1 SUBROUTINE ZGET07( TRANS, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
2 $ LDXACT, FERR, CHKFERR, 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 LOGICAL CHKFERR
11 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * )
15 COMPLEX*16 A( LDA, * ), B( LDB, * ), X( LDX, * ),
16 $ XACT( LDXACT, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * ZGET07 tests the error bounds from iterative refinement for the
23 * computed solution to a system of equations op(A)*X = B, where A is a
24 * general n by n matrix and op(A) = A or A**T, 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 / ( (n+1)*EPS + (*) ), where
33 * (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
34 *
35 * Arguments
36 * =========
37 *
38 * TRANS (input) CHARACTER*1
39 * Specifies the form of the system of equations.
40 * = 'N': A * X = B (No transpose)
41 * = 'T': A**T * X = B (Transpose)
42 * = 'C': A**H * X = B (Conjugate transpose = Transpose)
43 *
44 * N (input) INTEGER
45 * The number of rows of the matrices X and XACT. N >= 0.
46 *
47 * NRHS (input) INTEGER
48 * The number of columns of the matrices X and XACT. NRHS >= 0.
49 *
50 * A (input) COMPLEX*16 array, dimension (LDA,N)
51 * The original n by n matrix A.
52 *
53 * LDA (input) INTEGER
54 * The leading dimension of the array A. LDA >= max(1,N).
55 *
56 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
57 * The right hand side vectors for the system of linear
58 * equations.
59 *
60 * LDB (input) INTEGER
61 * The leading dimension of the array B. LDB >= max(1,N).
62 *
63 * X (input) COMPLEX*16 array, dimension (LDX,NRHS)
64 * The computed solution vectors. Each vector is stored as a
65 * column of the matrix X.
66 *
67 * LDX (input) INTEGER
68 * The leading dimension of the array X. LDX >= max(1,N).
69 *
70 * XACT (input) COMPLEX*16 array, dimension (LDX,NRHS)
71 * The exact solution vectors. Each vector is stored as a
72 * column of the matrix XACT.
73 *
74 * LDXACT (input) INTEGER
75 * The leading dimension of the array XACT. LDXACT >= max(1,N).
76 *
77 * FERR (input) DOUBLE PRECISION array, dimension (NRHS)
78 * The estimated forward error bounds for each solution vector
79 * X. If XTRUE is the true solution, FERR bounds the magnitude
80 * of the largest entry in (X - XTRUE) divided by the magnitude
81 * of the largest entry in X.
82 *
83 * CHKFERR (input) LOGICAL
84 * Set to .TRUE. to check FERR, .FALSE. not to check FERR.
85 * When the test system is ill-conditioned, the "true"
86 * solution in XACT may be incorrect.
87 *
88 * BERR (input) DOUBLE PRECISION array, dimension (NRHS)
89 * The componentwise relative backward error of each solution
90 * vector (i.e., the smallest relative change in any entry of A
91 * or B that makes X an exact solution).
92 *
93 * RESLTS (output) DOUBLE PRECISION array, dimension (2)
94 * The maximum over the NRHS solution vectors of the ratios:
95 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
96 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
97 *
98 * =====================================================================
99 *
100 * .. Parameters ..
101 DOUBLE PRECISION ZERO, ONE
102 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
103 * ..
104 * .. Local Scalars ..
105 LOGICAL NOTRAN
106 INTEGER I, IMAX, J, K
107 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
108 COMPLEX*16 ZDUM
109 * ..
110 * .. External Functions ..
111 LOGICAL LSAME
112 INTEGER IZAMAX
113 DOUBLE PRECISION DLAMCH
114 EXTERNAL LSAME, IZAMAX, DLAMCH
115 * ..
116 * .. Intrinsic Functions ..
117 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
118 * ..
119 * .. Statement Functions ..
120 DOUBLE PRECISION CABS1
121 * ..
122 * .. Statement Function definitions ..
123 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
124 * ..
125 * .. Executable Statements ..
126 *
127 * Quick exit if N = 0 or NRHS = 0.
128 *
129 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
130 RESLTS( 1 ) = ZERO
131 RESLTS( 2 ) = ZERO
132 RETURN
133 END IF
134 *
135 EPS = DLAMCH( 'Epsilon' )
136 UNFL = DLAMCH( 'Safe minimum' )
137 OVFL = ONE / UNFL
138 NOTRAN = LSAME( TRANS, 'N' )
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 IF( CHKFERR ) THEN
146 DO 30 J = 1, NRHS
147 IMAX = IZAMAX( N, X( 1, J ), 1 )
148 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
149 DIFF = ZERO
150 DO 10 I = 1, N
151 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
152 10 CONTINUE
153 *
154 IF( XNORM.GT.ONE ) THEN
155 GO TO 20
156 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
157 GO TO 20
158 ELSE
159 ERRBND = ONE / EPS
160 GO TO 30
161 END IF
162 *
163 20 CONTINUE
164 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
165 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
166 ELSE
167 ERRBND = ONE / EPS
168 END IF
169 30 CONTINUE
170 END IF
171 RESLTS( 1 ) = ERRBND
172 *
173 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
174 * (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
175 *
176 DO 70 K = 1, NRHS
177 DO 60 I = 1, N
178 TMP = CABS1( B( I, K ) )
179 IF( NOTRAN ) THEN
180 DO 40 J = 1, N
181 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) )
182 40 CONTINUE
183 ELSE
184 DO 50 J = 1, N
185 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) )
186 50 CONTINUE
187 END IF
188 IF( I.EQ.1 ) THEN
189 AXBI = TMP
190 ELSE
191 AXBI = MIN( AXBI, TMP )
192 END IF
193 60 CONTINUE
194 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
195 $ MAX( AXBI, ( N+1 )*UNFL ) )
196 IF( K.EQ.1 ) THEN
197 RESLTS( 2 ) = TMP
198 ELSE
199 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
200 END IF
201 70 CONTINUE
202 *
203 RETURN
204 *
205 * End of ZGET07
206 *
207 END
2 $ LDXACT, FERR, CHKFERR, 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 LOGICAL CHKFERR
11 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * )
15 COMPLEX*16 A( LDA, * ), B( LDB, * ), X( LDX, * ),
16 $ XACT( LDXACT, * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * ZGET07 tests the error bounds from iterative refinement for the
23 * computed solution to a system of equations op(A)*X = B, where A is a
24 * general n by n matrix and op(A) = A or A**T, 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 / ( (n+1)*EPS + (*) ), where
33 * (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
34 *
35 * Arguments
36 * =========
37 *
38 * TRANS (input) CHARACTER*1
39 * Specifies the form of the system of equations.
40 * = 'N': A * X = B (No transpose)
41 * = 'T': A**T * X = B (Transpose)
42 * = 'C': A**H * X = B (Conjugate transpose = Transpose)
43 *
44 * N (input) INTEGER
45 * The number of rows of the matrices X and XACT. N >= 0.
46 *
47 * NRHS (input) INTEGER
48 * The number of columns of the matrices X and XACT. NRHS >= 0.
49 *
50 * A (input) COMPLEX*16 array, dimension (LDA,N)
51 * The original n by n matrix A.
52 *
53 * LDA (input) INTEGER
54 * The leading dimension of the array A. LDA >= max(1,N).
55 *
56 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
57 * The right hand side vectors for the system of linear
58 * equations.
59 *
60 * LDB (input) INTEGER
61 * The leading dimension of the array B. LDB >= max(1,N).
62 *
63 * X (input) COMPLEX*16 array, dimension (LDX,NRHS)
64 * The computed solution vectors. Each vector is stored as a
65 * column of the matrix X.
66 *
67 * LDX (input) INTEGER
68 * The leading dimension of the array X. LDX >= max(1,N).
69 *
70 * XACT (input) COMPLEX*16 array, dimension (LDX,NRHS)
71 * The exact solution vectors. Each vector is stored as a
72 * column of the matrix XACT.
73 *
74 * LDXACT (input) INTEGER
75 * The leading dimension of the array XACT. LDXACT >= max(1,N).
76 *
77 * FERR (input) DOUBLE PRECISION array, dimension (NRHS)
78 * The estimated forward error bounds for each solution vector
79 * X. If XTRUE is the true solution, FERR bounds the magnitude
80 * of the largest entry in (X - XTRUE) divided by the magnitude
81 * of the largest entry in X.
82 *
83 * CHKFERR (input) LOGICAL
84 * Set to .TRUE. to check FERR, .FALSE. not to check FERR.
85 * When the test system is ill-conditioned, the "true"
86 * solution in XACT may be incorrect.
87 *
88 * BERR (input) DOUBLE PRECISION array, dimension (NRHS)
89 * The componentwise relative backward error of each solution
90 * vector (i.e., the smallest relative change in any entry of A
91 * or B that makes X an exact solution).
92 *
93 * RESLTS (output) DOUBLE PRECISION array, dimension (2)
94 * The maximum over the NRHS solution vectors of the ratios:
95 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
96 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
97 *
98 * =====================================================================
99 *
100 * .. Parameters ..
101 DOUBLE PRECISION ZERO, ONE
102 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
103 * ..
104 * .. Local Scalars ..
105 LOGICAL NOTRAN
106 INTEGER I, IMAX, J, K
107 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
108 COMPLEX*16 ZDUM
109 * ..
110 * .. External Functions ..
111 LOGICAL LSAME
112 INTEGER IZAMAX
113 DOUBLE PRECISION DLAMCH
114 EXTERNAL LSAME, IZAMAX, DLAMCH
115 * ..
116 * .. Intrinsic Functions ..
117 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
118 * ..
119 * .. Statement Functions ..
120 DOUBLE PRECISION CABS1
121 * ..
122 * .. Statement Function definitions ..
123 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
124 * ..
125 * .. Executable Statements ..
126 *
127 * Quick exit if N = 0 or NRHS = 0.
128 *
129 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
130 RESLTS( 1 ) = ZERO
131 RESLTS( 2 ) = ZERO
132 RETURN
133 END IF
134 *
135 EPS = DLAMCH( 'Epsilon' )
136 UNFL = DLAMCH( 'Safe minimum' )
137 OVFL = ONE / UNFL
138 NOTRAN = LSAME( TRANS, 'N' )
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 IF( CHKFERR ) THEN
146 DO 30 J = 1, NRHS
147 IMAX = IZAMAX( N, X( 1, J ), 1 )
148 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
149 DIFF = ZERO
150 DO 10 I = 1, N
151 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
152 10 CONTINUE
153 *
154 IF( XNORM.GT.ONE ) THEN
155 GO TO 20
156 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
157 GO TO 20
158 ELSE
159 ERRBND = ONE / EPS
160 GO TO 30
161 END IF
162 *
163 20 CONTINUE
164 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
165 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
166 ELSE
167 ERRBND = ONE / EPS
168 END IF
169 30 CONTINUE
170 END IF
171 RESLTS( 1 ) = ERRBND
172 *
173 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
174 * (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
175 *
176 DO 70 K = 1, NRHS
177 DO 60 I = 1, N
178 TMP = CABS1( B( I, K ) )
179 IF( NOTRAN ) THEN
180 DO 40 J = 1, N
181 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) )
182 40 CONTINUE
183 ELSE
184 DO 50 J = 1, N
185 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) )
186 50 CONTINUE
187 END IF
188 IF( I.EQ.1 ) THEN
189 AXBI = TMP
190 ELSE
191 AXBI = MIN( AXBI, TMP )
192 END IF
193 60 CONTINUE
194 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
195 $ MAX( AXBI, ( N+1 )*UNFL ) )
196 IF( K.EQ.1 ) THEN
197 RESLTS( 2 ) = TMP
198 ELSE
199 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
200 END IF
201 70 CONTINUE
202 *
203 RETURN
204 *
205 * End of ZGET07
206 *
207 END