1 SUBROUTINE DGTT05( TRANS, N, NRHS, DL, D, DU, 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 TRANS
10 INTEGER LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DL( * ),
14 $ DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ),
15 $ XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DGTT05 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 * general tridiagonal matrix of order n and op(A) = A or A**T,
24 * depending on TRANS.
25 *
26 * RESLTS(1) = test of the error bound
27 * = norm(X - XACT) / ( norm(X) * FERR )
28 *
29 * A large value is returned if this ratio is not less than one.
30 *
31 * RESLTS(2) = residual from the iterative refinement routine
32 * = the maximum of BERR / ( NZ*EPS + (*) ), where
33 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
34 * and NZ = max. number of nonzeros in any row of A, plus 1
35 *
36 * Arguments
37 * =========
38 *
39 * TRANS (input) CHARACTER*1
40 * Specifies the form of the system of equations.
41 * = 'N': A * X = B (No transpose)
42 * = 'T': A**T * X = B (Transpose)
43 * = 'C': A**H * X = B (Conjugate transpose = Transpose)
44 *
45 * N (input) INTEGER
46 * The number of rows of the matrices X and XACT. N >= 0.
47 *
48 * NRHS (input) INTEGER
49 * The number of columns of the matrices X and XACT. NRHS >= 0.
50 *
51 * DL (input) DOUBLE PRECISION array, dimension (N-1)
52 * The (n-1) sub-diagonal elements of A.
53 *
54 * D (input) DOUBLE PRECISION array, dimension (N)
55 * The diagonal elements of A.
56 *
57 * DU (input) DOUBLE PRECISION array, dimension (N-1)
58 * The (n-1) super-diagonal elements of A.
59 *
60 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
61 * The right hand side vectors for the system of linear
62 * equations.
63 *
64 * LDB (input) INTEGER
65 * The leading dimension of the array B. LDB >= max(1,N).
66 *
67 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
68 * The computed solution vectors. Each vector is stored as a
69 * column of the matrix X.
70 *
71 * LDX (input) INTEGER
72 * The leading dimension of the array X. LDX >= max(1,N).
73 *
74 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
75 * The exact solution vectors. Each vector is stored as a
76 * column of the matrix XACT.
77 *
78 * LDXACT (input) INTEGER
79 * The leading dimension of the array XACT. LDXACT >= max(1,N).
80 *
81 * FERR (input) DOUBLE PRECISION array, dimension (NRHS)
82 * The estimated forward error bounds for each solution vector
83 * X. If XTRUE is the true solution, FERR bounds the magnitude
84 * of the largest entry in (X - XTRUE) divided by the magnitude
85 * of the largest entry in X.
86 *
87 * BERR (input) DOUBLE PRECISION array, dimension (NRHS)
88 * The componentwise relative backward error of each solution
89 * vector (i.e., the smallest relative change in any entry of A
90 * or B that makes X an exact solution).
91 *
92 * RESLTS (output) DOUBLE PRECISION array, dimension (2)
93 * The maximum over the NRHS solution vectors of the ratios:
94 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
95 * RESLTS(2) = BERR / ( NZ*EPS + (*) )
96 *
97 * =====================================================================
98 *
99 * .. Parameters ..
100 DOUBLE PRECISION ZERO, ONE
101 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
102 * ..
103 * .. Local Scalars ..
104 LOGICAL NOTRAN
105 INTEGER I, IMAX, J, K, NZ
106 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
107 * ..
108 * .. External Functions ..
109 LOGICAL LSAME
110 INTEGER IDAMAX
111 DOUBLE PRECISION DLAMCH
112 EXTERNAL LSAME, IDAMAX, DLAMCH
113 * ..
114 * .. Intrinsic Functions ..
115 INTRINSIC ABS, MAX, MIN
116 * ..
117 * .. Executable Statements ..
118 *
119 * Quick exit if N = 0 or NRHS = 0.
120 *
121 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
122 RESLTS( 1 ) = ZERO
123 RESLTS( 2 ) = ZERO
124 RETURN
125 END IF
126 *
127 EPS = DLAMCH( 'Epsilon' )
128 UNFL = DLAMCH( 'Safe minimum' )
129 OVFL = ONE / UNFL
130 NOTRAN = LSAME( TRANS, 'N' )
131 NZ = 4
132 *
133 * Test 1: Compute the maximum of
134 * norm(X - XACT) / ( norm(X) * FERR )
135 * over all the vectors X and XACT using the infinity-norm.
136 *
137 ERRBND = ZERO
138 DO 30 J = 1, NRHS
139 IMAX = IDAMAX( N, X( 1, J ), 1 )
140 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
141 DIFF = ZERO
142 DO 10 I = 1, N
143 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
144 10 CONTINUE
145 *
146 IF( XNORM.GT.ONE ) THEN
147 GO TO 20
148 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
149 GO TO 20
150 ELSE
151 ERRBND = ONE / EPS
152 GO TO 30
153 END IF
154 *
155 20 CONTINUE
156 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
157 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
158 ELSE
159 ERRBND = ONE / EPS
160 END IF
161 30 CONTINUE
162 RESLTS( 1 ) = ERRBND
163 *
164 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
165 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
166 *
167 DO 60 K = 1, NRHS
168 IF( NOTRAN ) THEN
169 IF( N.EQ.1 ) THEN
170 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
171 ELSE
172 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
173 $ ABS( DU( 1 )*X( 2, K ) )
174 DO 40 I = 2, N - 1
175 TMP = ABS( B( I, K ) ) + ABS( DL( I-1 )*X( I-1, K ) )
176 $ + ABS( D( I )*X( I, K ) ) +
177 $ ABS( DU( I )*X( I+1, K ) )
178 AXBI = MIN( AXBI, TMP )
179 40 CONTINUE
180 TMP = ABS( B( N, K ) ) + ABS( DL( N-1 )*X( N-1, K ) ) +
181 $ ABS( D( N )*X( N, K ) )
182 AXBI = MIN( AXBI, TMP )
183 END IF
184 ELSE
185 IF( N.EQ.1 ) THEN
186 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
187 ELSE
188 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
189 $ ABS( DL( 1 )*X( 2, K ) )
190 DO 50 I = 2, N - 1
191 TMP = ABS( B( I, K ) ) + ABS( DU( I-1 )*X( I-1, K ) )
192 $ + ABS( D( I )*X( I, K ) ) +
193 $ ABS( DL( I )*X( I+1, K ) )
194 AXBI = MIN( AXBI, TMP )
195 50 CONTINUE
196 TMP = ABS( B( N, K ) ) + ABS( DU( N-1 )*X( N-1, K ) ) +
197 $ ABS( D( N )*X( N, K ) )
198 AXBI = MIN( AXBI, TMP )
199 END IF
200 END IF
201 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
202 IF( K.EQ.1 ) THEN
203 RESLTS( 2 ) = TMP
204 ELSE
205 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
206 END IF
207 60 CONTINUE
208 *
209 RETURN
210 *
211 * End of DGTT05
212 *
213 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 TRANS
10 INTEGER LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DL( * ),
14 $ DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ),
15 $ XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DGTT05 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 * general tridiagonal matrix of order n and op(A) = A or A**T,
24 * depending on TRANS.
25 *
26 * RESLTS(1) = test of the error bound
27 * = norm(X - XACT) / ( norm(X) * FERR )
28 *
29 * A large value is returned if this ratio is not less than one.
30 *
31 * RESLTS(2) = residual from the iterative refinement routine
32 * = the maximum of BERR / ( NZ*EPS + (*) ), where
33 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
34 * and NZ = max. number of nonzeros in any row of A, plus 1
35 *
36 * Arguments
37 * =========
38 *
39 * TRANS (input) CHARACTER*1
40 * Specifies the form of the system of equations.
41 * = 'N': A * X = B (No transpose)
42 * = 'T': A**T * X = B (Transpose)
43 * = 'C': A**H * X = B (Conjugate transpose = Transpose)
44 *
45 * N (input) INTEGER
46 * The number of rows of the matrices X and XACT. N >= 0.
47 *
48 * NRHS (input) INTEGER
49 * The number of columns of the matrices X and XACT. NRHS >= 0.
50 *
51 * DL (input) DOUBLE PRECISION array, dimension (N-1)
52 * The (n-1) sub-diagonal elements of A.
53 *
54 * D (input) DOUBLE PRECISION array, dimension (N)
55 * The diagonal elements of A.
56 *
57 * DU (input) DOUBLE PRECISION array, dimension (N-1)
58 * The (n-1) super-diagonal elements of A.
59 *
60 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
61 * The right hand side vectors for the system of linear
62 * equations.
63 *
64 * LDB (input) INTEGER
65 * The leading dimension of the array B. LDB >= max(1,N).
66 *
67 * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
68 * The computed solution vectors. Each vector is stored as a
69 * column of the matrix X.
70 *
71 * LDX (input) INTEGER
72 * The leading dimension of the array X. LDX >= max(1,N).
73 *
74 * XACT (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
75 * The exact solution vectors. Each vector is stored as a
76 * column of the matrix XACT.
77 *
78 * LDXACT (input) INTEGER
79 * The leading dimension of the array XACT. LDXACT >= max(1,N).
80 *
81 * FERR (input) DOUBLE PRECISION array, dimension (NRHS)
82 * The estimated forward error bounds for each solution vector
83 * X. If XTRUE is the true solution, FERR bounds the magnitude
84 * of the largest entry in (X - XTRUE) divided by the magnitude
85 * of the largest entry in X.
86 *
87 * BERR (input) DOUBLE PRECISION array, dimension (NRHS)
88 * The componentwise relative backward error of each solution
89 * vector (i.e., the smallest relative change in any entry of A
90 * or B that makes X an exact solution).
91 *
92 * RESLTS (output) DOUBLE PRECISION array, dimension (2)
93 * The maximum over the NRHS solution vectors of the ratios:
94 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
95 * RESLTS(2) = BERR / ( NZ*EPS + (*) )
96 *
97 * =====================================================================
98 *
99 * .. Parameters ..
100 DOUBLE PRECISION ZERO, ONE
101 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
102 * ..
103 * .. Local Scalars ..
104 LOGICAL NOTRAN
105 INTEGER I, IMAX, J, K, NZ
106 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
107 * ..
108 * .. External Functions ..
109 LOGICAL LSAME
110 INTEGER IDAMAX
111 DOUBLE PRECISION DLAMCH
112 EXTERNAL LSAME, IDAMAX, DLAMCH
113 * ..
114 * .. Intrinsic Functions ..
115 INTRINSIC ABS, MAX, MIN
116 * ..
117 * .. Executable Statements ..
118 *
119 * Quick exit if N = 0 or NRHS = 0.
120 *
121 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
122 RESLTS( 1 ) = ZERO
123 RESLTS( 2 ) = ZERO
124 RETURN
125 END IF
126 *
127 EPS = DLAMCH( 'Epsilon' )
128 UNFL = DLAMCH( 'Safe minimum' )
129 OVFL = ONE / UNFL
130 NOTRAN = LSAME( TRANS, 'N' )
131 NZ = 4
132 *
133 * Test 1: Compute the maximum of
134 * norm(X - XACT) / ( norm(X) * FERR )
135 * over all the vectors X and XACT using the infinity-norm.
136 *
137 ERRBND = ZERO
138 DO 30 J = 1, NRHS
139 IMAX = IDAMAX( N, X( 1, J ), 1 )
140 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
141 DIFF = ZERO
142 DO 10 I = 1, N
143 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
144 10 CONTINUE
145 *
146 IF( XNORM.GT.ONE ) THEN
147 GO TO 20
148 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
149 GO TO 20
150 ELSE
151 ERRBND = ONE / EPS
152 GO TO 30
153 END IF
154 *
155 20 CONTINUE
156 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
157 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
158 ELSE
159 ERRBND = ONE / EPS
160 END IF
161 30 CONTINUE
162 RESLTS( 1 ) = ERRBND
163 *
164 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
165 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
166 *
167 DO 60 K = 1, NRHS
168 IF( NOTRAN ) THEN
169 IF( N.EQ.1 ) THEN
170 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
171 ELSE
172 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
173 $ ABS( DU( 1 )*X( 2, K ) )
174 DO 40 I = 2, N - 1
175 TMP = ABS( B( I, K ) ) + ABS( DL( I-1 )*X( I-1, K ) )
176 $ + ABS( D( I )*X( I, K ) ) +
177 $ ABS( DU( I )*X( I+1, K ) )
178 AXBI = MIN( AXBI, TMP )
179 40 CONTINUE
180 TMP = ABS( B( N, K ) ) + ABS( DL( N-1 )*X( N-1, K ) ) +
181 $ ABS( D( N )*X( N, K ) )
182 AXBI = MIN( AXBI, TMP )
183 END IF
184 ELSE
185 IF( N.EQ.1 ) THEN
186 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
187 ELSE
188 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
189 $ ABS( DL( 1 )*X( 2, K ) )
190 DO 50 I = 2, N - 1
191 TMP = ABS( B( I, K ) ) + ABS( DU( I-1 )*X( I-1, K ) )
192 $ + ABS( D( I )*X( I, K ) ) +
193 $ ABS( DL( I )*X( I+1, K ) )
194 AXBI = MIN( AXBI, TMP )
195 50 CONTINUE
196 TMP = ABS( B( N, K ) ) + ABS( DU( N-1 )*X( N-1, K ) ) +
197 $ ABS( D( N )*X( N, K ) )
198 AXBI = MIN( AXBI, TMP )
199 END IF
200 END IF
201 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
202 IF( K.EQ.1 ) THEN
203 RESLTS( 2 ) = TMP
204 ELSE
205 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
206 END IF
207 60 CONTINUE
208 *
209 RETURN
210 *
211 * End of DGTT05
212 *
213 END