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
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
|
*> \brief \b SLAROT
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SLAROT( LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT,
* XRIGHT )
*
* .. Scalar Arguments ..
* LOGICAL LLEFT, LRIGHT, LROWS
* INTEGER LDA, NL
* REAL C, S, XLEFT, XRIGHT
* ..
* .. Array Arguments ..
* REAL A( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAROT applies a (Givens) rotation to two adjacent rows or
*> columns, where one element of the first and/or last column/row
*> for use on matrices stored in some format other than GE, so
*> that elements of the matrix may be used or modified for which
*> no array element is provided.
*>
*> One example is a symmetric matrix in SB format (bandwidth=4), for
*> which UPLO='L': Two adjacent rows will have the format:
*>
*> row j: C> C> C> C> C> . . . .
*> row j+1: C> C> C> C> C> . . . .
*>
*> '*' indicates elements for which storage is provided,
*> '.' indicates elements for which no storage is provided, but
*> are not necessarily zero; their values are determined by
*> symmetry. ' ' indicates elements which are necessarily zero,
*> and have no storage provided.
*>
*> Those columns which have two '*'s can be handled by SROT.
*> Those columns which have no '*'s can be ignored, since as long
*> as the Givens rotations are carefully applied to preserve
*> symmetry, their values are determined.
*> Those columns which have one '*' have to be handled separately,
*> by using separate variables "p" and "q":
*>
*> row j: C> C> C> C> C> p . . .
*> row j+1: q C> C> C> C> C> . . . .
*>
*> The element p would have to be set correctly, then that column
*> is rotated, setting p to its new value. The next call to
*> SLAROT would rotate columns j and j+1, using p, and restore
*> symmetry. The element q would start out being zero, and be
*> made non-zero by the rotation. Later, rotations would presumably
*> be chosen to zero q out.
*>
*> Typical Calling Sequences: rotating the i-th and (i+1)-st rows.
*> ------- ------- ---------
*>
*> General dense matrix:
*>
*> CALL SLAROT(.TRUE.,.FALSE.,.FALSE., N, C,S,
*> A(i,1),LDA, DUMMY, DUMMY)
*>
*> General banded matrix in GB format:
*>
*> j = MAX(1, i-KL )
*> NL = MIN( N, i+KU+1 ) + 1-j
*> CALL SLAROT( .TRUE., i-KL.GE.1, i+KU.LT.N, NL, C,S,
*> A(KU+i+1-j,j),LDA-1, XLEFT, XRIGHT )
*>
*> [ note that i+1-j is just MIN(i,KL+1) ]
*>
*> Symmetric banded matrix in SY format, bandwidth K,
*> lower triangle only:
*>
*> j = MAX(1, i-K )
*> NL = MIN( K+1, i ) + 1
*> CALL SLAROT( .TRUE., i-K.GE.1, .TRUE., NL, C,S,
*> A(i,j), LDA, XLEFT, XRIGHT )
*>
*> Same, but upper triangle only:
*>
*> NL = MIN( K+1, N-i ) + 1
*> CALL SLAROT( .TRUE., .TRUE., i+K.LT.N, NL, C,S,
*> A(i,i), LDA, XLEFT, XRIGHT )
*>
*> Symmetric banded matrix in SB format, bandwidth K,
*> lower triangle only:
*>
*> [ same as for SY, except:]
*> . . . .
*> A(i+1-j,j), LDA-1, XLEFT, XRIGHT )
*>
*> [ note that i+1-j is just MIN(i,K+1) ]
*>
*> Same, but upper triangle only:
*> . . .
*> A(K+1,i), LDA-1, XLEFT, XRIGHT )
*>
*> Rotating columns is just the transpose of rotating rows, except
*> for GB and SB: (rotating columns i and i+1)
*>
*> GB:
*> j = MAX(1, i-KU )
*> NL = MIN( N, i+KL+1 ) + 1-j
*> CALL SLAROT( .TRUE., i-KU.GE.1, i+KL.LT.N, NL, C,S,
*> A(KU+j+1-i,i),LDA-1, XTOP, XBOTTM )
*>
*> [note that KU+j+1-i is just MAX(1,KU+2-i)]
*>
*> SB: (upper triangle)
*>
*> . . . . . .
*> A(K+j+1-i,i),LDA-1, XTOP, XBOTTM )
*>
*> SB: (lower triangle)
*>
*> . . . . . .
*> A(1,i),LDA-1, XTOP, XBOTTM )
*> \endverbatim
*
* Arguments:
* ==========
*
*> \verbatim
*> LROWS - LOGICAL
*> If .TRUE., then SLAROT will rotate two rows. If .FALSE.,
*> then it will rotate two columns.
*> Not modified.
*>
*> LLEFT - LOGICAL
*> If .TRUE., then XLEFT will be used instead of the
*> corresponding element of A for the first element in the
*> second row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.)
*> If .FALSE., then the corresponding element of A will be
*> used.
*> Not modified.
*>
*> LRIGHT - LOGICAL
*> If .TRUE., then XRIGHT will be used instead of the
*> corresponding element of A for the last element in the
*> first row (if LROWS=.FALSE.) or column (if LROWS=.TRUE.) If
*> .FALSE., then the corresponding element of A will be used.
*> Not modified.
*>
*> NL - INTEGER
*> The length of the rows (if LROWS=.TRUE.) or columns (if
*> LROWS=.FALSE.) to be rotated. If XLEFT and/or XRIGHT are
*> used, the columns/rows they are in should be included in
*> NL, e.g., if LLEFT = LRIGHT = .TRUE., then NL must be at
*> least 2. The number of rows/columns to be rotated
*> exclusive of those involving XLEFT and/or XRIGHT may
*> not be negative, i.e., NL minus how many of LLEFT and
*> LRIGHT are .TRUE. must be at least zero; if not, XERBLA
*> will be called.
*> Not modified.
*>
*> C, S - REAL
*> Specify the Givens rotation to be applied. If LROWS is
*> true, then the matrix ( c s )
*> (-s c ) is applied from the left;
*> if false, then the transpose thereof is applied from the
*> right. For a Givens rotation, C**2 + S**2 should be 1,
*> but this is not checked.
*> Not modified.
*>
*> A - REAL array.
*> The array containing the rows/columns to be rotated. The
*> first element of A should be the upper left element to
*> be rotated.
*> Read and modified.
*>
*> LDA - INTEGER
*> The "effective" leading dimension of A. If A contains
*> a matrix stored in GE or SY format, then this is just
*> the leading dimension of A as dimensioned in the calling
*> routine. If A contains a matrix stored in band (GB or SB)
*> format, then this should be *one less* than the leading
*> dimension used in the calling routine. Thus, if
*> A were dimensioned A(LDA,*) in SLAROT, then A(1,j) would
*> be the j-th element in the first of the two rows
*> to be rotated, and A(2,j) would be the j-th in the second,
*> regardless of how the array may be stored in the calling
*> routine. [A cannot, however, actually be dimensioned thus,
*> since for band format, the row number may exceed LDA, which
*> is not legal FORTRAN.]
*> If LROWS=.TRUE., then LDA must be at least 1, otherwise
*> it must be at least NL minus the number of .TRUE. values
*> in XLEFT and XRIGHT.
*> Not modified.
*>
*> XLEFT - REAL
*> If LLEFT is .TRUE., then XLEFT will be used and modified
*> instead of A(2,1) (if LROWS=.TRUE.) or A(1,2)
*> (if LROWS=.FALSE.).
*> Read and modified.
*>
*> XRIGHT - REAL
*> If LRIGHT is .TRUE., then XRIGHT will be used and modified
*> instead of A(1,NL) (if LROWS=.TRUE.) or A(NL,1)
*> (if LROWS=.FALSE.).
*> Read and modified.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup real_matgen
*
* =====================================================================
SUBROUTINE SLAROT( LROWS, LLEFT, LRIGHT, NL, C, S, A, LDA, XLEFT,
$ XRIGHT )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
LOGICAL LLEFT, LRIGHT, LROWS
INTEGER LDA, NL
REAL C, S, XLEFT, XRIGHT
* ..
* .. Array Arguments ..
REAL A( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER IINC, INEXT, IX, IY, IYT, NT
* ..
* .. Local Arrays ..
REAL XT( 2 ), YT( 2 )
* ..
* .. External Subroutines ..
EXTERNAL SROT, XERBLA
* ..
* .. Executable Statements ..
*
* Set up indices, arrays for ends
*
IF( LROWS ) THEN
IINC = LDA
INEXT = 1
ELSE
IINC = 1
INEXT = LDA
END IF
*
IF( LLEFT ) THEN
NT = 1
IX = 1 + IINC
IY = 2 + LDA
XT( 1 ) = A( 1 )
YT( 1 ) = XLEFT
ELSE
NT = 0
IX = 1
IY = 1 + INEXT
END IF
*
IF( LRIGHT ) THEN
IYT = 1 + INEXT + ( NL-1 )*IINC
NT = NT + 1
XT( NT ) = XRIGHT
YT( NT ) = A( IYT )
END IF
*
* Check for errors
*
IF( NL.LT.NT ) THEN
CALL XERBLA( 'SLAROT', 4 )
RETURN
END IF
IF( LDA.LE.0 .OR. ( .NOT.LROWS .AND. LDA.LT.NL-NT ) ) THEN
CALL XERBLA( 'SLAROT', 8 )
RETURN
END IF
*
* Rotate
*
CALL SROT( NL-NT, A( IX ), IINC, A( IY ), IINC, C, S )
CALL SROT( NT, XT, 1, YT, 1, C, S )
*
* Stuff values back into XLEFT, XRIGHT, etc.
*
IF( LLEFT ) THEN
A( 1 ) = XT( 1 )
XLEFT = YT( 1 )
END IF
*
IF( LRIGHT ) THEN
XRIGHT = XT( NT )
A( IYT ) = YT( NT )
END IF
*
RETURN
*
* End of SLAROT
*
END
|
