LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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◆ zlahilb()

 subroutine zlahilb ( integer N, integer NRHS, complex*16, dimension(lda,n) A, integer LDA, complex*16, dimension(ldx, nrhs) X, integer LDX, complex*16, dimension(ldb, nrhs) B, integer LDB, double precision, dimension(n) WORK, integer INFO, character*3 PATH )

ZLAHILB

Purpose:
``` 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.```
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 COMPLEX 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 COMPLEX 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``` [in] PATH ``` PATH is CHARACTER*3 The LAPACK path name.```

Definition at line 132 of file zlahilb.f.

134*
135* -- LAPACK test routine --
136* -- LAPACK is a software package provided by Univ. of Tennessee, --
137* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
138*
139* .. Scalar Arguments ..
140 INTEGER N, NRHS, LDA, LDX, LDB, INFO
141* .. Array Arguments ..
142 DOUBLE PRECISION WORK(N)
143 COMPLEX*16 A(LDA,N), X(LDX, NRHS), B(LDB, NRHS)
144 CHARACTER*3 PATH
145* ..
146*
147* =====================================================================
148* .. Local Scalars ..
149 INTEGER TM, TI, R
150 INTEGER M
151 INTEGER I, J
152 COMPLEX*16 TMP
153 CHARACTER*2 C2
154* ..
155* .. Parameters ..
156* NMAX_EXACT the largest dimension where the generated data is
157* exact.
158* NMAX_APPROX the largest dimension where the generated data has
159* a small componentwise relative error.
160* ??? complex uses how many bits ???
161 INTEGER NMAX_EXACT, NMAX_APPROX, SIZE_D
162 parameter(nmax_exact = 6, nmax_approx = 11, size_d = 8)
163*
164* d's are generated from random permutation of those eight elements.
165 COMPLEX*16 d1(8), d2(8), invd1(8), invd2(8)
166 DATA d1 /(-1,0),(0,1),(-1,-1),(0,-1),(1,0),(-1,1),(1,1),(1,-1)/
167 DATA d2 /(-1,0),(0,-1),(-1,1),(0,1),(1,0),(-1,-1),(1,-1),(1,1)/
168
169 DATA invd1 /(-1,0),(0,-1),(-.5,.5),(0,1),(1,0),
170 \$ (-.5,-.5),(.5,-.5),(.5,.5)/
171 DATA invd2 /(-1,0),(0,1),(-.5,-.5),(0,-1),(1,0),
172 \$ (-.5,.5),(.5,.5),(.5,-.5)/
173* ..
174* .. External Functions
175 EXTERNAL zlaset, lsamen
176 INTRINSIC dble
177 LOGICAL LSAMEN
178* ..
179* .. Executable Statements ..
180 c2 = path( 2: 3 )
181*
182* Test the input arguments
183*
184 info = 0
185 IF (n .LT. 0 .OR. n .GT. nmax_approx) THEN
186 info = -1
187 ELSE IF (nrhs .LT. 0) THEN
188 info = -2
189 ELSE IF (lda .LT. n) THEN
190 info = -4
191 ELSE IF (ldx .LT. n) THEN
192 info = -6
193 ELSE IF (ldb .LT. n) THEN
194 info = -8
195 END IF
196 IF (info .LT. 0) THEN
197 CALL xerbla('ZLAHILB', -info)
198 RETURN
199 END IF
200 IF (n .GT. nmax_exact) THEN
201 info = 1
202 END IF
203*
204* Compute M = the LCM of the integers [1, 2*N-1]. The largest
205* reasonable N is small enough that integers suffice (up to N = 11).
206 m = 1
207 DO i = 2, (2*n-1)
208 tm = m
209 ti = i
210 r = mod(tm, ti)
211 DO WHILE (r .NE. 0)
212 tm = ti
213 ti = r
214 r = mod(tm, ti)
215 END DO
216 m = (m / ti) * i
217 END DO
218*
219* Generate the scaled Hilbert matrix in A
220* If we are testing SY routines,
221* take D1_i = D2_i, else, D1_i = D2_i*
222 IF ( lsamen( 2, c2, 'SY' ) ) THEN
223 DO j = 1, n
224 DO i = 1, n
225 a(i, j) = d1(mod(j,size_d)+1) * (dble(m) / (i + j - 1))
226 \$ * d1(mod(i,size_d)+1)
227 END DO
228 END DO
229 ELSE
230 DO j = 1, n
231 DO i = 1, n
232 a(i, j) = d1(mod(j,size_d)+1) * (dble(m) / (i + j - 1))
233 \$ * d2(mod(i,size_d)+1)
234 END DO
235 END DO
236 END IF
237*
238* Generate matrix B as simply the first NRHS columns of M * the
239* identity.
240 tmp = dble(m)
241 CALL zlaset('Full', n, nrhs, (0.0d+0,0.0d+0), tmp, b, ldb)
242*
243* Generate the true solutions in X. Because B = the first NRHS
244* columns of M*I, the true solutions are just the first NRHS columns
245* of the inverse Hilbert matrix.
246 work(1) = n
247 DO j = 2, n
248 work(j) = ( ( (work(j-1)/(j-1)) * (j-1 - n) ) /(j-1) )
249 \$ * (n +j -1)
250 END DO
251
252* If we are testing SY routines,
253* take D1_i = D2_i, else, D1_i = D2_i*
254 IF ( lsamen( 2, c2, 'SY' ) ) THEN
255 DO j = 1, nrhs
256 DO i = 1, n
257 x(i, j) = invd1(mod(j,size_d)+1) *
258 \$ ((work(i)*work(j)) / (i + j - 1))
259 \$ * invd1(mod(i,size_d)+1)
260 END DO
261 END DO
262 ELSE
263 DO j = 1, nrhs
264 DO i = 1, n
265 x(i, j) = invd2(mod(j,size_d)+1) *
266 \$ ((work(i)*work(j)) / (i + j - 1))
267 \$ * invd1(mod(i,size_d)+1)
268 END DO
269 END DO
270 END IF
logical function lsamen(N, CA, CB)
LSAMEN
Definition: lsamen.f:74
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
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