1 SUBROUTINE CGTT05( 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 REAL BERR( * ), FERR( * ), RESLTS( * )
14 COMPLEX B( LDB, * ), D( * ), DL( * ), DU( * ),
15 $ X( LDX, * ), XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * CGTT05 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) COMPLEX array, dimension (N-1)
52 * The (n-1) sub-diagonal elements of A.
53 *
54 * D (input) COMPLEX array, dimension (N)
55 * The diagonal elements of A.
56 *
57 * DU (input) COMPLEX array, dimension (N-1)
58 * The (n-1) super-diagonal elements of A.
59 *
60 * B (input) COMPLEX 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) COMPLEX 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) COMPLEX 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) REAL 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) REAL 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) REAL 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 REAL ZERO, ONE
101 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
102 * ..
103 * .. Local Scalars ..
104 LOGICAL NOTRAN
105 INTEGER I, IMAX, J, K, NZ
106 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
107 COMPLEX ZDUM
108 * ..
109 * .. External Functions ..
110 LOGICAL LSAME
111 INTEGER ICAMAX
112 REAL SLAMCH
113 EXTERNAL LSAME, ICAMAX, SLAMCH
114 * ..
115 * .. Intrinsic Functions ..
116 INTRINSIC ABS, AIMAG, MAX, MIN, REAL
117 * ..
118 * .. Statement Functions ..
119 REAL CABS1
120 * ..
121 * .. Statement Function definitions ..
122 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
123 * ..
124 * .. Executable Statements ..
125 *
126 * Quick exit if N = 0 or NRHS = 0.
127 *
128 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
129 RESLTS( 1 ) = ZERO
130 RESLTS( 2 ) = ZERO
131 RETURN
132 END IF
133 *
134 EPS = SLAMCH( 'Epsilon' )
135 UNFL = SLAMCH( 'Safe minimum' )
136 OVFL = ONE / UNFL
137 NOTRAN = LSAME( TRANS, 'N' )
138 NZ = 4
139 *
140 * Test 1: Compute the maximum of
141 * norm(X - XACT) / ( norm(X) * FERR )
142 * over all the vectors X and XACT using the infinity-norm.
143 *
144 ERRBND = ZERO
145 DO 30 J = 1, NRHS
146 IMAX = ICAMAX( N, X( 1, J ), 1 )
147 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
148 DIFF = ZERO
149 DO 10 I = 1, N
150 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
151 10 CONTINUE
152 *
153 IF( XNORM.GT.ONE ) THEN
154 GO TO 20
155 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
156 GO TO 20
157 ELSE
158 ERRBND = ONE / EPS
159 GO TO 30
160 END IF
161 *
162 20 CONTINUE
163 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
164 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
165 ELSE
166 ERRBND = ONE / EPS
167 END IF
168 30 CONTINUE
169 RESLTS( 1 ) = ERRBND
170 *
171 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
172 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
173 *
174 DO 60 K = 1, NRHS
175 IF( NOTRAN ) THEN
176 IF( N.EQ.1 ) THEN
177 AXBI = CABS1( B( 1, K ) ) +
178 $ CABS1( D( 1 ) )*CABS1( X( 1, K ) )
179 ELSE
180 AXBI = CABS1( B( 1, K ) ) +
181 $ CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
182 $ CABS1( DU( 1 ) )*CABS1( X( 2, K ) )
183 DO 40 I = 2, N - 1
184 TMP = CABS1( B( I, K ) ) +
185 $ CABS1( DL( I-1 ) )*CABS1( X( I-1, K ) ) +
186 $ CABS1( D( I ) )*CABS1( X( I, K ) ) +
187 $ CABS1( DU( I ) )*CABS1( X( I+1, K ) )
188 AXBI = MIN( AXBI, TMP )
189 40 CONTINUE
190 TMP = CABS1( B( N, K ) ) + CABS1( DL( N-1 ) )*
191 $ CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
192 $ CABS1( X( N, K ) )
193 AXBI = MIN( AXBI, TMP )
194 END IF
195 ELSE
196 IF( N.EQ.1 ) THEN
197 AXBI = CABS1( B( 1, K ) ) +
198 $ CABS1( D( 1 ) )*CABS1( X( 1, K ) )
199 ELSE
200 AXBI = CABS1( B( 1, K ) ) +
201 $ CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
202 $ CABS1( DL( 1 ) )*CABS1( X( 2, K ) )
203 DO 50 I = 2, N - 1
204 TMP = CABS1( B( I, K ) ) +
205 $ CABS1( DU( I-1 ) )*CABS1( X( I-1, K ) ) +
206 $ CABS1( D( I ) )*CABS1( X( I, K ) ) +
207 $ CABS1( DL( I ) )*CABS1( X( I+1, K ) )
208 AXBI = MIN( AXBI, TMP )
209 50 CONTINUE
210 TMP = CABS1( B( N, K ) ) + CABS1( DU( N-1 ) )*
211 $ CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
212 $ CABS1( X( N, K ) )
213 AXBI = MIN( AXBI, TMP )
214 END IF
215 END IF
216 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
217 IF( K.EQ.1 ) THEN
218 RESLTS( 2 ) = TMP
219 ELSE
220 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
221 END IF
222 60 CONTINUE
223 *
224 RETURN
225 *
226 * End of CGTT05
227 *
228 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 REAL BERR( * ), FERR( * ), RESLTS( * )
14 COMPLEX B( LDB, * ), D( * ), DL( * ), DU( * ),
15 $ X( LDX, * ), XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * CGTT05 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) COMPLEX array, dimension (N-1)
52 * The (n-1) sub-diagonal elements of A.
53 *
54 * D (input) COMPLEX array, dimension (N)
55 * The diagonal elements of A.
56 *
57 * DU (input) COMPLEX array, dimension (N-1)
58 * The (n-1) super-diagonal elements of A.
59 *
60 * B (input) COMPLEX 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) COMPLEX 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) COMPLEX 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) REAL 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) REAL 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) REAL 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 REAL ZERO, ONE
101 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
102 * ..
103 * .. Local Scalars ..
104 LOGICAL NOTRAN
105 INTEGER I, IMAX, J, K, NZ
106 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
107 COMPLEX ZDUM
108 * ..
109 * .. External Functions ..
110 LOGICAL LSAME
111 INTEGER ICAMAX
112 REAL SLAMCH
113 EXTERNAL LSAME, ICAMAX, SLAMCH
114 * ..
115 * .. Intrinsic Functions ..
116 INTRINSIC ABS, AIMAG, MAX, MIN, REAL
117 * ..
118 * .. Statement Functions ..
119 REAL CABS1
120 * ..
121 * .. Statement Function definitions ..
122 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
123 * ..
124 * .. Executable Statements ..
125 *
126 * Quick exit if N = 0 or NRHS = 0.
127 *
128 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
129 RESLTS( 1 ) = ZERO
130 RESLTS( 2 ) = ZERO
131 RETURN
132 END IF
133 *
134 EPS = SLAMCH( 'Epsilon' )
135 UNFL = SLAMCH( 'Safe minimum' )
136 OVFL = ONE / UNFL
137 NOTRAN = LSAME( TRANS, 'N' )
138 NZ = 4
139 *
140 * Test 1: Compute the maximum of
141 * norm(X - XACT) / ( norm(X) * FERR )
142 * over all the vectors X and XACT using the infinity-norm.
143 *
144 ERRBND = ZERO
145 DO 30 J = 1, NRHS
146 IMAX = ICAMAX( N, X( 1, J ), 1 )
147 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
148 DIFF = ZERO
149 DO 10 I = 1, N
150 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
151 10 CONTINUE
152 *
153 IF( XNORM.GT.ONE ) THEN
154 GO TO 20
155 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
156 GO TO 20
157 ELSE
158 ERRBND = ONE / EPS
159 GO TO 30
160 END IF
161 *
162 20 CONTINUE
163 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
164 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
165 ELSE
166 ERRBND = ONE / EPS
167 END IF
168 30 CONTINUE
169 RESLTS( 1 ) = ERRBND
170 *
171 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
172 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
173 *
174 DO 60 K = 1, NRHS
175 IF( NOTRAN ) THEN
176 IF( N.EQ.1 ) THEN
177 AXBI = CABS1( B( 1, K ) ) +
178 $ CABS1( D( 1 ) )*CABS1( X( 1, K ) )
179 ELSE
180 AXBI = CABS1( B( 1, K ) ) +
181 $ CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
182 $ CABS1( DU( 1 ) )*CABS1( X( 2, K ) )
183 DO 40 I = 2, N - 1
184 TMP = CABS1( B( I, K ) ) +
185 $ CABS1( DL( I-1 ) )*CABS1( X( I-1, K ) ) +
186 $ CABS1( D( I ) )*CABS1( X( I, K ) ) +
187 $ CABS1( DU( I ) )*CABS1( X( I+1, K ) )
188 AXBI = MIN( AXBI, TMP )
189 40 CONTINUE
190 TMP = CABS1( B( N, K ) ) + CABS1( DL( N-1 ) )*
191 $ CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
192 $ CABS1( X( N, K ) )
193 AXBI = MIN( AXBI, TMP )
194 END IF
195 ELSE
196 IF( N.EQ.1 ) THEN
197 AXBI = CABS1( B( 1, K ) ) +
198 $ CABS1( D( 1 ) )*CABS1( X( 1, K ) )
199 ELSE
200 AXBI = CABS1( B( 1, K ) ) +
201 $ CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
202 $ CABS1( DL( 1 ) )*CABS1( X( 2, K ) )
203 DO 50 I = 2, N - 1
204 TMP = CABS1( B( I, K ) ) +
205 $ CABS1( DU( I-1 ) )*CABS1( X( I-1, K ) ) +
206 $ CABS1( D( I ) )*CABS1( X( I, K ) ) +
207 $ CABS1( DL( I ) )*CABS1( X( I+1, K ) )
208 AXBI = MIN( AXBI, TMP )
209 50 CONTINUE
210 TMP = CABS1( B( N, K ) ) + CABS1( DU( N-1 ) )*
211 $ CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
212 $ CABS1( X( N, K ) )
213 AXBI = MIN( AXBI, TMP )
214 END IF
215 END IF
216 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
217 IF( K.EQ.1 ) THEN
218 RESLTS( 2 ) = TMP
219 ELSE
220 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
221 END IF
222 60 CONTINUE
223 *
224 RETURN
225 *
226 * End of CGTT05
227 *
228 END