1 SUBROUTINE CGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X,
2 $ LDX, 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 KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 REAL BERR( * ), FERR( * ), RESLTS( * )
14 COMPLEX AB( LDAB, * ), B( LDB, * ), X( LDX, * ),
15 $ XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * CGBT05 tests the error bounds from iterative refinement for the
22 * computed solution to a system of equations op(A)*X = B, where A is a
23 * general band matrix of order n with kl subdiagonals and ku
24 * superdiagonals 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 / ( 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, B, and XACT, and the
47 * order of the matrix A. N >= 0.
48 *
49 * KL (input) INTEGER
50 * The number of subdiagonals within the band of A. KL >= 0.
51 *
52 * KU (input) INTEGER
53 * The number of superdiagonals within the band of A. KU >= 0.
54 *
55 * NRHS (input) INTEGER
56 * The number of columns of the matrices X, B, and XACT.
57 * NRHS >= 0.
58 *
59 * AB (input) COMPLEX array, dimension (LDAB,N)
60 * The original band matrix A, stored in rows 1 to KL+KU+1.
61 * The j-th column of A is stored in the j-th column of the
62 * array AB as follows:
63 * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
64 *
65 * LDAB (input) INTEGER
66 * The leading dimension of the array AB. LDAB >= KL+KU+1.
67 *
68 * B (input) COMPLEX array, dimension (LDB,NRHS)
69 * The right hand side vectors for the system of linear
70 * equations.
71 *
72 * LDB (input) INTEGER
73 * The leading dimension of the array B. LDB >= max(1,N).
74 *
75 * X (input) COMPLEX array, dimension (LDX,NRHS)
76 * The computed solution vectors. Each vector is stored as a
77 * column of the matrix X.
78 *
79 * LDX (input) INTEGER
80 * The leading dimension of the array X. LDX >= max(1,N).
81 *
82 * XACT (input) COMPLEX array, dimension (LDX,NRHS)
83 * The exact solution vectors. Each vector is stored as a
84 * column of the matrix XACT.
85 *
86 * LDXACT (input) INTEGER
87 * The leading dimension of the array XACT. LDXACT >= max(1,N).
88 *
89 * FERR (input) REAL array, dimension (NRHS)
90 * The estimated forward error bounds for each solution vector
91 * X. If XTRUE is the true solution, FERR bounds the magnitude
92 * of the largest entry in (X - XTRUE) divided by the magnitude
93 * of the largest entry in X.
94 *
95 * BERR (input) REAL array, dimension (NRHS)
96 * The componentwise relative backward error of each solution
97 * vector (i.e., the smallest relative change in any entry of A
98 * or B that makes X an exact solution).
99 *
100 * RESLTS (output) REAL array, dimension (2)
101 * The maximum over the NRHS solution vectors of the ratios:
102 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
103 * RESLTS(2) = BERR / ( NZ*EPS + (*) )
104 *
105 * =====================================================================
106 *
107 * .. Parameters ..
108 REAL ZERO, ONE
109 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
110 * ..
111 * .. Local Scalars ..
112 LOGICAL NOTRAN
113 INTEGER I, IMAX, J, K, NZ
114 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
115 COMPLEX ZDUM
116 * ..
117 * .. External Functions ..
118 LOGICAL LSAME
119 INTEGER ICAMAX
120 REAL SLAMCH
121 EXTERNAL LSAME, ICAMAX, SLAMCH
122 * ..
123 * .. Intrinsic Functions ..
124 INTRINSIC ABS, AIMAG, MAX, MIN, REAL
125 * ..
126 * .. Statement Functions ..
127 REAL CABS1
128 * ..
129 * .. Statement Function definitions ..
130 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
131 * ..
132 * .. Executable Statements ..
133 *
134 * Quick exit if N = 0 or NRHS = 0.
135 *
136 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
137 RESLTS( 1 ) = ZERO
138 RESLTS( 2 ) = ZERO
139 RETURN
140 END IF
141 *
142 EPS = SLAMCH( 'Epsilon' )
143 UNFL = SLAMCH( 'Safe minimum' )
144 OVFL = ONE / UNFL
145 NOTRAN = LSAME( TRANS, 'N' )
146 NZ = MIN( KL+KU+2, N+1 )
147 *
148 * Test 1: Compute the maximum of
149 * norm(X - XACT) / ( norm(X) * FERR )
150 * over all the vectors X and XACT using the infinity-norm.
151 *
152 ERRBND = ZERO
153 DO 30 J = 1, NRHS
154 IMAX = ICAMAX( N, X( 1, J ), 1 )
155 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
156 DIFF = ZERO
157 DO 10 I = 1, N
158 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
159 10 CONTINUE
160 *
161 IF( XNORM.GT.ONE ) THEN
162 GO TO 20
163 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
164 GO TO 20
165 ELSE
166 ERRBND = ONE / EPS
167 GO TO 30
168 END IF
169 *
170 20 CONTINUE
171 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
172 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
173 ELSE
174 ERRBND = ONE / EPS
175 END IF
176 30 CONTINUE
177 RESLTS( 1 ) = ERRBND
178 *
179 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
180 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
181 *
182 DO 70 K = 1, NRHS
183 DO 60 I = 1, N
184 TMP = CABS1( B( I, K ) )
185 IF( NOTRAN ) THEN
186 DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N )
187 TMP = TMP + CABS1( AB( KU+1+I-J, J ) )*
188 $ CABS1( X( J, K ) )
189 40 CONTINUE
190 ELSE
191 DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N )
192 TMP = TMP + CABS1( AB( KU+1+J-I, I ) )*
193 $ CABS1( X( J, K ) )
194 50 CONTINUE
195 END IF
196 IF( I.EQ.1 ) THEN
197 AXBI = TMP
198 ELSE
199 AXBI = MIN( AXBI, TMP )
200 END IF
201 60 CONTINUE
202 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
203 IF( K.EQ.1 ) THEN
204 RESLTS( 2 ) = TMP
205 ELSE
206 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
207 END IF
208 70 CONTINUE
209 *
210 RETURN
211 *
212 * End of CGBT05
213 *
214 END
2 $ LDX, 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 KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 REAL BERR( * ), FERR( * ), RESLTS( * )
14 COMPLEX AB( LDAB, * ), B( LDB, * ), X( LDX, * ),
15 $ XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * CGBT05 tests the error bounds from iterative refinement for the
22 * computed solution to a system of equations op(A)*X = B, where A is a
23 * general band matrix of order n with kl subdiagonals and ku
24 * superdiagonals 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 / ( 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, B, and XACT, and the
47 * order of the matrix A. N >= 0.
48 *
49 * KL (input) INTEGER
50 * The number of subdiagonals within the band of A. KL >= 0.
51 *
52 * KU (input) INTEGER
53 * The number of superdiagonals within the band of A. KU >= 0.
54 *
55 * NRHS (input) INTEGER
56 * The number of columns of the matrices X, B, and XACT.
57 * NRHS >= 0.
58 *
59 * AB (input) COMPLEX array, dimension (LDAB,N)
60 * The original band matrix A, stored in rows 1 to KL+KU+1.
61 * The j-th column of A is stored in the j-th column of the
62 * array AB as follows:
63 * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
64 *
65 * LDAB (input) INTEGER
66 * The leading dimension of the array AB. LDAB >= KL+KU+1.
67 *
68 * B (input) COMPLEX array, dimension (LDB,NRHS)
69 * The right hand side vectors for the system of linear
70 * equations.
71 *
72 * LDB (input) INTEGER
73 * The leading dimension of the array B. LDB >= max(1,N).
74 *
75 * X (input) COMPLEX array, dimension (LDX,NRHS)
76 * The computed solution vectors. Each vector is stored as a
77 * column of the matrix X.
78 *
79 * LDX (input) INTEGER
80 * The leading dimension of the array X. LDX >= max(1,N).
81 *
82 * XACT (input) COMPLEX array, dimension (LDX,NRHS)
83 * The exact solution vectors. Each vector is stored as a
84 * column of the matrix XACT.
85 *
86 * LDXACT (input) INTEGER
87 * The leading dimension of the array XACT. LDXACT >= max(1,N).
88 *
89 * FERR (input) REAL array, dimension (NRHS)
90 * The estimated forward error bounds for each solution vector
91 * X. If XTRUE is the true solution, FERR bounds the magnitude
92 * of the largest entry in (X - XTRUE) divided by the magnitude
93 * of the largest entry in X.
94 *
95 * BERR (input) REAL array, dimension (NRHS)
96 * The componentwise relative backward error of each solution
97 * vector (i.e., the smallest relative change in any entry of A
98 * or B that makes X an exact solution).
99 *
100 * RESLTS (output) REAL array, dimension (2)
101 * The maximum over the NRHS solution vectors of the ratios:
102 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
103 * RESLTS(2) = BERR / ( NZ*EPS + (*) )
104 *
105 * =====================================================================
106 *
107 * .. Parameters ..
108 REAL ZERO, ONE
109 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
110 * ..
111 * .. Local Scalars ..
112 LOGICAL NOTRAN
113 INTEGER I, IMAX, J, K, NZ
114 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
115 COMPLEX ZDUM
116 * ..
117 * .. External Functions ..
118 LOGICAL LSAME
119 INTEGER ICAMAX
120 REAL SLAMCH
121 EXTERNAL LSAME, ICAMAX, SLAMCH
122 * ..
123 * .. Intrinsic Functions ..
124 INTRINSIC ABS, AIMAG, MAX, MIN, REAL
125 * ..
126 * .. Statement Functions ..
127 REAL CABS1
128 * ..
129 * .. Statement Function definitions ..
130 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
131 * ..
132 * .. Executable Statements ..
133 *
134 * Quick exit if N = 0 or NRHS = 0.
135 *
136 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
137 RESLTS( 1 ) = ZERO
138 RESLTS( 2 ) = ZERO
139 RETURN
140 END IF
141 *
142 EPS = SLAMCH( 'Epsilon' )
143 UNFL = SLAMCH( 'Safe minimum' )
144 OVFL = ONE / UNFL
145 NOTRAN = LSAME( TRANS, 'N' )
146 NZ = MIN( KL+KU+2, N+1 )
147 *
148 * Test 1: Compute the maximum of
149 * norm(X - XACT) / ( norm(X) * FERR )
150 * over all the vectors X and XACT using the infinity-norm.
151 *
152 ERRBND = ZERO
153 DO 30 J = 1, NRHS
154 IMAX = ICAMAX( N, X( 1, J ), 1 )
155 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
156 DIFF = ZERO
157 DO 10 I = 1, N
158 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
159 10 CONTINUE
160 *
161 IF( XNORM.GT.ONE ) THEN
162 GO TO 20
163 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
164 GO TO 20
165 ELSE
166 ERRBND = ONE / EPS
167 GO TO 30
168 END IF
169 *
170 20 CONTINUE
171 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
172 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
173 ELSE
174 ERRBND = ONE / EPS
175 END IF
176 30 CONTINUE
177 RESLTS( 1 ) = ERRBND
178 *
179 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
180 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
181 *
182 DO 70 K = 1, NRHS
183 DO 60 I = 1, N
184 TMP = CABS1( B( I, K ) )
185 IF( NOTRAN ) THEN
186 DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N )
187 TMP = TMP + CABS1( AB( KU+1+I-J, J ) )*
188 $ CABS1( X( J, K ) )
189 40 CONTINUE
190 ELSE
191 DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N )
192 TMP = TMP + CABS1( AB( KU+1+J-I, I ) )*
193 $ CABS1( X( J, K ) )
194 50 CONTINUE
195 END IF
196 IF( I.EQ.1 ) THEN
197 AXBI = TMP
198 ELSE
199 AXBI = MIN( AXBI, TMP )
200 END IF
201 60 CONTINUE
202 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
203 IF( K.EQ.1 ) THEN
204 RESLTS( 2 ) = TMP
205 ELSE
206 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
207 END IF
208 70 CONTINUE
209 *
210 RETURN
211 *
212 * End of CGBT05
213 *
214 END