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
|
SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
$ JPIV )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER IJOB, LDZ, N
DOUBLE PRECISION RDSCAL, RDSUM
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), JPIV( * )
DOUBLE PRECISION RHS( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* DLATDF uses the LU factorization of the n-by-n matrix Z computed by
* DGETC2 and computes a contribution to the reciprocal Dif-estimate
* by solving Z * x = b for x, and choosing the r.h.s. b such that
* the norm of x is as large as possible. On entry RHS = b holds the
* contribution from earlier solved sub-systems, and on return RHS = x.
*
* The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
* where P and Q are permutation matrices. L is lower triangular with
* unit diagonal elements and U is upper triangular.
*
* Arguments
* =========
*
* IJOB (input) INTEGER
* IJOB = 2: First compute an approximative null-vector e
* of Z using DGECON, e is normalized and solve for
* Zx = +-e - f with the sign giving the greater value
* of 2-norm(x). About 5 times as expensive as Default.
* IJOB .ne. 2: Local look ahead strategy where all entries of
* the r.h.s. b is choosen as either +1 or -1 (Default).
*
* N (input) INTEGER
* The number of columns of the matrix Z.
*
* Z (input) DOUBLE PRECISION array, dimension (LDZ, N)
* On entry, the LU part of the factorization of the n-by-n
* matrix Z computed by DGETC2: Z = P * L * U * Q
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDA >= max(1, N).
*
* RHS (input/output) DOUBLE PRECISION array, dimension N.
* On entry, RHS contains contributions from other subsystems.
* On exit, RHS contains the solution of the subsystem with
* entries acoording to the value of IJOB (see above).
*
* RDSUM (input/output) DOUBLE PRECISION
* On entry, the sum of squares of computed contributions to
* the Dif-estimate under computation by DTGSYL, where the
* scaling factor RDSCAL (see below) has been factored out.
* On exit, the corresponding sum of squares updated with the
* contributions from the current sub-system.
* If TRANS = 'T' RDSUM is not touched.
* NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
*
* RDSCAL (input/output) DOUBLE PRECISION
* On entry, scaling factor used to prevent overflow in RDSUM.
* On exit, RDSCAL is updated w.r.t. the current contributions
* in RDSUM.
* If TRANS = 'T', RDSCAL is not touched.
* NOTE: RDSCAL only makes sense when DTGSY2 is called by
* DTGSYL.
*
* IPIV (input) INTEGER array, dimension (N).
* The pivot indices; for 1 <= i <= N, row i of the
* matrix has been interchanged with row IPIV(i).
*
* JPIV (input) INTEGER array, dimension (N).
* The pivot indices; for 1 <= j <= N, column j of the
* matrix has been interchanged with column JPIV(j).
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* This routine is a further developed implementation of algorithm
* BSOLVE in [1] using complete pivoting in the LU factorization.
*
* [1] Bo Kagstrom and Lars Westin,
* Generalized Schur Methods with Condition Estimators for
* Solving the Generalized Sylvester Equation, IEEE Transactions
* on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
*
* [2] Peter Poromaa,
* On Efficient and Robust Estimators for the Separation
* between two Regular Matrix Pairs with Applications in
* Condition Estimation. Report IMINF-95.05, Departement of
* Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
*
* =====================================================================
*
* .. Parameters ..
INTEGER MAXDIM
PARAMETER ( MAXDIM = 8 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J, K
DOUBLE PRECISION BM, BP, PMONE, SMINU, SPLUS, TEMP
* ..
* .. Local Arrays ..
INTEGER IWORK( MAXDIM )
DOUBLE PRECISION WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP,
$ DSCAL
* ..
* .. External Functions ..
DOUBLE PRECISION DASUM, DDOT
EXTERNAL DASUM, DDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
IF( IJOB.NE.2 ) THEN
*
* Apply permutations IPIV to RHS
*
CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
*
* Solve for L-part choosing RHS either to +1 or -1.
*
PMONE = -ONE
*
DO 10 J = 1, N - 1
BP = RHS( J ) + ONE
BM = RHS( J ) - ONE
SPLUS = ONE
*
* Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
* SMIN computed more efficiently than in BSOLVE [1].
*
SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 )
SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
SPLUS = SPLUS*RHS( J )
IF( SPLUS.GT.SMINU ) THEN
RHS( J ) = BP
ELSE IF( SMINU.GT.SPLUS ) THEN
RHS( J ) = BM
ELSE
*
* In this case the updating sums are equal and we can
* choose RHS(J) +1 or -1. The first time this happens
* we choose -1, thereafter +1. This is a simple way to
* get good estimates of matrices like Byers well-known
* example (see [1]). (Not done in BSOLVE.)
*
RHS( J ) = RHS( J ) + PMONE
PMONE = ONE
END IF
*
* Compute the remaining r.h.s.
*
TEMP = -RHS( J )
CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
*
10 CONTINUE
*
* Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
* in BSOLVE and will hopefully give us a better estimate because
* any ill-conditioning of the original matrix is transfered to U
* and not to L. U(N, N) is an approximation to sigma_min(LU).
*
CALL DCOPY( N-1, RHS, 1, XP, 1 )
XP( N ) = RHS( N ) + ONE
RHS( N ) = RHS( N ) - ONE
SPLUS = ZERO
SMINU = ZERO
DO 30 I = N, 1, -1
TEMP = ONE / Z( I, I )
XP( I ) = XP( I )*TEMP
RHS( I ) = RHS( I )*TEMP
DO 20 K = I + 1, N
XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
20 CONTINUE
SPLUS = SPLUS + ABS( XP( I ) )
SMINU = SMINU + ABS( RHS( I ) )
30 CONTINUE
IF( SPLUS.GT.SMINU )
$ CALL DCOPY( N, XP, 1, RHS, 1 )
*
* Apply the permutations JPIV to the computed solution (RHS)
*
CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
*
* Compute the sum of squares
*
CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
*
ELSE
*
* IJOB = 2, Compute approximate nullvector XM of Z
*
CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
*
* Compute RHS
*
CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
CALL DSCAL( N, TEMP, XM, 1 )
CALL DCOPY( N, XM, 1, XP, 1 )
CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
$ CALL DCOPY( N, XP, 1, RHS, 1 )
*
* Compute the sum of squares
*
CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
*
END IF
*
RETURN
*
* End of DLATDF
*
END
|
