1 SUBROUTINE CPPT05( UPLO, N, NRHS, AP, 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 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 * CPPT05 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 * Hermitian 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 upper or lower triangular part of the
39 * Hermitian matrix A is stored.
40 * = 'U': Upper triangular
41 * = 'L': Lower triangular
42 *
43 * N (input) INTEGER
44 * The number of rows of the matrices X, B, and XACT, and the
45 * order of the matrix A. N >= 0.
46 *
47 * NRHS (input) INTEGER
48 * The number of columns of the matrices X, B, and XACT.
49 * NRHS >= 0.
50 *
51 * AP (input) COMPLEX array, dimension (N*(N+1)/2)
52 * The upper or lower triangle of the Hermitian matrix A, packed
53 * columnwise in a linear array. The j-th column of A is stored
54 * in the array AP as follows:
55 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
56 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
57 *
58 * B (input) COMPLEX array, dimension (LDB,NRHS)
59 * The right hand side vectors for the system of linear
60 * equations.
61 *
62 * LDB (input) INTEGER
63 * The leading dimension of the array B. LDB >= max(1,N).
64 *
65 * X (input) COMPLEX array, dimension (LDX,NRHS)
66 * The computed solution vectors. Each vector is stored as a
67 * column of the matrix X.
68 *
69 * LDX (input) INTEGER
70 * The leading dimension of the array X. LDX >= max(1,N).
71 *
72 * XACT (input) COMPLEX array, dimension (LDX,NRHS)
73 * The exact solution vectors. Each vector is stored as a
74 * column of the matrix XACT.
75 *
76 * LDXACT (input) INTEGER
77 * The leading dimension of the array XACT. LDXACT >= max(1,N).
78 *
79 * FERR (input) REAL array, dimension (NRHS)
80 * The estimated forward error bounds for each solution vector
81 * X. If XTRUE is the true solution, FERR bounds the magnitude
82 * of the largest entry in (X - XTRUE) divided by the magnitude
83 * of the largest entry in X.
84 *
85 * BERR (input) REAL array, dimension (NRHS)
86 * The componentwise relative backward error of each solution
87 * vector (i.e., the smallest relative change in any entry of A
88 * or B that makes X an exact solution).
89 *
90 * RESLTS (output) REAL array, dimension (2)
91 * The maximum over the NRHS solution vectors of the ratios:
92 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
93 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
94 *
95 * =====================================================================
96 *
97 * .. Parameters ..
98 REAL ZERO, ONE
99 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
100 * ..
101 * .. Local Scalars ..
102 LOGICAL UPPER
103 INTEGER I, IMAX, J, JC, K
104 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
105 COMPLEX ZDUM
106 * ..
107 * .. External Functions ..
108 LOGICAL LSAME
109 INTEGER ICAMAX
110 REAL SLAMCH
111 EXTERNAL LSAME, ICAMAX, SLAMCH
112 * ..
113 * .. Intrinsic Functions ..
114 INTRINSIC ABS, AIMAG, MAX, MIN, REAL
115 * ..
116 * .. Statement Functions ..
117 REAL CABS1
118 * ..
119 * .. Statement Function definitions ..
120 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
121 * ..
122 * .. Executable Statements ..
123 *
124 * Quick exit if N = 0 or NRHS = 0.
125 *
126 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
127 RESLTS( 1 ) = ZERO
128 RESLTS( 2 ) = ZERO
129 RETURN
130 END IF
131 *
132 EPS = SLAMCH( 'Epsilon' )
133 UNFL = SLAMCH( 'Safe minimum' )
134 OVFL = ONE / UNFL
135 UPPER = LSAME( UPLO, 'U' )
136 *
137 * Test 1: Compute the maximum of
138 * norm(X - XACT) / ( norm(X) * FERR )
139 * over all the vectors X and XACT using the infinity-norm.
140 *
141 ERRBND = ZERO
142 DO 30 J = 1, NRHS
143 IMAX = ICAMAX( N, X( 1, J ), 1 )
144 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
145 DIFF = ZERO
146 DO 10 I = 1, N
147 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
148 10 CONTINUE
149 *
150 IF( XNORM.GT.ONE ) THEN
151 GO TO 20
152 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
153 GO TO 20
154 ELSE
155 ERRBND = ONE / EPS
156 GO TO 30
157 END IF
158 *
159 20 CONTINUE
160 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
161 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
162 ELSE
163 ERRBND = ONE / EPS
164 END IF
165 30 CONTINUE
166 RESLTS( 1 ) = ERRBND
167 *
168 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
169 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
170 *
171 DO 90 K = 1, NRHS
172 DO 80 I = 1, N
173 TMP = CABS1( B( I, K ) )
174 IF( UPPER ) THEN
175 JC = ( ( I-1 )*I ) / 2
176 DO 40 J = 1, I - 1
177 TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
178 40 CONTINUE
179 TMP = TMP + ABS( REAL( AP( JC+I ) ) )*CABS1( X( I, K ) )
180 JC = JC + I + I
181 DO 50 J = I + 1, N
182 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
183 JC = JC + J
184 50 CONTINUE
185 ELSE
186 JC = I
187 DO 60 J = 1, I - 1
188 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
189 JC = JC + N - J
190 60 CONTINUE
191 TMP = TMP + ABS( REAL( AP( JC ) ) )*CABS1( X( I, K ) )
192 DO 70 J = I + 1, N
193 TMP = TMP + CABS1( AP( JC+J-I ) )*CABS1( X( J, K ) )
194 70 CONTINUE
195 END IF
196 IF( I.EQ.1 ) THEN
197 AXBI = TMP
198 ELSE
199 AXBI = MIN( AXBI, TMP )
200 END IF
201 80 CONTINUE
202 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
203 $ MAX( AXBI, ( N+1 )*UNFL ) )
204 IF( K.EQ.1 ) THEN
205 RESLTS( 2 ) = TMP
206 ELSE
207 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
208 END IF
209 90 CONTINUE
210 *
211 RETURN
212 *
213 * End of CPPT05
214 *
215 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 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 * CPPT05 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 * Hermitian 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 upper or lower triangular part of the
39 * Hermitian matrix A is stored.
40 * = 'U': Upper triangular
41 * = 'L': Lower triangular
42 *
43 * N (input) INTEGER
44 * The number of rows of the matrices X, B, and XACT, and the
45 * order of the matrix A. N >= 0.
46 *
47 * NRHS (input) INTEGER
48 * The number of columns of the matrices X, B, and XACT.
49 * NRHS >= 0.
50 *
51 * AP (input) COMPLEX array, dimension (N*(N+1)/2)
52 * The upper or lower triangle of the Hermitian matrix A, packed
53 * columnwise in a linear array. The j-th column of A is stored
54 * in the array AP as follows:
55 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
56 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
57 *
58 * B (input) COMPLEX array, dimension (LDB,NRHS)
59 * The right hand side vectors for the system of linear
60 * equations.
61 *
62 * LDB (input) INTEGER
63 * The leading dimension of the array B. LDB >= max(1,N).
64 *
65 * X (input) COMPLEX array, dimension (LDX,NRHS)
66 * The computed solution vectors. Each vector is stored as a
67 * column of the matrix X.
68 *
69 * LDX (input) INTEGER
70 * The leading dimension of the array X. LDX >= max(1,N).
71 *
72 * XACT (input) COMPLEX array, dimension (LDX,NRHS)
73 * The exact solution vectors. Each vector is stored as a
74 * column of the matrix XACT.
75 *
76 * LDXACT (input) INTEGER
77 * The leading dimension of the array XACT. LDXACT >= max(1,N).
78 *
79 * FERR (input) REAL array, dimension (NRHS)
80 * The estimated forward error bounds for each solution vector
81 * X. If XTRUE is the true solution, FERR bounds the magnitude
82 * of the largest entry in (X - XTRUE) divided by the magnitude
83 * of the largest entry in X.
84 *
85 * BERR (input) REAL array, dimension (NRHS)
86 * The componentwise relative backward error of each solution
87 * vector (i.e., the smallest relative change in any entry of A
88 * or B that makes X an exact solution).
89 *
90 * RESLTS (output) REAL array, dimension (2)
91 * The maximum over the NRHS solution vectors of the ratios:
92 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
93 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
94 *
95 * =====================================================================
96 *
97 * .. Parameters ..
98 REAL ZERO, ONE
99 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
100 * ..
101 * .. Local Scalars ..
102 LOGICAL UPPER
103 INTEGER I, IMAX, J, JC, K
104 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
105 COMPLEX ZDUM
106 * ..
107 * .. External Functions ..
108 LOGICAL LSAME
109 INTEGER ICAMAX
110 REAL SLAMCH
111 EXTERNAL LSAME, ICAMAX, SLAMCH
112 * ..
113 * .. Intrinsic Functions ..
114 INTRINSIC ABS, AIMAG, MAX, MIN, REAL
115 * ..
116 * .. Statement Functions ..
117 REAL CABS1
118 * ..
119 * .. Statement Function definitions ..
120 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
121 * ..
122 * .. Executable Statements ..
123 *
124 * Quick exit if N = 0 or NRHS = 0.
125 *
126 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
127 RESLTS( 1 ) = ZERO
128 RESLTS( 2 ) = ZERO
129 RETURN
130 END IF
131 *
132 EPS = SLAMCH( 'Epsilon' )
133 UNFL = SLAMCH( 'Safe minimum' )
134 OVFL = ONE / UNFL
135 UPPER = LSAME( UPLO, 'U' )
136 *
137 * Test 1: Compute the maximum of
138 * norm(X - XACT) / ( norm(X) * FERR )
139 * over all the vectors X and XACT using the infinity-norm.
140 *
141 ERRBND = ZERO
142 DO 30 J = 1, NRHS
143 IMAX = ICAMAX( N, X( 1, J ), 1 )
144 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
145 DIFF = ZERO
146 DO 10 I = 1, N
147 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
148 10 CONTINUE
149 *
150 IF( XNORM.GT.ONE ) THEN
151 GO TO 20
152 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
153 GO TO 20
154 ELSE
155 ERRBND = ONE / EPS
156 GO TO 30
157 END IF
158 *
159 20 CONTINUE
160 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
161 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
162 ELSE
163 ERRBND = ONE / EPS
164 END IF
165 30 CONTINUE
166 RESLTS( 1 ) = ERRBND
167 *
168 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
169 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
170 *
171 DO 90 K = 1, NRHS
172 DO 80 I = 1, N
173 TMP = CABS1( B( I, K ) )
174 IF( UPPER ) THEN
175 JC = ( ( I-1 )*I ) / 2
176 DO 40 J = 1, I - 1
177 TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
178 40 CONTINUE
179 TMP = TMP + ABS( REAL( AP( JC+I ) ) )*CABS1( X( I, K ) )
180 JC = JC + I + I
181 DO 50 J = I + 1, N
182 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
183 JC = JC + J
184 50 CONTINUE
185 ELSE
186 JC = I
187 DO 60 J = 1, I - 1
188 TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
189 JC = JC + N - J
190 60 CONTINUE
191 TMP = TMP + ABS( REAL( AP( JC ) ) )*CABS1( X( I, K ) )
192 DO 70 J = I + 1, N
193 TMP = TMP + CABS1( AP( JC+J-I ) )*CABS1( X( J, K ) )
194 70 CONTINUE
195 END IF
196 IF( I.EQ.1 ) THEN
197 AXBI = TMP
198 ELSE
199 AXBI = MIN( AXBI, TMP )
200 END IF
201 80 CONTINUE
202 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
203 $ MAX( AXBI, ( N+1 )*UNFL ) )
204 IF( K.EQ.1 ) THEN
205 RESLTS( 2 ) = TMP
206 ELSE
207 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
208 END IF
209 90 CONTINUE
210 *
211 RETURN
212 *
213 * End of CPPT05
214 *
215 END