1 SUBROUTINE DTRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
2 $ LDX, XACT, 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 DIAG, TRANS, UPLO
10 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
14 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DTRT05 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 * triangular 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 matrix A is upper or lower triangular.
38 * = 'U': Upper triangular
39 * = 'L': Lower triangular
40 *
41 * TRANS (input) CHARACTER*1
42 * Specifies the form of the system of equations.
43 * = 'N': A * X = B (No transpose)
44 * = 'T': A'* X = B (Transpose)
45 * = 'C': A'* X = B (Conjugate transpose = Transpose)
46 *
47 * DIAG (input) CHARACTER*1
48 * Specifies whether or not the matrix A is unit triangular.
49 * = 'N': Non-unit triangular
50 * = 'U': Unit triangular
51 *
52 * N (input) INTEGER
53 * The number of rows of the matrices X, B, and XACT, and the
54 * order of the matrix A. N >= 0.
55 *
56 * NRHS (input) INTEGER
57 * The number of columns of the matrices X, B, and XACT.
58 * NRHS >= 0.
59 *
60 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
61 * The triangular matrix A. If UPLO = 'U', the leading n by n
62 * upper triangular part of the array A contains the upper
63 * triangular matrix, and the strictly lower triangular part of
64 * A is not referenced. If UPLO = 'L', the leading n by n lower
65 * triangular part of the array A contains the lower triangular
66 * matrix, and the strictly upper triangular part of A is not
67 * referenced. If DIAG = 'U', the diagonal elements of A are
68 * also not referenced and are assumed to be 1.
69 *
70 * LDA (input) INTEGER
71 * The leading dimension of the array A. LDA >= max(1,N).
72 *
73 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
74 * The right hand side vectors for the system of linear
75 * equations.
76 *
77 * LDB (input) INTEGER
78 * The leading dimension of the array B. LDB >= max(1,N).
79 *
80 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
81 * The computed solution vectors. Each vector is stored as a
82 * column of the matrix X.
83 *
84 * LDX (input) INTEGER
85 * The leading dimension of the array X. LDX >= max(1,N).
86 *
87 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
88 * The exact solution vectors. Each vector is stored as a
89 * column of the matrix XACT.
90 *
91 * LDXACT (input) INTEGER
92 * The leading dimension of the array XACT. LDXACT >= max(1,N).
93 *
94 * FERR (input) DOUBLE PRECISION array, dimension (NRHS)
95 * The estimated forward error bounds for each solution vector
96 * X. If XTRUE is the true solution, FERR bounds the magnitude
97 * of the largest entry in (X - XTRUE) divided by the magnitude
98 * of the largest entry in X.
99 *
100 * BERR (input) DOUBLE PRECISION array, dimension (NRHS)
101 * The componentwise relative backward error of each solution
102 * vector (i.e., the smallest relative change in any entry of A
103 * or B that makes X an exact solution).
104 *
105 * RESLTS (output) DOUBLE PRECISION array, dimension (2)
106 * The maximum over the NRHS solution vectors of the ratios:
107 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
108 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
109 *
110 * =====================================================================
111 *
112 * .. Parameters ..
113 DOUBLE PRECISION ZERO, ONE
114 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
115 * ..
116 * .. Local Scalars ..
117 LOGICAL NOTRAN, UNIT, UPPER
118 INTEGER I, IFU, IMAX, J, K
119 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
120 * ..
121 * .. External Functions ..
122 LOGICAL LSAME
123 INTEGER IDAMAX
124 DOUBLE PRECISION DLAMCH
125 EXTERNAL LSAME, IDAMAX, DLAMCH
126 * ..
127 * .. Intrinsic Functions ..
128 INTRINSIC ABS, MAX, MIN
129 * ..
130 * .. Executable Statements ..
131 *
132 * Quick exit if N = 0 or NRHS = 0.
133 *
134 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
135 RESLTS( 1 ) = ZERO
136 RESLTS( 2 ) = ZERO
137 RETURN
138 END IF
139 *
140 EPS = DLAMCH( 'Epsilon' )
141 UNFL = DLAMCH( 'Safe minimum' )
142 OVFL = ONE / UNFL
143 UPPER = LSAME( UPLO, 'U' )
144 NOTRAN = LSAME( TRANS, 'N' )
145 UNIT = LSAME( DIAG, 'U' )
146 *
147 * Test 1: Compute the maximum of
148 * norm(X - XACT) / ( norm(X) * FERR )
149 * over all the vectors X and XACT using the infinity-norm.
150 *
151 ERRBND = ZERO
152 DO 30 J = 1, NRHS
153 IMAX = IDAMAX( N, X( 1, J ), 1 )
154 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
155 DIFF = ZERO
156 DO 10 I = 1, N
157 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
158 10 CONTINUE
159 *
160 IF( XNORM.GT.ONE ) THEN
161 GO TO 20
162 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
163 GO TO 20
164 ELSE
165 ERRBND = ONE / EPS
166 GO TO 30
167 END IF
168 *
169 20 CONTINUE
170 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
171 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
172 ELSE
173 ERRBND = ONE / EPS
174 END IF
175 30 CONTINUE
176 RESLTS( 1 ) = ERRBND
177 *
178 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
179 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
180 *
181 IFU = 0
182 IF( UNIT )
183 $ IFU = 1
184 DO 90 K = 1, NRHS
185 DO 80 I = 1, N
186 TMP = ABS( B( I, K ) )
187 IF( UPPER ) THEN
188 IF( .NOT.NOTRAN ) THEN
189 DO 40 J = 1, I - IFU
190 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
191 40 CONTINUE
192 IF( UNIT )
193 $ TMP = TMP + ABS( X( I, K ) )
194 ELSE
195 IF( UNIT )
196 $ TMP = TMP + ABS( X( I, K ) )
197 DO 50 J = I + IFU, N
198 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
199 50 CONTINUE
200 END IF
201 ELSE
202 IF( NOTRAN ) THEN
203 DO 60 J = 1, I - IFU
204 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
205 60 CONTINUE
206 IF( UNIT )
207 $ TMP = TMP + ABS( X( I, K ) )
208 ELSE
209 IF( UNIT )
210 $ TMP = TMP + ABS( X( I, K ) )
211 DO 70 J = I + IFU, N
212 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
213 70 CONTINUE
214 END IF
215 END IF
216 IF( I.EQ.1 ) THEN
217 AXBI = TMP
218 ELSE
219 AXBI = MIN( AXBI, TMP )
220 END IF
221 80 CONTINUE
222 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
223 $ MAX( AXBI, ( N+1 )*UNFL ) )
224 IF( K.EQ.1 ) THEN
225 RESLTS( 2 ) = TMP
226 ELSE
227 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
228 END IF
229 90 CONTINUE
230 *
231 RETURN
232 *
233 * End of DTRT05
234 *
235 END
2 $ LDX, XACT, 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 DIAG, TRANS, UPLO
10 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
14 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DTRT05 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 * triangular 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 matrix A is upper or lower triangular.
38 * = 'U': Upper triangular
39 * = 'L': Lower triangular
40 *
41 * TRANS (input) CHARACTER*1
42 * Specifies the form of the system of equations.
43 * = 'N': A * X = B (No transpose)
44 * = 'T': A'* X = B (Transpose)
45 * = 'C': A'* X = B (Conjugate transpose = Transpose)
46 *
47 * DIAG (input) CHARACTER*1
48 * Specifies whether or not the matrix A is unit triangular.
49 * = 'N': Non-unit triangular
50 * = 'U': Unit triangular
51 *
52 * N (input) INTEGER
53 * The number of rows of the matrices X, B, and XACT, and the
54 * order of the matrix A. N >= 0.
55 *
56 * NRHS (input) INTEGER
57 * The number of columns of the matrices X, B, and XACT.
58 * NRHS >= 0.
59 *
60 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
61 * The triangular matrix A. If UPLO = 'U', the leading n by n
62 * upper triangular part of the array A contains the upper
63 * triangular matrix, and the strictly lower triangular part of
64 * A is not referenced. If UPLO = 'L', the leading n by n lower
65 * triangular part of the array A contains the lower triangular
66 * matrix, and the strictly upper triangular part of A is not
67 * referenced. If DIAG = 'U', the diagonal elements of A are
68 * also not referenced and are assumed to be 1.
69 *
70 * LDA (input) INTEGER
71 * The leading dimension of the array A. LDA >= max(1,N).
72 *
73 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
74 * The right hand side vectors for the system of linear
75 * equations.
76 *
77 * LDB (input) INTEGER
78 * The leading dimension of the array B. LDB >= max(1,N).
79 *
80 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
81 * The computed solution vectors. Each vector is stored as a
82 * column of the matrix X.
83 *
84 * LDX (input) INTEGER
85 * The leading dimension of the array X. LDX >= max(1,N).
86 *
87 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
88 * The exact solution vectors. Each vector is stored as a
89 * column of the matrix XACT.
90 *
91 * LDXACT (input) INTEGER
92 * The leading dimension of the array XACT. LDXACT >= max(1,N).
93 *
94 * FERR (input) DOUBLE PRECISION array, dimension (NRHS)
95 * The estimated forward error bounds for each solution vector
96 * X. If XTRUE is the true solution, FERR bounds the magnitude
97 * of the largest entry in (X - XTRUE) divided by the magnitude
98 * of the largest entry in X.
99 *
100 * BERR (input) DOUBLE PRECISION array, dimension (NRHS)
101 * The componentwise relative backward error of each solution
102 * vector (i.e., the smallest relative change in any entry of A
103 * or B that makes X an exact solution).
104 *
105 * RESLTS (output) DOUBLE PRECISION array, dimension (2)
106 * The maximum over the NRHS solution vectors of the ratios:
107 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
108 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
109 *
110 * =====================================================================
111 *
112 * .. Parameters ..
113 DOUBLE PRECISION ZERO, ONE
114 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
115 * ..
116 * .. Local Scalars ..
117 LOGICAL NOTRAN, UNIT, UPPER
118 INTEGER I, IFU, IMAX, J, K
119 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
120 * ..
121 * .. External Functions ..
122 LOGICAL LSAME
123 INTEGER IDAMAX
124 DOUBLE PRECISION DLAMCH
125 EXTERNAL LSAME, IDAMAX, DLAMCH
126 * ..
127 * .. Intrinsic Functions ..
128 INTRINSIC ABS, MAX, MIN
129 * ..
130 * .. Executable Statements ..
131 *
132 * Quick exit if N = 0 or NRHS = 0.
133 *
134 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
135 RESLTS( 1 ) = ZERO
136 RESLTS( 2 ) = ZERO
137 RETURN
138 END IF
139 *
140 EPS = DLAMCH( 'Epsilon' )
141 UNFL = DLAMCH( 'Safe minimum' )
142 OVFL = ONE / UNFL
143 UPPER = LSAME( UPLO, 'U' )
144 NOTRAN = LSAME( TRANS, 'N' )
145 UNIT = LSAME( DIAG, 'U' )
146 *
147 * Test 1: Compute the maximum of
148 * norm(X - XACT) / ( norm(X) * FERR )
149 * over all the vectors X and XACT using the infinity-norm.
150 *
151 ERRBND = ZERO
152 DO 30 J = 1, NRHS
153 IMAX = IDAMAX( N, X( 1, J ), 1 )
154 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
155 DIFF = ZERO
156 DO 10 I = 1, N
157 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
158 10 CONTINUE
159 *
160 IF( XNORM.GT.ONE ) THEN
161 GO TO 20
162 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
163 GO TO 20
164 ELSE
165 ERRBND = ONE / EPS
166 GO TO 30
167 END IF
168 *
169 20 CONTINUE
170 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
171 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
172 ELSE
173 ERRBND = ONE / EPS
174 END IF
175 30 CONTINUE
176 RESLTS( 1 ) = ERRBND
177 *
178 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
179 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
180 *
181 IFU = 0
182 IF( UNIT )
183 $ IFU = 1
184 DO 90 K = 1, NRHS
185 DO 80 I = 1, N
186 TMP = ABS( B( I, K ) )
187 IF( UPPER ) THEN
188 IF( .NOT.NOTRAN ) THEN
189 DO 40 J = 1, I - IFU
190 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
191 40 CONTINUE
192 IF( UNIT )
193 $ TMP = TMP + ABS( X( I, K ) )
194 ELSE
195 IF( UNIT )
196 $ TMP = TMP + ABS( X( I, K ) )
197 DO 50 J = I + IFU, N
198 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
199 50 CONTINUE
200 END IF
201 ELSE
202 IF( NOTRAN ) THEN
203 DO 60 J = 1, I - IFU
204 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
205 60 CONTINUE
206 IF( UNIT )
207 $ TMP = TMP + ABS( X( I, K ) )
208 ELSE
209 IF( UNIT )
210 $ TMP = TMP + ABS( X( I, K ) )
211 DO 70 J = I + IFU, N
212 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
213 70 CONTINUE
214 END IF
215 END IF
216 IF( I.EQ.1 ) THEN
217 AXBI = TMP
218 ELSE
219 AXBI = MIN( AXBI, TMP )
220 END IF
221 80 CONTINUE
222 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
223 $ MAX( AXBI, ( N+1 )*UNFL ) )
224 IF( K.EQ.1 ) THEN
225 RESLTS( 2 ) = TMP
226 ELSE
227 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
228 END IF
229 90 CONTINUE
230 *
231 RETURN
232 *
233 * End of DTRT05
234 *
235 END