1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
|
*> \brief \b SGBT01
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SGBT01( M, N, KL, KU, A, LDA, AFAC, LDAFAC, IPIV, WORK,
* RESID )
*
* .. Scalar Arguments ..
* INTEGER KL, KU, LDA, LDAFAC, M, N
* REAL RESID
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* REAL A( LDA, * ), AFAC( LDAFAC, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGBT01 reconstructs a band matrix A from its L*U factorization and
*> computes the residual:
*> norm(L*U - A) / ( N * norm(A) * EPS ),
*> where EPS is the machine epsilon.
*>
*> The expression L*U - A is computed one column at a time, so A and
*> AFAC are not modified.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KL
*> \verbatim
*> KL is INTEGER
*> The number of subdiagonals within the band of A. KL >= 0.
*> \endverbatim
*>
*> \param[in] KU
*> \verbatim
*> KU is INTEGER
*> The number of superdiagonals within the band of A. KU >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> The original matrix A in band storage, stored in rows 1 to
*> KL+KU+1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER.
*> The leading dimension of the array A. LDA >= max(1,KL+KU+1).
*> \endverbatim
*>
*> \param[in] AFAC
*> \verbatim
*> AFAC is REAL array, dimension (LDAFAC,N)
*> The factored form of the matrix A. AFAC contains the banded
*> factors L and U from the L*U factorization, as computed by
*> SGBTRF. U is stored as an upper triangular band matrix with
*> KL+KU superdiagonals in rows 1 to KL+KU+1, and the
*> multipliers used during the factorization are stored in rows
*> KL+KU+2 to 2*KL+KU+1. See SGBTRF for further details.
*> \endverbatim
*>
*> \param[in] LDAFAC
*> \verbatim
*> LDAFAC is INTEGER
*> The leading dimension of the array AFAC.
*> LDAFAC >= max(1,2*KL*KU+1).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (min(M,N))
*> The pivot indices from SGBTRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (2*KL+KU+1)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is REAL
*> norm(L*U - A) / ( N * norm(A) * EPS )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup single_lin
*
* =====================================================================
SUBROUTINE SGBT01( M, N, KL, KU, A, LDA, AFAC, LDAFAC, IPIV, WORK,
$ RESID )
*
* -- LAPACK test routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER KL, KU, LDA, LDAFAC, M, N
REAL RESID
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
REAL A( LDA, * ), AFAC( LDAFAC, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, I1, I2, IL, IP, IW, J, JL, JU, JUA, KD, LENJ
REAL ANORM, EPS, T
* ..
* .. External Functions ..
REAL SASUM, SLAMCH
EXTERNAL SASUM, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SCOPY
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
* Quick exit if M = 0 or N = 0.
*
RESID = ZERO
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
* Determine EPS and the norm of A.
*
EPS = SLAMCH( 'Epsilon' )
KD = KU + 1
ANORM = ZERO
DO 10 J = 1, N
I1 = MAX( KD+1-J, 1 )
I2 = MIN( KD+M-J, KL+KD )
IF( I2.GE.I1 )
$ ANORM = MAX( ANORM, SASUM( I2-I1+1, A( I1, J ), 1 ) )
10 CONTINUE
*
* Compute one column at a time of L*U - A.
*
KD = KL + KU + 1
DO 40 J = 1, N
*
* Copy the J-th column of U to WORK.
*
JU = MIN( KL+KU, J-1 )
JL = MIN( KL, M-J )
LENJ = MIN( M, J ) - J + JU + 1
IF( LENJ.GT.0 ) THEN
CALL SCOPY( LENJ, AFAC( KD-JU, J ), 1, WORK, 1 )
DO 20 I = LENJ + 1, JU + JL + 1
WORK( I ) = ZERO
20 CONTINUE
*
* Multiply by the unit lower triangular matrix L. Note that L
* is stored as a product of transformations and permutations.
*
DO 30 I = MIN( M-1, J ), J - JU, -1
IL = MIN( KL, M-I )
IF( IL.GT.0 ) THEN
IW = I - J + JU + 1
T = WORK( IW )
CALL SAXPY( IL, T, AFAC( KD+1, I ), 1, WORK( IW+1 ),
$ 1 )
IP = IPIV( I )
IF( I.NE.IP ) THEN
IP = IP - J + JU + 1
WORK( IW ) = WORK( IP )
WORK( IP ) = T
END IF
END IF
30 CONTINUE
*
* Subtract the corresponding column of A.
*
JUA = MIN( JU, KU )
IF( JUA+JL+1.GT.0 )
$ CALL SAXPY( JUA+JL+1, -ONE, A( KU+1-JUA, J ), 1,
$ WORK( JU+1-JUA ), 1 )
*
* Compute the 1-norm of the column.
*
RESID = MAX( RESID, SASUM( JU+JL+1, WORK, 1 ) )
END IF
40 CONTINUE
*
* Compute norm( L*U - A ) / ( N * norm(A) * EPS )
*
IF( ANORM.LE.ZERO ) THEN
IF( RESID.NE.ZERO )
$ RESID = ONE / EPS
ELSE
RESID = ( ( RESID / REAL( N ) ) / ANORM ) / EPS
END IF
*
RETURN
*
* End of SGBT01
*
END
|
