1 SUBROUTINE CTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
2 $ 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 LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 REAL BERR( * ), FERR( * ), RESLTS( * )
14 COMPLEX AP( * ), B( LDB, * ), X( LDX, * ),
15 $ XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * CTPT05 tests the error bounds from iterative refinement for the
22 * computed solution to a system of equations A*X = B, where A is a
23 * triangular matrix in packed storage format.
24 *
25 * RESLTS(1) = test of the error bound
26 * = norm(X - XACT) / ( norm(X) * FERR )
27 *
28 * A large value is returned if this ratio is not less than one.
29 *
30 * RESLTS(2) = residual from the iterative refinement routine
31 * = the maximum of BERR / ( (n+1)*EPS + (*) ), where
32 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
33 *
34 * Arguments
35 * =========
36 *
37 * UPLO (input) CHARACTER*1
38 * Specifies whether the matrix A is upper or lower triangular.
39 * = 'U': Upper triangular
40 * = 'L': Lower triangular
41 *
42 * TRANS (input) CHARACTER*1
43 * Specifies the form of the system of equations.
44 * = 'N': A * X = B (No transpose)
45 * = 'T': A'* X = B (Transpose)
46 * = 'C': A'* X = B (Conjugate transpose = Transpose)
47 *
48 * DIAG (input) CHARACTER*1
49 * Specifies whether or not the matrix A is unit triangular.
50 * = 'N': Non-unit triangular
51 * = 'U': Unit triangular
52 *
53 * N (input) INTEGER
54 * The number of rows of the matrices X, B, and XACT, and the
55 * order of the matrix A. N >= 0.
56 *
57 * NRHS (input) INTEGER
58 * The number of columns of the matrices X, B, and XACT.
59 * NRHS >= 0.
60 *
61 * AP (input) COMPLEX array, dimension (N*(N+1)/2)
62 * The upper or lower triangular matrix A, packed columnwise in
63 * a linear array. The j-th column of A is stored in the array
64 * AP as follows:
65 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
66 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
67 * If DIAG = 'U', the diagonal elements of A are not referenced
68 * and are assumed to be 1.
69 *
70 * B (input) COMPLEX array, dimension (LDB,NRHS)
71 * The right hand side vectors for the system of linear
72 * equations.
73 *
74 * LDB (input) INTEGER
75 * The leading dimension of the array B. LDB >= max(1,N).
76 *
77 * X (input) COMPLEX array, dimension (LDX,NRHS)
78 * The computed solution vectors. Each vector is stored as a
79 * column of the matrix X.
80 *
81 * LDX (input) INTEGER
82 * The leading dimension of the array X. LDX >= max(1,N).
83 *
84 * XACT (input) COMPLEX array, dimension (LDX,NRHS)
85 * The exact solution vectors. Each vector is stored as a
86 * column of the matrix XACT.
87 *
88 * LDXACT (input) INTEGER
89 * The leading dimension of the array XACT. LDXACT >= max(1,N).
90 *
91 * FERR (input) REAL array, dimension (NRHS)
92 * The estimated forward error bounds for each solution vector
93 * X. If XTRUE is the true solution, FERR bounds the magnitude
94 * of the largest entry in (X - XTRUE) divided by the magnitude
95 * of the largest entry in X.
96 *
97 * BERR (input) REAL array, dimension (NRHS)
98 * The componentwise relative backward error of each solution
99 * vector (i.e., the smallest relative change in any entry of A
100 * or B that makes X an exact solution).
101 *
102 * RESLTS (output) REAL array, dimension (2)
103 * The maximum over the NRHS solution vectors of the ratios:
104 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
105 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
106 *
107 * =====================================================================
108 *
109 * .. Parameters ..
110 REAL ZERO, ONE
111 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
112 * ..
113 * .. Local Scalars ..
114 LOGICAL NOTRAN, UNIT, UPPER
115 INTEGER I, IFU, IMAX, J, JC, K
116 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
117 COMPLEX ZDUM
118 * ..
119 * .. External Functions ..
120 LOGICAL LSAME
121 INTEGER ICAMAX
122 REAL SLAMCH
123 EXTERNAL LSAME, ICAMAX, SLAMCH
124 * ..
125 * .. Intrinsic Functions ..
126 INTRINSIC ABS, AIMAG, MAX, MIN, REAL
127 * ..
128 * .. Statement Functions ..
129 REAL CABS1
130 * ..
131 * .. Statement Function definitions ..
132 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
133 * ..
134 * .. Executable Statements ..
135 *
136 * Quick exit if N = 0 or NRHS = 0.
137 *
138 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
139 RESLTS( 1 ) = ZERO
140 RESLTS( 2 ) = ZERO
141 RETURN
142 END IF
143 *
144 EPS = SLAMCH( 'Epsilon' )
145 UNFL = SLAMCH( 'Safe minimum' )
146 OVFL = ONE / UNFL
147 UPPER = LSAME( UPLO, 'U' )
148 NOTRAN = LSAME( TRANS, 'N' )
149 UNIT = LSAME( DIAG, 'U' )
150 *
151 * Test 1: Compute the maximum of
152 * norm(X - XACT) / ( norm(X) * FERR )
153 * over all the vectors X and XACT using the infinity-norm.
154 *
155 ERRBND = ZERO
156 DO 30 J = 1, NRHS
157 IMAX = ICAMAX( N, X( 1, J ), 1 )
158 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
159 DIFF = ZERO
160 DO 10 I = 1, N
161 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
162 10 CONTINUE
163 *
164 IF( XNORM.GT.ONE ) THEN
165 GO TO 20
166 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
167 GO TO 20
168 ELSE
169 ERRBND = ONE / EPS
170 GO TO 30
171 END IF
172 *
173 20 CONTINUE
174 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
175 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
176 ELSE
177 ERRBND = ONE / EPS
178 END IF
179 30 CONTINUE
180 RESLTS( 1 ) = ERRBND
181 *
182 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
183 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
184 *
185 IFU = 0
186 IF( UNIT )
187 $ IFU = 1
188 DO 90 K = 1, NRHS
189 DO 80 I = 1, N
190 TMP = CABS1( B( I, K ) )
191 IF( UPPER ) THEN
192 JC = ( ( I-1 )*I ) / 2
193 IF( .NOT.NOTRAN ) THEN
194 DO 40 J = 1, I - IFU
195 TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
196 40 CONTINUE
197 IF( UNIT )
198 $ TMP = TMP + CABS1( X( I, K ) )
199 ELSE
200 JC = JC + I
201 IF( UNIT ) THEN
202 TMP = TMP + CABS1( X( I, K ) )
203 JC = JC + I
204 END IF
205 DO 50 J = I + IFU, N
206 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
207 JC = JC + J
208 50 CONTINUE
209 END IF
210 ELSE
211 IF( NOTRAN ) THEN
212 JC = I
213 DO 60 J = 1, I - IFU
214 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
215 JC = JC + N - J
216 60 CONTINUE
217 IF( UNIT )
218 $ TMP = TMP + CABS1( X( I, K ) )
219 ELSE
220 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2
221 IF( UNIT )
222 $ TMP = TMP + CABS1( X( I, K ) )
223 DO 70 J = I + IFU, N
224 TMP = TMP + CABS1( AP( JC+J-I ) )*
225 $ CABS1( X( J, K ) )
226 70 CONTINUE
227 END IF
228 END IF
229 IF( I.EQ.1 ) THEN
230 AXBI = TMP
231 ELSE
232 AXBI = MIN( AXBI, TMP )
233 END IF
234 80 CONTINUE
235 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
236 $ MAX( AXBI, ( N+1 )*UNFL ) )
237 IF( K.EQ.1 ) THEN
238 RESLTS( 2 ) = TMP
239 ELSE
240 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
241 END IF
242 90 CONTINUE
243 *
244 RETURN
245 *
246 * End of CTPT05
247 *
248 END
2 $ 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 LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 REAL BERR( * ), FERR( * ), RESLTS( * )
14 COMPLEX AP( * ), B( LDB, * ), X( LDX, * ),
15 $ XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * CTPT05 tests the error bounds from iterative refinement for the
22 * computed solution to a system of equations A*X = B, where A is a
23 * triangular matrix in packed storage format.
24 *
25 * RESLTS(1) = test of the error bound
26 * = norm(X - XACT) / ( norm(X) * FERR )
27 *
28 * A large value is returned if this ratio is not less than one.
29 *
30 * RESLTS(2) = residual from the iterative refinement routine
31 * = the maximum of BERR / ( (n+1)*EPS + (*) ), where
32 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
33 *
34 * Arguments
35 * =========
36 *
37 * UPLO (input) CHARACTER*1
38 * Specifies whether the matrix A is upper or lower triangular.
39 * = 'U': Upper triangular
40 * = 'L': Lower triangular
41 *
42 * TRANS (input) CHARACTER*1
43 * Specifies the form of the system of equations.
44 * = 'N': A * X = B (No transpose)
45 * = 'T': A'* X = B (Transpose)
46 * = 'C': A'* X = B (Conjugate transpose = Transpose)
47 *
48 * DIAG (input) CHARACTER*1
49 * Specifies whether or not the matrix A is unit triangular.
50 * = 'N': Non-unit triangular
51 * = 'U': Unit triangular
52 *
53 * N (input) INTEGER
54 * The number of rows of the matrices X, B, and XACT, and the
55 * order of the matrix A. N >= 0.
56 *
57 * NRHS (input) INTEGER
58 * The number of columns of the matrices X, B, and XACT.
59 * NRHS >= 0.
60 *
61 * AP (input) COMPLEX array, dimension (N*(N+1)/2)
62 * The upper or lower triangular matrix A, packed columnwise in
63 * a linear array. The j-th column of A is stored in the array
64 * AP as follows:
65 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
66 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
67 * If DIAG = 'U', the diagonal elements of A are not referenced
68 * and are assumed to be 1.
69 *
70 * B (input) COMPLEX array, dimension (LDB,NRHS)
71 * The right hand side vectors for the system of linear
72 * equations.
73 *
74 * LDB (input) INTEGER
75 * The leading dimension of the array B. LDB >= max(1,N).
76 *
77 * X (input) COMPLEX array, dimension (LDX,NRHS)
78 * The computed solution vectors. Each vector is stored as a
79 * column of the matrix X.
80 *
81 * LDX (input) INTEGER
82 * The leading dimension of the array X. LDX >= max(1,N).
83 *
84 * XACT (input) COMPLEX array, dimension (LDX,NRHS)
85 * The exact solution vectors. Each vector is stored as a
86 * column of the matrix XACT.
87 *
88 * LDXACT (input) INTEGER
89 * The leading dimension of the array XACT. LDXACT >= max(1,N).
90 *
91 * FERR (input) REAL array, dimension (NRHS)
92 * The estimated forward error bounds for each solution vector
93 * X. If XTRUE is the true solution, FERR bounds the magnitude
94 * of the largest entry in (X - XTRUE) divided by the magnitude
95 * of the largest entry in X.
96 *
97 * BERR (input) REAL array, dimension (NRHS)
98 * The componentwise relative backward error of each solution
99 * vector (i.e., the smallest relative change in any entry of A
100 * or B that makes X an exact solution).
101 *
102 * RESLTS (output) REAL array, dimension (2)
103 * The maximum over the NRHS solution vectors of the ratios:
104 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
105 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
106 *
107 * =====================================================================
108 *
109 * .. Parameters ..
110 REAL ZERO, ONE
111 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
112 * ..
113 * .. Local Scalars ..
114 LOGICAL NOTRAN, UNIT, UPPER
115 INTEGER I, IFU, IMAX, J, JC, K
116 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
117 COMPLEX ZDUM
118 * ..
119 * .. External Functions ..
120 LOGICAL LSAME
121 INTEGER ICAMAX
122 REAL SLAMCH
123 EXTERNAL LSAME, ICAMAX, SLAMCH
124 * ..
125 * .. Intrinsic Functions ..
126 INTRINSIC ABS, AIMAG, MAX, MIN, REAL
127 * ..
128 * .. Statement Functions ..
129 REAL CABS1
130 * ..
131 * .. Statement Function definitions ..
132 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
133 * ..
134 * .. Executable Statements ..
135 *
136 * Quick exit if N = 0 or NRHS = 0.
137 *
138 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
139 RESLTS( 1 ) = ZERO
140 RESLTS( 2 ) = ZERO
141 RETURN
142 END IF
143 *
144 EPS = SLAMCH( 'Epsilon' )
145 UNFL = SLAMCH( 'Safe minimum' )
146 OVFL = ONE / UNFL
147 UPPER = LSAME( UPLO, 'U' )
148 NOTRAN = LSAME( TRANS, 'N' )
149 UNIT = LSAME( DIAG, 'U' )
150 *
151 * Test 1: Compute the maximum of
152 * norm(X - XACT) / ( norm(X) * FERR )
153 * over all the vectors X and XACT using the infinity-norm.
154 *
155 ERRBND = ZERO
156 DO 30 J = 1, NRHS
157 IMAX = ICAMAX( N, X( 1, J ), 1 )
158 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
159 DIFF = ZERO
160 DO 10 I = 1, N
161 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
162 10 CONTINUE
163 *
164 IF( XNORM.GT.ONE ) THEN
165 GO TO 20
166 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
167 GO TO 20
168 ELSE
169 ERRBND = ONE / EPS
170 GO TO 30
171 END IF
172 *
173 20 CONTINUE
174 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
175 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
176 ELSE
177 ERRBND = ONE / EPS
178 END IF
179 30 CONTINUE
180 RESLTS( 1 ) = ERRBND
181 *
182 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
183 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
184 *
185 IFU = 0
186 IF( UNIT )
187 $ IFU = 1
188 DO 90 K = 1, NRHS
189 DO 80 I = 1, N
190 TMP = CABS1( B( I, K ) )
191 IF( UPPER ) THEN
192 JC = ( ( I-1 )*I ) / 2
193 IF( .NOT.NOTRAN ) THEN
194 DO 40 J = 1, I - IFU
195 TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
196 40 CONTINUE
197 IF( UNIT )
198 $ TMP = TMP + CABS1( X( I, K ) )
199 ELSE
200 JC = JC + I
201 IF( UNIT ) THEN
202 TMP = TMP + CABS1( X( I, K ) )
203 JC = JC + I
204 END IF
205 DO 50 J = I + IFU, N
206 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
207 JC = JC + J
208 50 CONTINUE
209 END IF
210 ELSE
211 IF( NOTRAN ) THEN
212 JC = I
213 DO 60 J = 1, I - IFU
214 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
215 JC = JC + N - J
216 60 CONTINUE
217 IF( UNIT )
218 $ TMP = TMP + CABS1( X( I, K ) )
219 ELSE
220 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2
221 IF( UNIT )
222 $ TMP = TMP + CABS1( X( I, K ) )
223 DO 70 J = I + IFU, N
224 TMP = TMP + CABS1( AP( JC+J-I ) )*
225 $ CABS1( X( J, K ) )
226 70 CONTINUE
227 END IF
228 END IF
229 IF( I.EQ.1 ) THEN
230 AXBI = TMP
231 ELSE
232 AXBI = MIN( AXBI, TMP )
233 END IF
234 80 CONTINUE
235 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
236 $ MAX( AXBI, ( N+1 )*UNFL ) )
237 IF( K.EQ.1 ) THEN
238 RESLTS( 2 ) = TMP
239 ELSE
240 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
241 END IF
242 90 CONTINUE
243 *
244 RETURN
245 *
246 * End of CTPT05
247 *
248 END