1 SUBROUTINE SLAHILB(N, NRHS, A, LDA, X, LDX, B, LDB, WORK, INFO)
2 !
3 ! -- LAPACK auxiliary test routine (version 3.0) --
4 ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
5 ! Courant Institute, Argonne National Lab, and Rice University
6 ! 28 August, 2006
7 !
8 ! David Vu <dtv@cs.berkeley.edu>
9 ! Yozo Hida <yozo@cs.berkeley.edu>
10 ! Jason Riedy <ejr@cs.berkeley.edu>
11 ! D. Halligan <dhalligan@berkeley.edu>
12 !
13 IMPLICIT NONE
14 ! .. Scalar Arguments ..
15 INTEGER N, NRHS, LDA, LDX, LDB, INFO
16 ! .. Array Arguments ..
17 REAL A(LDA, N), X(LDX, NRHS), B(LDB, NRHS), WORK(N)
18 ! ..
19 !
20 ! Purpose
21 ! =======
22 !
23 ! SLAHILB generates an N by N scaled Hilbert matrix in A along with
24 ! NRHS right-hand sides in B and solutions in X such that A*X=B.
25 !
26 ! The Hilbert matrix is scaled by M = LCM(1, 2, ..., 2*N-1) so that all
27 ! entries are integers. The right-hand sides are the first NRHS
28 ! columns of M * the identity matrix, and the solutions are the
29 ! first NRHS columns of the inverse Hilbert matrix.
30 !
31 ! The condition number of the Hilbert matrix grows exponentially with
32 ! its size, roughly as O(e ** (3.5*N)). Additionally, the inverse
33 ! Hilbert matrices beyond a relatively small dimension cannot be
34 ! generated exactly without extra precision. Precision is exhausted
35 ! when the largest entry in the inverse Hilbert matrix is greater than
36 ! 2 to the power of the number of bits in the fraction of the data type
37 ! used plus one, which is 24 for single precision.
38 !
39 ! In single, the generated solution is exact for N <= 6 and has
40 ! small componentwise error for 7 <= N <= 11.
41 !
42 ! Arguments
43 ! =========
44 !
45 ! N (input) INTEGER
46 ! The dimension of the matrix A.
47 !
48 ! NRHS (input) NRHS
49 ! The requested number of right-hand sides.
50 !
51 ! A (output) REAL array, dimension (LDA, N)
52 ! The generated scaled Hilbert matrix.
53 !
54 ! LDA (input) INTEGER
55 ! The leading dimension of the array A. LDA >= N.
56 !
57 ! X (output) REAL array, dimension (LDX, NRHS)
58 ! The generated exact solutions. Currently, the first NRHS
59 ! columns of the inverse Hilbert matrix.
60 !
61 ! LDX (input) INTEGER
62 ! The leading dimension of the array X. LDX >= N.
63 !
64 ! B (output) REAL array, dimension (LDB, NRHS)
65 ! The generated right-hand sides. Currently, the first NRHS
66 ! columns of LCM(1, 2, ..., 2*N-1) * the identity matrix.
67 !
68 ! LDB (input) INTEGER
69 ! The leading dimension of the array B. LDB >= N.
70 !
71 ! WORK (workspace) REAL array, dimension (N)
72 !
73 !
74 ! INFO (output) INTEGER
75 ! = 0: successful exit
76 ! = 1: N is too large; the data is still generated but may not
77 ! be not exact.
78 ! < 0: if INFO = -i, the i-th argument had an illegal value
79 !
80 ! =====================================================================
81
82 ! .. Local Scalars ..
83 INTEGER TM, TI, R
84 INTEGER M
85 INTEGER I, J
86
87 ! .. Parameters ..
88 ! NMAX_EXACT the largest dimension where the generated data is
89 ! exact.
90 ! NMAX_APPROX the largest dimension where the generated data has
91 ! a small componentwise relative error.
92 INTEGER NMAX_EXACT, NMAX_APPROX
93 PARAMETER (NMAX_EXACT = 6, NMAX_APPROX = 11)
94
95 ! ..
96 ! .. External Functions
97 EXTERNAL SLASET
98 INTRINSIC REAL
99 ! ..
100 ! .. Executable Statements ..
101 !
102 ! Test the input arguments
103 !
104 INFO = 0
105 IF (N .LT. 0 .OR. N .GT. NMAX_APPROX) THEN
106 INFO = -1
107 ELSE IF (NRHS .LT. 0) THEN
108 INFO = -2
109 ELSE IF (LDA .LT. N) THEN
110 INFO = -4
111 ELSE IF (LDX .LT. N) THEN
112 INFO = -6
113 ELSE IF (LDB .LT. N) THEN
114 INFO = -8
115 END IF
116 IF (INFO .LT. 0) THEN
117 CALL XERBLA('SLAHILB', -INFO)
118 RETURN
119 END IF
120 IF (N .GT. NMAX_EXACT) THEN
121 INFO = 1
122 END IF
123
124 ! Compute M = the LCM of the integers [1, 2*N-1]. The largest
125 ! reasonable N is small enough that integers suffice (up to N = 11).
126 M = 1
127 DO I = 2, (2*N-1)
128 TM = M
129 TI = I
130 R = MOD(TM, TI)
131 DO WHILE (R .NE. 0)
132 TM = TI
133 TI = R
134 R = MOD(TM, TI)
135 END DO
136 M = (M / TI) * I
137 END DO
138
139 ! Generate the scaled Hilbert matrix in A
140 DO J = 1, N
141 DO I = 1, N
142 A(I, J) = REAL(M) / (I + J - 1)
143 END DO
144 END DO
145
146 ! Generate matrix B as simply the first NRHS columns of M * the
147 ! identity.
148 CALL SLASET('Full', N, NRHS, 0.0, REAL(M), B, LDB)
149
150 ! Generate the true solutions in X. Because B = the first NRHS
151 ! columns of M*I, the true solutions are just the first NRHS columns
152 ! of the inverse Hilbert matrix.
153 WORK(1) = N
154 DO J = 2, N
155 WORK(J) = ( ( (WORK(J-1)/(J-1)) * (J-1 - N) ) /(J-1) )
156 $ * (N +J -1)
157 END DO
158
159 DO J = 1, NRHS
160 DO I = 1, N
161 X(I, J) = (WORK(I)*WORK(J)) / (I + J - 1)
162 END DO
163 END DO
164
165 END
166
2 !
3 ! -- LAPACK auxiliary test routine (version 3.0) --
4 ! Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
5 ! Courant Institute, Argonne National Lab, and Rice University
6 ! 28 August, 2006
7 !
8 ! David Vu <dtv@cs.berkeley.edu>
9 ! Yozo Hida <yozo@cs.berkeley.edu>
10 ! Jason Riedy <ejr@cs.berkeley.edu>
11 ! D. Halligan <dhalligan@berkeley.edu>
12 !
13 IMPLICIT NONE
14 ! .. Scalar Arguments ..
15 INTEGER N, NRHS, LDA, LDX, LDB, INFO
16 ! .. Array Arguments ..
17 REAL A(LDA, N), X(LDX, NRHS), B(LDB, NRHS), WORK(N)
18 ! ..
19 !
20 ! Purpose
21 ! =======
22 !
23 ! SLAHILB generates an N by N scaled Hilbert matrix in A along with
24 ! NRHS right-hand sides in B and solutions in X such that A*X=B.
25 !
26 ! The Hilbert matrix is scaled by M = LCM(1, 2, ..., 2*N-1) so that all
27 ! entries are integers. The right-hand sides are the first NRHS
28 ! columns of M * the identity matrix, and the solutions are the
29 ! first NRHS columns of the inverse Hilbert matrix.
30 !
31 ! The condition number of the Hilbert matrix grows exponentially with
32 ! its size, roughly as O(e ** (3.5*N)). Additionally, the inverse
33 ! Hilbert matrices beyond a relatively small dimension cannot be
34 ! generated exactly without extra precision. Precision is exhausted
35 ! when the largest entry in the inverse Hilbert matrix is greater than
36 ! 2 to the power of the number of bits in the fraction of the data type
37 ! used plus one, which is 24 for single precision.
38 !
39 ! In single, the generated solution is exact for N <= 6 and has
40 ! small componentwise error for 7 <= N <= 11.
41 !
42 ! Arguments
43 ! =========
44 !
45 ! N (input) INTEGER
46 ! The dimension of the matrix A.
47 !
48 ! NRHS (input) NRHS
49 ! The requested number of right-hand sides.
50 !
51 ! A (output) REAL array, dimension (LDA, N)
52 ! The generated scaled Hilbert matrix.
53 !
54 ! LDA (input) INTEGER
55 ! The leading dimension of the array A. LDA >= N.
56 !
57 ! X (output) REAL array, dimension (LDX, NRHS)
58 ! The generated exact solutions. Currently, the first NRHS
59 ! columns of the inverse Hilbert matrix.
60 !
61 ! LDX (input) INTEGER
62 ! The leading dimension of the array X. LDX >= N.
63 !
64 ! B (output) REAL array, dimension (LDB, NRHS)
65 ! The generated right-hand sides. Currently, the first NRHS
66 ! columns of LCM(1, 2, ..., 2*N-1) * the identity matrix.
67 !
68 ! LDB (input) INTEGER
69 ! The leading dimension of the array B. LDB >= N.
70 !
71 ! WORK (workspace) REAL array, dimension (N)
72 !
73 !
74 ! INFO (output) INTEGER
75 ! = 0: successful exit
76 ! = 1: N is too large; the data is still generated but may not
77 ! be not exact.
78 ! < 0: if INFO = -i, the i-th argument had an illegal value
79 !
80 ! =====================================================================
81
82 ! .. Local Scalars ..
83 INTEGER TM, TI, R
84 INTEGER M
85 INTEGER I, J
86
87 ! .. Parameters ..
88 ! NMAX_EXACT the largest dimension where the generated data is
89 ! exact.
90 ! NMAX_APPROX the largest dimension where the generated data has
91 ! a small componentwise relative error.
92 INTEGER NMAX_EXACT, NMAX_APPROX
93 PARAMETER (NMAX_EXACT = 6, NMAX_APPROX = 11)
94
95 ! ..
96 ! .. External Functions
97 EXTERNAL SLASET
98 INTRINSIC REAL
99 ! ..
100 ! .. Executable Statements ..
101 !
102 ! Test the input arguments
103 !
104 INFO = 0
105 IF (N .LT. 0 .OR. N .GT. NMAX_APPROX) THEN
106 INFO = -1
107 ELSE IF (NRHS .LT. 0) THEN
108 INFO = -2
109 ELSE IF (LDA .LT. N) THEN
110 INFO = -4
111 ELSE IF (LDX .LT. N) THEN
112 INFO = -6
113 ELSE IF (LDB .LT. N) THEN
114 INFO = -8
115 END IF
116 IF (INFO .LT. 0) THEN
117 CALL XERBLA('SLAHILB', -INFO)
118 RETURN
119 END IF
120 IF (N .GT. NMAX_EXACT) THEN
121 INFO = 1
122 END IF
123
124 ! Compute M = the LCM of the integers [1, 2*N-1]. The largest
125 ! reasonable N is small enough that integers suffice (up to N = 11).
126 M = 1
127 DO I = 2, (2*N-1)
128 TM = M
129 TI = I
130 R = MOD(TM, TI)
131 DO WHILE (R .NE. 0)
132 TM = TI
133 TI = R
134 R = MOD(TM, TI)
135 END DO
136 M = (M / TI) * I
137 END DO
138
139 ! Generate the scaled Hilbert matrix in A
140 DO J = 1, N
141 DO I = 1, N
142 A(I, J) = REAL(M) / (I + J - 1)
143 END DO
144 END DO
145
146 ! Generate matrix B as simply the first NRHS columns of M * the
147 ! identity.
148 CALL SLASET('Full', N, NRHS, 0.0, REAL(M), B, LDB)
149
150 ! Generate the true solutions in X. Because B = the first NRHS
151 ! columns of M*I, the true solutions are just the first NRHS columns
152 ! of the inverse Hilbert matrix.
153 WORK(1) = N
154 DO J = 2, N
155 WORK(J) = ( ( (WORK(J-1)/(J-1)) * (J-1 - N) ) /(J-1) )
156 $ * (N +J -1)
157 END DO
158
159 DO J = 1, NRHS
160 DO I = 1, N
161 X(I, J) = (WORK(I)*WORK(J)) / (I + J - 1)
162 END DO
163 END DO
164
165 END
166