1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
|
*> \brief \b SLAHILB
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SLAHILB( N, NRHS, A, LDA, X, LDX, B, LDB, WORK, INFO)
*
* .. Scalar Arguments ..
* INTEGER N, NRHS, LDA, LDX, LDB, INFO
* .. Array Arguments ..
* REAL A(LDA, N), X(LDX, NRHS), B(LDB, NRHS), WORK(N)
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \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 REAL 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 REAL 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
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup real_matgen
*
* =====================================================================
SUBROUTINE SLAHILB( N, NRHS, A, LDA, X, LDX, B, LDB, WORK, INFO)
*
* -- 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 ..
REAL A(LDA, N), X(LDX, NRHS), B(LDB, NRHS), WORK(N)
* ..
*
* =====================================================================
* .. Local Scalars ..
INTEGER TM, TI, R
INTEGER M
INTEGER I, J
* .. 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.
INTEGER NMAX_EXACT, NMAX_APPROX
PARAMETER (NMAX_EXACT = 6, NMAX_APPROX = 11)
* ..
* .. External Functions
EXTERNAL SLASET
INTRINSIC REAL
* ..
* .. Executable Statements ..
*
* 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('SLAHILB', -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
DO J = 1, N
DO I = 1, N
A(I, J) = REAL(M) / (I + J - 1)
END DO
END DO
* Generate matrix B as simply the first NRHS columns of M * the
* identity.
CALL SLASET('Full', N, NRHS, 0.0, REAL(M), 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
DO J = 1, NRHS
DO I = 1, N
X(I, J) = (WORK(I)*WORK(J)) / (I + J - 1)
END DO
END DO
END
|
