LAPACK  3.10.0 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```

Definition at line 123 of file slahilb.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  REAL 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 * .. External Functions
150  EXTERNAL slaset
151  INTRINSIC real
152 * ..
153 * .. Executable Statements ..
154 *
155 * Test the input arguments
156 *
157  info = 0
158  IF (n .LT. 0 .OR. n .GT. nmax_approx) THEN
159  info = -1
160  ELSE IF (nrhs .LT. 0) THEN
161  info = -2
162  ELSE IF (lda .LT. n) THEN
163  info = -4
164  ELSE IF (ldx .LT. n) THEN
165  info = -6
166  ELSE IF (ldb .LT. n) THEN
167  info = -8
168  END IF
169  IF (info .LT. 0) THEN
170  CALL xerbla('SLAHILB', -info)
171  RETURN
172  END IF
173  IF (n .GT. nmax_exact) THEN
174  info = 1
175  END IF
176 *
177 * Compute M = the LCM of the integers [1, 2*N-1]. The largest
178 * reasonable N is small enough that integers suffice (up to N = 11).
179  m = 1
180  DO i = 2, (2*n-1)
181  tm = m
182  ti = i
183  r = mod(tm, ti)
184  DO WHILE (r .NE. 0)
185  tm = ti
186  ti = r
187  r = mod(tm, ti)
188  END DO
189  m = (m / ti) * i
190  END DO
191 *
192 * Generate the scaled Hilbert matrix in A
193  DO j = 1, n
194  DO i = 1, n
195  a(i, j) = real(m) / (i + j - 1)
196  END DO
197  END DO
198 *
199 * Generate matrix B as simply the first NRHS columns of M * the
200 * identity.
201  CALL slaset('Full', n, nrhs, 0.0, real(m), b, ldb)
202 *
203 * Generate the true solutions in X. Because B = the first NRHS
204 * columns of M*I, the true solutions are just the first NRHS columns
205 * of the inverse Hilbert matrix.
206  work(1) = n
207  DO j = 2, n
208  work(j) = ( ( (work(j-1)/(j-1)) * (j-1 - n) ) /(j-1) )
209  \$ * (n +j -1)
210  END DO
211 *
212  DO j = 1, nrhs
213  DO i = 1, n
214  x(i, j) = (work(i)*work(j)) / (i + j - 1)
215  END DO
216  END DO
217 *
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:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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