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
|
SUBROUTINE SB04NY( RC, UL, M, A, LDA, LAMBDA, D, TOL, IWORK,
$ DWORK, LDDWOR, INFO )
C
C RELEASE 4.0, WGS COPYRIGHT 1999.
C
C PURPOSE
C
C To solve a system of equations in Hessenberg form with one
C offdiagonal and one right-hand side.
C
C ARGUMENTS
C
C Mode Parameters
C
C RC CHARACTER*1
C Indicates processing by columns or rows, as follows:
C = 'R': Row transformations are applied;
C = 'C': Column transformations are applied.
C
C UL CHARACTER*1
C Indicates whether AB is upper or lower Hessenberg matrix,
C as follows:
C = 'U': AB is upper Hessenberg;
C = 'L': AB is lower Hessenberg.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The order of the matrix A. M >= 0.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,M)
C The leading M-by-M part of this array must contain a
C matrix A in Hessenberg form.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,M).
C
C LAMBDA (input) DOUBLE PRECISION
C This variable must contain the value to be added to the
C diagonal elements of A.
C
C D (input/output) DOUBLE PRECISION array, dimension (M)
C On entry, this array must contain the right-hand side
C vector of the Hessenberg system.
C On exit, if INFO = 0, this array contains the solution
C vector of the Hessenberg system.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used to test for near singularity of
C the triangular factor R of the Hessenberg matrix. A matrix
C whose estimated condition number is less than 1/TOL is
C considered to be nonsingular.
C
C Workspace
C
C IWORK INTEGER array, dimension (M)
C
C DWORK DOUBLE PRECISION array, dimension (LDDWOR,M+3)
C The leading M-by-M part of this array is used for
C computing the triangular factor of the QR decomposition
C of the Hessenberg matrix. The remaining 3*M elements are
C used as workspace for the computation of the reciprocal
C condition estimate.
C
C LDDWOR INTEGER
C The leading dimension of array DWORK. LDDWOR >= MAX(1,M).
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C = 1: if the Hessenberg matrix is (numerically) singular.
C That is, its estimated reciprocal condition number
C is less than or equal to TOL.
C
C NUMERICAL ASPECTS
C
C None.
C
C CONTRIBUTORS
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Aug. 1997.
C Supersedes Release 2.0 routine SB04BY by M. Vanbegin, and
C P. Van Dooren, Philips Research Laboratory, Brussels, Belgium.
C
C REVISIONS
C
C -
C
C Note that RC, UL, M and LDA must be such that the value of the
C LOGICAL variable OK in the following statement is true.
C
C OK = ( ( UL.EQ.'U' ) .OR. ( UL.EQ.'u' ) .OR.
C ( UL.EQ.'L' ) .OR. ( UL.EQ.'l' ) )
C .AND.
C ( ( RC.EQ.'R' ) .OR. ( RC.EQ.'r' ) .OR.
C ( RC.EQ.'C' ) .OR. ( RC.EQ.'c' ) )
C .AND.
C ( M.GE.0 )
C .AND.
C ( LDA.GE.MAX( 1, M ) )
C .AND.
C ( LDDWOR.GE.MAX( 1, M ) )
C
C KEYWORDS
C
C Hessenberg form, orthogonal transformation, real Schur form,
C Sylvester equation.
C
C ******************************************************************
C
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
C .. Scalar Arguments ..
CHARACTER RC, UL
INTEGER INFO, LDA, LDDWOR, M
DOUBLE PRECISION LAMBDA, TOL
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), D(*), DWORK(LDDWOR,*)
C .. Local Scalars ..
CHARACTER TRANS
INTEGER J, J1, MJ
DOUBLE PRECISION C, R, RCOND, S
C .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DLARTG, DROT, DTRCON, DTRSV
C .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
C .. Executable Statements ..
C
INFO = 0
C
C For speed, no tests on the input scalar arguments are made.
C Quick return if possible.
C
IF ( M.EQ.0 )
$ RETURN
C
IF ( LSAME( UL, 'U' ) ) THEN
C
DO 20 J = 1, M
CALL DCOPY( MIN( J+1, M ), A(1,J), 1, DWORK(1,J), 1 )
DWORK(J,J) = DWORK(J,J) + LAMBDA
20 CONTINUE
C
IF ( LSAME( RC, 'R' ) ) THEN
TRANS = 'N'
C
C A is an upper Hessenberg matrix, row transformations.
C
DO 40 J = 1, M - 1
MJ = M - J
IF ( DWORK(J+1,J).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J), DWORK(J+1,J), C, S, R )
DWORK(J,J) = R
DWORK(J+1,J) = ZERO
CALL DROT( MJ, DWORK(J,J+1), LDDWOR, DWORK(J+1,J+1),
$ LDDWOR, C, S )
CALL DROT( 1, D(J), 1, D(J+1), 1, C, S )
END IF
40 CONTINUE
C
ELSE
TRANS = 'T'
C
C A is an upper Hessenberg matrix, column transformations.
C
DO 60 J = 1, M - 1
MJ = M - J
IF ( DWORK(MJ+1,MJ).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ+1), DWORK(MJ+1,MJ), C, S,
$ R )
DWORK(MJ+1,MJ+1) = R
DWORK(MJ+1,MJ) = ZERO
CALL DROT( MJ, DWORK(1,MJ+1), 1, DWORK(1,MJ), 1, C,
$ S )
CALL DROT( 1, D(MJ+1), 1, D(MJ), 1, C, S )
END IF
60 CONTINUE
C
END IF
ELSE
C
DO 80 J = 1, M
J1 = MAX( J - 1, 1 )
CALL DCOPY( M-J1+1, A(J1,J), 1, DWORK(J1,J), 1 )
DWORK(J,J) = DWORK(J,J) + LAMBDA
80 CONTINUE
C
IF ( LSAME( RC, 'R' ) ) THEN
TRANS = 'N'
C
C A is a lower Hessenberg matrix, row transformations.
C
DO 100 J = 1, M - 1
MJ = M - J
IF ( DWORK(MJ,MJ+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(MJ+1,MJ+1), DWORK(MJ,MJ+1), C, S,
$ R )
DWORK(MJ+1,MJ+1) = R
DWORK(MJ,MJ+1) = ZERO
CALL DROT( MJ, DWORK(MJ+1,1), LDDWOR, DWORK(MJ,1),
$ LDDWOR, C, S )
CALL DROT( 1, D(MJ+1), 1, D(MJ), 1, C, S )
END IF
100 CONTINUE
C
ELSE
TRANS = 'T'
C
C A is a lower Hessenberg matrix, column transformations.
C
DO 120 J = 1, M - 1
MJ = M - J
IF ( DWORK(J,J+1).NE.ZERO ) THEN
CALL DLARTG( DWORK(J,J), DWORK(J,J+1), C, S, R )
DWORK(J,J) = R
DWORK(J,J+1) = ZERO
CALL DROT( MJ, DWORK(J+1,J), 1, DWORK(J+1,J+1), 1, C,
$ S )
C
CALL DROT( 1, D(J), 1, D(J+1), 1, C, S )
END IF
120 CONTINUE
C
END IF
END IF
C
CALL DTRCON( '1-norm', UL, 'Non-unit', M, DWORK, LDDWOR, RCOND,
$ DWORK(1,M+1), IWORK, INFO )
IF ( RCOND.LE.TOL ) THEN
INFO = 1
ELSE
CALL DTRSV( UL, TRANS, 'Non-unit', M, DWORK, LDDWOR, D, 1 )
END IF
C
RETURN
C *** Last line of SB04NY ***
END
|
