1 SUBROUTINE SGBT05( 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 AB( LDAB, * ), B( LDB, * ), BERR( * ),
14 $ FERR( * ), RESLTS( * ), X( LDX, * ),
15 $ XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * SGBT05 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) REAL 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) REAL 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) REAL 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) REAL 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 * ..
116 * .. External Functions ..
117 LOGICAL LSAME
118 INTEGER ISAMAX
119 REAL SLAMCH
120 EXTERNAL LSAME, ISAMAX, SLAMCH
121 * ..
122 * .. Intrinsic Functions ..
123 INTRINSIC ABS, MAX, MIN
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 = SLAMCH( 'Epsilon' )
136 UNFL = SLAMCH( 'Safe minimum' )
137 OVFL = ONE / UNFL
138 NOTRAN = LSAME( TRANS, 'N' )
139 NZ = MIN( KL+KU+2, N+1 )
140 *
141 * Test 1: Compute the maximum of
142 * norm(X - XACT) / ( norm(X) * FERR )
143 * over all the vectors X and XACT using the infinity-norm.
144 *
145 ERRBND = ZERO
146 DO 30 J = 1, NRHS
147 IMAX = ISAMAX( N, X( 1, J ), 1 )
148 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
149 DIFF = ZERO
150 DO 10 I = 1, N
151 DIFF = MAX( DIFF, ABS( 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 RESLTS( 1 ) = ERRBND
171 *
172 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
173 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
174 *
175 DO 70 K = 1, NRHS
176 DO 60 I = 1, N
177 TMP = ABS( B( I, K ) )
178 IF( NOTRAN ) THEN
179 DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N )
180 TMP = TMP + ABS( AB( KU+1+I-J, J ) )*ABS( X( J, K ) )
181 40 CONTINUE
182 ELSE
183 DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N )
184 TMP = TMP + ABS( AB( KU+1+J-I, I ) )*ABS( X( J, K ) )
185 50 CONTINUE
186 END IF
187 IF( I.EQ.1 ) THEN
188 AXBI = TMP
189 ELSE
190 AXBI = MIN( AXBI, TMP )
191 END IF
192 60 CONTINUE
193 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
194 IF( K.EQ.1 ) THEN
195 RESLTS( 2 ) = TMP
196 ELSE
197 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
198 END IF
199 70 CONTINUE
200 *
201 RETURN
202 *
203 * End of SGBT05
204 *
205 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 AB( LDAB, * ), B( LDB, * ), BERR( * ),
14 $ FERR( * ), RESLTS( * ), X( LDX, * ),
15 $ XACT( LDXACT, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * SGBT05 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) REAL 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) REAL 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) REAL 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) REAL 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 * ..
116 * .. External Functions ..
117 LOGICAL LSAME
118 INTEGER ISAMAX
119 REAL SLAMCH
120 EXTERNAL LSAME, ISAMAX, SLAMCH
121 * ..
122 * .. Intrinsic Functions ..
123 INTRINSIC ABS, MAX, MIN
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 = SLAMCH( 'Epsilon' )
136 UNFL = SLAMCH( 'Safe minimum' )
137 OVFL = ONE / UNFL
138 NOTRAN = LSAME( TRANS, 'N' )
139 NZ = MIN( KL+KU+2, N+1 )
140 *
141 * Test 1: Compute the maximum of
142 * norm(X - XACT) / ( norm(X) * FERR )
143 * over all the vectors X and XACT using the infinity-norm.
144 *
145 ERRBND = ZERO
146 DO 30 J = 1, NRHS
147 IMAX = ISAMAX( N, X( 1, J ), 1 )
148 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
149 DIFF = ZERO
150 DO 10 I = 1, N
151 DIFF = MAX( DIFF, ABS( 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 RESLTS( 1 ) = ERRBND
171 *
172 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
173 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
174 *
175 DO 70 K = 1, NRHS
176 DO 60 I = 1, N
177 TMP = ABS( B( I, K ) )
178 IF( NOTRAN ) THEN
179 DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N )
180 TMP = TMP + ABS( AB( KU+1+I-J, J ) )*ABS( X( J, K ) )
181 40 CONTINUE
182 ELSE
183 DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N )
184 TMP = TMP + ABS( AB( KU+1+J-I, I ) )*ABS( X( J, K ) )
185 50 CONTINUE
186 END IF
187 IF( I.EQ.1 ) THEN
188 AXBI = TMP
189 ELSE
190 AXBI = MIN( AXBI, TMP )
191 END IF
192 60 CONTINUE
193 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
194 IF( K.EQ.1 ) THEN
195 RESLTS( 2 ) = TMP
196 ELSE
197 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
198 END IF
199 70 CONTINUE
200 *
201 RETURN
202 *
203 * End of SGBT05
204 *
205 END