER i, j
143 * ..
144 * .. Parameters ..
145 * NMAX_EXACT the largest dimension where the generated data is
146 * exact.
147 * NMAX_APPROX the largest dimension where the generated data has
148 * a small componentwise relative error.
149  INTEGER nmax_exact, nmax_approx
150  parameter(nmax_exact = 6, nmax_approx = 11)
151 * ..
152 * .. External Functions
153  EXTERNAL slaset
154  INTRINSIC real
155 * ..
156 * .. Executable Statements ..
157 *
158 * Test the input arguments
159 *
160  info = 0
161  IF (n .LT. 0 .OR. n .GT. nmax_approx) THEN
162  info = -1
163  ELSE IF (nrhs .LT. 0) THEN
164  info = -2
165  ELSE IF (lda .LT. n) THEN
166  info = -4
167  ELSE IF (ldx .LT. n) THEN
168  info = -6
169  ELSE IF (ldb .LT. n) THEN
170  info = -8
171  END IF
172  IF (info .LT. 0) THEN
173  CALL xerbla('SLAHILB', -info)
174  RETURN
175  END IF
176  IF (n .GT. nmax_exact) THEN
177  info = 1
178  END IF
179 *
180 * Compute M = the LCM of the integers [1, 2*N-1]. The largest
181 * reasonable N is small enough that integers suffice (up to N = 11).
182  m = 1
183  DO i = 2, (2*n-1)
184  tm = m
185  ti = i
186  r = mod(tm, ti)
187  DO WHILE (r .NE. 0)
188  tm = ti
189  ti = r
190  r = mod(tm, ti)
191  END DO
192  m = (m / ti) * i
193  END DO
194 *
195 * Generate the scaled Hilbert matrix in A
196  DO j = 1, n
197  DO i = 1, n
198  a(i, j) = REAL(M) / (i + j - 1)
199  END DO
200  END DO
201 *
202 * Generate matrix B as simply the first NRHS columns of M * the
203 * identity.
204  CALL slaset('Full', n, nrhs, 0.0, REAL(M), b, ldb)
205 *
206 * Generate the true solutions in X. Because B = the first NRHS
207 * columns of M*I, the true solutions are just the first NRHS columns
208 * of the inverse Hilbert matrix.
209  work(1) = n
210  DO j = 2, n
211  work(j) = ( ( (work(j-1)/(j-1)) * (j-1 - n) ) /(j-1) )
212  \$ * (n +j -1)
213  END DO
214 *
215  DO j = 1, nrhs
216  DO i = 1, n
217  x(i, j) = (work(i)*work(j)) / (i + j - 1)
218  END DO
219  END DO
220 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: slaset.f:112
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