1 SUBROUTINE DPTT05( N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT,
2 $ 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 INTEGER LDB, LDX, LDXACT, N, NRHS
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), E( * ),
13 $ FERR( * ), RESLTS( * ), X( LDX, * ),
14 $ XACT( LDXACT, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DPTT05 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 tridiagonal matrix of order n.
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 / ( NZ*EPS + (*) ), where
31 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
32 * and NZ = max. number of nonzeros in any row of A, plus 1
33 *
34 * Arguments
35 * =========
36 *
37 * N (input) INTEGER
38 * The number of rows of the matrices X, B, and XACT, and the
39 * order of the matrix A. N >= 0.
40 *
41 * NRHS (input) INTEGER
42 * The number of columns of the matrices X, B, and XACT.
43 * NRHS >= 0.
44 *
45 * D (input) DOUBLE PRECISION array, dimension (N)
46 * The n diagonal elements of the tridiagonal matrix A.
47 *
48 * E (input) DOUBLE PRECISION array, dimension (N-1)
49 * The (n-1) subdiagonal elements of the tridiagonal matrix A.
50 *
51 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
52 * The right hand side vectors for the system of linear
53 * equations.
54 *
55 * LDB (input) INTEGER
56 * The leading dimension of the array B. LDB >= max(1,N).
57 *
58 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
59 * The computed solution vectors. Each vector is stored as a
60 * column of the matrix X.
61 *
62 * LDX (input) INTEGER
63 * The leading dimension of the array X. LDX >= max(1,N).
64 *
65 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
66 * The exact solution vectors. Each vector is stored as a
67 * column of the matrix XACT.
68 *
69 * LDXACT (input) INTEGER
70 * The leading dimension of the array XACT. LDXACT >= max(1,N).
71 *
72 * FERR (input) DOUBLE PRECISION array, dimension (NRHS)
73 * The estimated forward error bounds for each solution vector
74 * X. If XTRUE is the true solution, FERR bounds the magnitude
75 * of the largest entry in (X - XTRUE) divided by the magnitude
76 * of the largest entry in X.
77 *
78 * BERR (input) DOUBLE PRECISION array, dimension (NRHS)
79 * The componentwise relative backward error of each solution
80 * vector (i.e., the smallest relative change in any entry of A
81 * or B that makes X an exact solution).
82 *
83 * RESLTS (output) DOUBLE PRECISION array, dimension (2)
84 * The maximum over the NRHS solution vectors of the ratios:
85 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
86 * RESLTS(2) = BERR / ( NZ*EPS + (*) )
87 *
88 * =====================================================================
89 *
90 * .. Parameters ..
91 DOUBLE PRECISION ZERO, ONE
92 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
93 * ..
94 * .. Local Scalars ..
95 INTEGER I, IMAX, J, K, NZ
96 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
97 * ..
98 * .. External Functions ..
99 INTEGER IDAMAX
100 DOUBLE PRECISION DLAMCH
101 EXTERNAL IDAMAX, DLAMCH
102 * ..
103 * .. Intrinsic Functions ..
104 INTRINSIC ABS, MAX, MIN
105 * ..
106 * .. Executable Statements ..
107 *
108 * Quick exit if N = 0 or NRHS = 0.
109 *
110 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
111 RESLTS( 1 ) = ZERO
112 RESLTS( 2 ) = ZERO
113 RETURN
114 END IF
115 *
116 EPS = DLAMCH( 'Epsilon' )
117 UNFL = DLAMCH( 'Safe minimum' )
118 OVFL = ONE / UNFL
119 NZ = 4
120 *
121 * Test 1: Compute the maximum of
122 * norm(X - XACT) / ( norm(X) * FERR )
123 * over all the vectors X and XACT using the infinity-norm.
124 *
125 ERRBND = ZERO
126 DO 30 J = 1, NRHS
127 IMAX = IDAMAX( N, X( 1, J ), 1 )
128 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
129 DIFF = ZERO
130 DO 10 I = 1, N
131 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
132 10 CONTINUE
133 *
134 IF( XNORM.GT.ONE ) THEN
135 GO TO 20
136 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
137 GO TO 20
138 ELSE
139 ERRBND = ONE / EPS
140 GO TO 30
141 END IF
142 *
143 20 CONTINUE
144 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
145 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
146 ELSE
147 ERRBND = ONE / EPS
148 END IF
149 30 CONTINUE
150 RESLTS( 1 ) = ERRBND
151 *
152 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
153 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
154 *
155 DO 50 K = 1, NRHS
156 IF( N.EQ.1 ) THEN
157 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
158 ELSE
159 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
160 $ ABS( E( 1 )*X( 2, K ) )
161 DO 40 I = 2, N - 1
162 TMP = ABS( B( I, K ) ) + ABS( E( I-1 )*X( I-1, K ) ) +
163 $ ABS( D( I )*X( I, K ) ) + ABS( E( I )*X( I+1, K ) )
164 AXBI = MIN( AXBI, TMP )
165 40 CONTINUE
166 TMP = ABS( B( N, K ) ) + ABS( E( N-1 )*X( N-1, K ) ) +
167 $ ABS( D( N )*X( N, K ) )
168 AXBI = MIN( AXBI, TMP )
169 END IF
170 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
171 IF( K.EQ.1 ) THEN
172 RESLTS( 2 ) = TMP
173 ELSE
174 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
175 END IF
176 50 CONTINUE
177 *
178 RETURN
179 *
180 * End of DPTT05
181 *
182 END
2 $ 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 INTEGER LDB, LDX, LDXACT, N, NRHS
10 * ..
11 * .. Array Arguments ..
12 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), E( * ),
13 $ FERR( * ), RESLTS( * ), X( LDX, * ),
14 $ XACT( LDXACT, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * DPTT05 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 tridiagonal matrix of order n.
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 / ( NZ*EPS + (*) ), where
31 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
32 * and NZ = max. number of nonzeros in any row of A, plus 1
33 *
34 * Arguments
35 * =========
36 *
37 * N (input) INTEGER
38 * The number of rows of the matrices X, B, and XACT, and the
39 * order of the matrix A. N >= 0.
40 *
41 * NRHS (input) INTEGER
42 * The number of columns of the matrices X, B, and XACT.
43 * NRHS >= 0.
44 *
45 * D (input) DOUBLE PRECISION array, dimension (N)
46 * The n diagonal elements of the tridiagonal matrix A.
47 *
48 * E (input) DOUBLE PRECISION array, dimension (N-1)
49 * The (n-1) subdiagonal elements of the tridiagonal matrix A.
50 *
51 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
52 * The right hand side vectors for the system of linear
53 * equations.
54 *
55 * LDB (input) INTEGER
56 * The leading dimension of the array B. LDB >= max(1,N).
57 *
58 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
59 * The computed solution vectors. Each vector is stored as a
60 * column of the matrix X.
61 *
62 * LDX (input) INTEGER
63 * The leading dimension of the array X. LDX >= max(1,N).
64 *
65 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
66 * The exact solution vectors. Each vector is stored as a
67 * column of the matrix XACT.
68 *
69 * LDXACT (input) INTEGER
70 * The leading dimension of the array XACT. LDXACT >= max(1,N).
71 *
72 * FERR (input) DOUBLE PRECISION array, dimension (NRHS)
73 * The estimated forward error bounds for each solution vector
74 * X. If XTRUE is the true solution, FERR bounds the magnitude
75 * of the largest entry in (X - XTRUE) divided by the magnitude
76 * of the largest entry in X.
77 *
78 * BERR (input) DOUBLE PRECISION array, dimension (NRHS)
79 * The componentwise relative backward error of each solution
80 * vector (i.e., the smallest relative change in any entry of A
81 * or B that makes X an exact solution).
82 *
83 * RESLTS (output) DOUBLE PRECISION array, dimension (2)
84 * The maximum over the NRHS solution vectors of the ratios:
85 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
86 * RESLTS(2) = BERR / ( NZ*EPS + (*) )
87 *
88 * =====================================================================
89 *
90 * .. Parameters ..
91 DOUBLE PRECISION ZERO, ONE
92 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
93 * ..
94 * .. Local Scalars ..
95 INTEGER I, IMAX, J, K, NZ
96 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
97 * ..
98 * .. External Functions ..
99 INTEGER IDAMAX
100 DOUBLE PRECISION DLAMCH
101 EXTERNAL IDAMAX, DLAMCH
102 * ..
103 * .. Intrinsic Functions ..
104 INTRINSIC ABS, MAX, MIN
105 * ..
106 * .. Executable Statements ..
107 *
108 * Quick exit if N = 0 or NRHS = 0.
109 *
110 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
111 RESLTS( 1 ) = ZERO
112 RESLTS( 2 ) = ZERO
113 RETURN
114 END IF
115 *
116 EPS = DLAMCH( 'Epsilon' )
117 UNFL = DLAMCH( 'Safe minimum' )
118 OVFL = ONE / UNFL
119 NZ = 4
120 *
121 * Test 1: Compute the maximum of
122 * norm(X - XACT) / ( norm(X) * FERR )
123 * over all the vectors X and XACT using the infinity-norm.
124 *
125 ERRBND = ZERO
126 DO 30 J = 1, NRHS
127 IMAX = IDAMAX( N, X( 1, J ), 1 )
128 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
129 DIFF = ZERO
130 DO 10 I = 1, N
131 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
132 10 CONTINUE
133 *
134 IF( XNORM.GT.ONE ) THEN
135 GO TO 20
136 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
137 GO TO 20
138 ELSE
139 ERRBND = ONE / EPS
140 GO TO 30
141 END IF
142 *
143 20 CONTINUE
144 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
145 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
146 ELSE
147 ERRBND = ONE / EPS
148 END IF
149 30 CONTINUE
150 RESLTS( 1 ) = ERRBND
151 *
152 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
153 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
154 *
155 DO 50 K = 1, NRHS
156 IF( N.EQ.1 ) THEN
157 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
158 ELSE
159 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
160 $ ABS( E( 1 )*X( 2, K ) )
161 DO 40 I = 2, N - 1
162 TMP = ABS( B( I, K ) ) + ABS( E( I-1 )*X( I-1, K ) ) +
163 $ ABS( D( I )*X( I, K ) ) + ABS( E( I )*X( I+1, K ) )
164 AXBI = MIN( AXBI, TMP )
165 40 CONTINUE
166 TMP = ABS( B( N, K ) ) + ABS( E( N-1 )*X( N-1, K ) ) +
167 $ ABS( D( N )*X( N, K ) )
168 AXBI = MIN( AXBI, TMP )
169 END IF
170 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
171 IF( K.EQ.1 ) THEN
172 RESLTS( 2 ) = TMP
173 ELSE
174 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
175 END IF
176 50 CONTINUE
177 *
178 RETURN
179 *
180 * End of DPTT05
181 *
182 END