1 SUBROUTINE SPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
2 $ LDXACT, FERR, BERR, RESLTS )
3 *
4 * -- LAPACK test routine (version 3.1) --
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
6 * November 2006
7 *
8 * .. Scalar Arguments ..
9 CHARACTER UPLO
10 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
14 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * SPOT05 tests the error bounds from iterative refinement for the
21 * computed solution to a system of equations A*X = B, where A is a
22 * symmetric n by n matrix.
23 *
24 * RESLTS(1) = test of the error bound
25 * = norm(X - XACT) / ( norm(X) * FERR )
26 *
27 * A large value is returned if this ratio is not less than one.
28 *
29 * RESLTS(2) = residual from the iterative refinement routine
30 * = the maximum of BERR / ( (n+1)*EPS + (*) ), where
31 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
32 *
33 * Arguments
34 * =========
35 *
36 * UPLO (input) CHARACTER*1
37 * Specifies whether the upper or lower triangular part of the
38 * symmetric matrix A is stored.
39 * = 'U': Upper triangular
40 * = 'L': Lower triangular
41 *
42 * N (input) INTEGER
43 * The number of rows of the matrices X, B, and XACT, and the
44 * order of the matrix A. N >= 0.
45 *
46 * NRHS (input) INTEGER
47 * The number of columns of the matrices X, B, and XACT.
48 * NRHS >= 0.
49 *
50 * A (input) REAL array, dimension (LDA,N)
51 * The symmetric matrix A. If UPLO = 'U', the leading n by n
52 * upper triangular part of A contains the upper triangular part
53 * of the matrix A, and the strictly lower triangular part of A
54 * is not referenced. If UPLO = 'L', the leading n by n lower
55 * triangular part of A contains the lower triangular part of
56 * the matrix A, and the strictly upper triangular part of A is
57 * not referenced.
58 *
59 * LDA (input) INTEGER
60 * The leading dimension of the array A. LDA >= max(1,N).
61 *
62 * B (input) REAL array, dimension (LDB,NRHS)
63 * The right hand side vectors for the system of linear
64 * equations.
65 *
66 * LDB (input) INTEGER
67 * The leading dimension of the array B. LDB >= max(1,N).
68 *
69 * X (input) REAL array, dimension (LDX,NRHS)
70 * The computed solution vectors. Each vector is stored as a
71 * column of the matrix X.
72 *
73 * LDX (input) INTEGER
74 * The leading dimension of the array X. LDX >= max(1,N).
75 *
76 * XACT (input) REAL array, dimension (LDX,NRHS)
77 * The exact solution vectors. Each vector is stored as a
78 * column of the matrix XACT.
79 *
80 * LDXACT (input) INTEGER
81 * The leading dimension of the array XACT. LDXACT >= max(1,N).
82 *
83 * FERR (input) REAL array, dimension (NRHS)
84 * The estimated forward error bounds for each solution vector
85 * X. If XTRUE is the true solution, FERR bounds the magnitude
86 * of the largest entry in (X - XTRUE) divided by the magnitude
87 * of the largest entry in X.
88 *
89 * BERR (input) REAL array, dimension (NRHS)
90 * The componentwise relative backward error of each solution
91 * vector (i.e., the smallest relative change in any entry of A
92 * or B that makes X an exact solution).
93 *
94 * RESLTS (output) REAL array, dimension (2)
95 * The maximum over the NRHS solution vectors of the ratios:
96 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
97 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
98 *
99 * =====================================================================
100 *
101 * .. Parameters ..
102 REAL ZERO, ONE
103 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
104 * ..
105 * .. Local Scalars ..
106 LOGICAL UPPER
107 INTEGER I, IMAX, J, K
108 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
109 * ..
110 * .. External Functions ..
111 LOGICAL LSAME
112 INTEGER ISAMAX
113 REAL SLAMCH
114 EXTERNAL LSAME, ISAMAX, SLAMCH
115 * ..
116 * .. Intrinsic Functions ..
117 INTRINSIC ABS, MAX, MIN
118 * ..
119 * .. Executable Statements ..
120 *
121 * Quick exit if N = 0 or NRHS = 0.
122 *
123 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
124 RESLTS( 1 ) = ZERO
125 RESLTS( 2 ) = ZERO
126 RETURN
127 END IF
128 *
129 EPS = SLAMCH( 'Epsilon' )
130 UNFL = SLAMCH( 'Safe minimum' )
131 OVFL = ONE / UNFL
132 UPPER = LSAME( UPLO, 'U' )
133 *
134 * Test 1: Compute the maximum of
135 * norm(X - XACT) / ( norm(X) * FERR )
136 * over all the vectors X and XACT using the infinity-norm.
137 *
138 ERRBND = ZERO
139 DO 30 J = 1, NRHS
140 IMAX = ISAMAX( N, X( 1, J ), 1 )
141 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
142 DIFF = ZERO
143 DO 10 I = 1, N
144 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
145 10 CONTINUE
146 *
147 IF( XNORM.GT.ONE ) THEN
148 GO TO 20
149 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
150 GO TO 20
151 ELSE
152 ERRBND = ONE / EPS
153 GO TO 30
154 END IF
155 *
156 20 CONTINUE
157 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
158 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
159 ELSE
160 ERRBND = ONE / EPS
161 END IF
162 30 CONTINUE
163 RESLTS( 1 ) = ERRBND
164 *
165 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
166 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
167 *
168 DO 90 K = 1, NRHS
169 DO 80 I = 1, N
170 TMP = ABS( B( I, K ) )
171 IF( UPPER ) THEN
172 DO 40 J = 1, I
173 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
174 40 CONTINUE
175 DO 50 J = I + 1, N
176 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
177 50 CONTINUE
178 ELSE
179 DO 60 J = 1, I - 1
180 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
181 60 CONTINUE
182 DO 70 J = I, N
183 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
184 70 CONTINUE
185 END IF
186 IF( I.EQ.1 ) THEN
187 AXBI = TMP
188 ELSE
189 AXBI = MIN( AXBI, TMP )
190 END IF
191 80 CONTINUE
192 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
193 $ MAX( AXBI, ( N+1 )*UNFL ) )
194 IF( K.EQ.1 ) THEN
195 RESLTS( 2 ) = TMP
196 ELSE
197 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
198 END IF
199 90 CONTINUE
200 *
201 RETURN
202 *
203 * End of SPOT05
204 *
205 END
2 $ LDXACT, FERR, BERR, RESLTS )
3 *
4 * -- LAPACK test routine (version 3.1) --
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
6 * November 2006
7 *
8 * .. Scalar Arguments ..
9 CHARACTER UPLO
10 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
14 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * SPOT05 tests the error bounds from iterative refinement for the
21 * computed solution to a system of equations A*X = B, where A is a
22 * symmetric n by n matrix.
23 *
24 * RESLTS(1) = test of the error bound
25 * = norm(X - XACT) / ( norm(X) * FERR )
26 *
27 * A large value is returned if this ratio is not less than one.
28 *
29 * RESLTS(2) = residual from the iterative refinement routine
30 * = the maximum of BERR / ( (n+1)*EPS + (*) ), where
31 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
32 *
33 * Arguments
34 * =========
35 *
36 * UPLO (input) CHARACTER*1
37 * Specifies whether the upper or lower triangular part of the
38 * symmetric matrix A is stored.
39 * = 'U': Upper triangular
40 * = 'L': Lower triangular
41 *
42 * N (input) INTEGER
43 * The number of rows of the matrices X, B, and XACT, and the
44 * order of the matrix A. N >= 0.
45 *
46 * NRHS (input) INTEGER
47 * The number of columns of the matrices X, B, and XACT.
48 * NRHS >= 0.
49 *
50 * A (input) REAL array, dimension (LDA,N)
51 * The symmetric matrix A. If UPLO = 'U', the leading n by n
52 * upper triangular part of A contains the upper triangular part
53 * of the matrix A, and the strictly lower triangular part of A
54 * is not referenced. If UPLO = 'L', the leading n by n lower
55 * triangular part of A contains the lower triangular part of
56 * the matrix A, and the strictly upper triangular part of A is
57 * not referenced.
58 *
59 * LDA (input) INTEGER
60 * The leading dimension of the array A. LDA >= max(1,N).
61 *
62 * B (input) REAL array, dimension (LDB,NRHS)
63 * The right hand side vectors for the system of linear
64 * equations.
65 *
66 * LDB (input) INTEGER
67 * The leading dimension of the array B. LDB >= max(1,N).
68 *
69 * X (input) REAL array, dimension (LDX,NRHS)
70 * The computed solution vectors. Each vector is stored as a
71 * column of the matrix X.
72 *
73 * LDX (input) INTEGER
74 * The leading dimension of the array X. LDX >= max(1,N).
75 *
76 * XACT (input) REAL array, dimension (LDX,NRHS)
77 * The exact solution vectors. Each vector is stored as a
78 * column of the matrix XACT.
79 *
80 * LDXACT (input) INTEGER
81 * The leading dimension of the array XACT. LDXACT >= max(1,N).
82 *
83 * FERR (input) REAL array, dimension (NRHS)
84 * The estimated forward error bounds for each solution vector
85 * X. If XTRUE is the true solution, FERR bounds the magnitude
86 * of the largest entry in (X - XTRUE) divided by the magnitude
87 * of the largest entry in X.
88 *
89 * BERR (input) REAL array, dimension (NRHS)
90 * The componentwise relative backward error of each solution
91 * vector (i.e., the smallest relative change in any entry of A
92 * or B that makes X an exact solution).
93 *
94 * RESLTS (output) REAL array, dimension (2)
95 * The maximum over the NRHS solution vectors of the ratios:
96 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
97 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
98 *
99 * =====================================================================
100 *
101 * .. Parameters ..
102 REAL ZERO, ONE
103 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
104 * ..
105 * .. Local Scalars ..
106 LOGICAL UPPER
107 INTEGER I, IMAX, J, K
108 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
109 * ..
110 * .. External Functions ..
111 LOGICAL LSAME
112 INTEGER ISAMAX
113 REAL SLAMCH
114 EXTERNAL LSAME, ISAMAX, SLAMCH
115 * ..
116 * .. Intrinsic Functions ..
117 INTRINSIC ABS, MAX, MIN
118 * ..
119 * .. Executable Statements ..
120 *
121 * Quick exit if N = 0 or NRHS = 0.
122 *
123 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
124 RESLTS( 1 ) = ZERO
125 RESLTS( 2 ) = ZERO
126 RETURN
127 END IF
128 *
129 EPS = SLAMCH( 'Epsilon' )
130 UNFL = SLAMCH( 'Safe minimum' )
131 OVFL = ONE / UNFL
132 UPPER = LSAME( UPLO, 'U' )
133 *
134 * Test 1: Compute the maximum of
135 * norm(X - XACT) / ( norm(X) * FERR )
136 * over all the vectors X and XACT using the infinity-norm.
137 *
138 ERRBND = ZERO
139 DO 30 J = 1, NRHS
140 IMAX = ISAMAX( N, X( 1, J ), 1 )
141 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
142 DIFF = ZERO
143 DO 10 I = 1, N
144 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
145 10 CONTINUE
146 *
147 IF( XNORM.GT.ONE ) THEN
148 GO TO 20
149 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
150 GO TO 20
151 ELSE
152 ERRBND = ONE / EPS
153 GO TO 30
154 END IF
155 *
156 20 CONTINUE
157 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
158 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
159 ELSE
160 ERRBND = ONE / EPS
161 END IF
162 30 CONTINUE
163 RESLTS( 1 ) = ERRBND
164 *
165 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
166 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
167 *
168 DO 90 K = 1, NRHS
169 DO 80 I = 1, N
170 TMP = ABS( B( I, K ) )
171 IF( UPPER ) THEN
172 DO 40 J = 1, I
173 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
174 40 CONTINUE
175 DO 50 J = I + 1, N
176 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
177 50 CONTINUE
178 ELSE
179 DO 60 J = 1, I - 1
180 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
181 60 CONTINUE
182 DO 70 J = I, N
183 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
184 70 CONTINUE
185 END IF
186 IF( I.EQ.1 ) THEN
187 AXBI = TMP
188 ELSE
189 AXBI = MIN( AXBI, TMP )
190 END IF
191 80 CONTINUE
192 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
193 $ MAX( AXBI, ( N+1 )*UNFL ) )
194 IF( K.EQ.1 ) THEN
195 RESLTS( 2 ) = TMP
196 ELSE
197 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
198 END IF
199 90 CONTINUE
200 *
201 RETURN
202 *
203 * End of SPOT05
204 *
205 END