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
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
|
*> \brief \b CLAROR
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition
* ==========
*
* SUBROUTINE CLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO )
*
* .. Scalar Arguments ..
* CHARACTER INIT, SIDE
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* INTEGER ISEED( 4 )
* COMPLEX A( LDA, * ), X( * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> CLAROR pre- or post-multiplies an M by N matrix A by a random
*> unitary matrix U, overwriting A. A may optionally be
*> initialized to the identity matrix before multiplying by U.
*> U is generated using the method of G.W. Stewart
*> ( SIAM J. Numer. Anal. 17, 1980, pp. 403-409 ).
*> (BLAS-2 version)
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \param[in] SIDE
*> \verbatim
*> SIDE is CHARACTER*1
*> SIDE specifies whether A is multiplied on the left or right
*> by U.
*> SIDE = 'L' Multiply A on the left (premultiply) by U
*> SIDE = 'R' Multiply A on the right (postmultiply) by UC> SIDE = 'C' Multiply A on the left by U and the right by UC> SIDE = 'T' Multiply A on the left by U and the right by U'
*> Not modified.
*> \endverbatim
*>
*> \param[in] INIT
*> \verbatim
*> INIT is CHARACTER*1
*> INIT specifies whether or not A should be initialized to
*> the identity matrix.
*> INIT = 'I' Initialize A to (a section of) the
*> identity matrix before applying U.
*> INIT = 'N' No initialization. Apply U to the
*> input matrix A.
*>
*> INIT = 'I' may be used to generate square (i.e., unitary)
*> or rectangular orthogonal matrices (orthogonality being
*> in the sense of CDOTC):
*>
*> For square matrices, M=N, and SIDE many be either 'L' or
*> 'R'; the rows will be orthogonal to each other, as will the
*> columns.
*> For rectangular matrices where M < N, SIDE = 'R' will
*> produce a dense matrix whose rows will be orthogonal and
*> whose columns will not, while SIDE = 'L' will produce a
*> matrix whose rows will be orthogonal, and whose first M
*> columns will be orthogonal, the remaining columns being
*> zero.
*> For matrices where M > N, just use the previous
*> explaination, interchanging 'L' and 'R' and "rows" and
*> "columns".
*>
*> Not modified.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> Number of rows of A. Not modified.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> Number of columns of A. Not modified.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension ( LDA, N )
*> Input and output array. Overwritten by U A ( if SIDE = 'L' )
*> or by A U ( if SIDE = 'R' )
*> or by U A U* ( if SIDE = 'C')
*> or by U A U' ( if SIDE = 'T') on exit.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> Leading dimension of A. Must be at least MAX ( 1, M ).
*> Not modified.
*> \endverbatim
*>
*> \param[in,out] ISEED
*> \verbatim
*> ISEED is INTEGER array, dimension ( 4 )
*> On entry ISEED specifies the seed of the random number
*> generator. The array elements should be between 0 and 4095;
*> if not they will be reduced mod 4096. Also, ISEED(4) must
*> be odd. The random number generator uses a linear
*> congruential sequence limited to small integers, and so
*> should produce machine independent random numbers. The
*> values of ISEED are changed on exit, and can be used in the
*> next call to CLAROR to continue the same random number
*> sequence.
*> Modified.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is COMPLEX array, dimension ( 3*MAX( M, N ) )
*> Workspace. Of length:
*> 2*M + N if SIDE = 'L',
*> 2*N + M if SIDE = 'R',
*> 3*N if SIDE = 'C' or 'T'.
*> Modified.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> An error flag. It is set to:
*> 0 if no error.
*> 1 if CLARND returned a bad random number (installation
*> problem)
*> -1 if SIDE is not L, R, C, or T.
*> -3 if M is negative.
*> -4 if N is negative or if SIDE is C or T and N is not equal
*> to M.
*> -6 if LDA is less than M.
*> \endverbatim
*>
*
* Authors
* =======
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex_matgen
*
* =====================================================================
SUBROUTINE CLAROR( SIDE, INIT, M, N, A, LDA, ISEED, X, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* -- 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 ..
CHARACTER INIT, SIDE
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER ISEED( 4 )
COMPLEX A( LDA, * ), X( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TOOSML
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0,
$ TOOSML = 1.0E-20 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
INTEGER IROW, ITYPE, IXFRM, J, JCOL, KBEG, NXFRM
REAL FACTOR, XABS, XNORM
COMPLEX CSIGN, XNORMS
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SCNRM2
COMPLEX CLARND
EXTERNAL LSAME, SCNRM2, CLARND
* ..
* .. External Subroutines ..
EXTERNAL CGEMV, CGERC, CLACGV, CLASET, CSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CMPLX, CONJG
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.EQ.0 .OR. M.EQ.0 )
$ RETURN
*
ITYPE = 0
IF( LSAME( SIDE, 'L' ) ) THEN
ITYPE = 1
ELSE IF( LSAME( SIDE, 'R' ) ) THEN
ITYPE = 2
ELSE IF( LSAME( SIDE, 'C' ) ) THEN
ITYPE = 3
ELSE IF( LSAME( SIDE, 'T' ) ) THEN
ITYPE = 4
END IF
*
* Check for argument errors.
*
IF( ITYPE.EQ.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 .OR. ( ITYPE.EQ.3 .AND. N.NE.M ) ) THEN
INFO = -4
ELSE IF( LDA.LT.M ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CLAROR', -INFO )
RETURN
END IF
*
IF( ITYPE.EQ.1 ) THEN
NXFRM = M
ELSE
NXFRM = N
END IF
*
* Initialize A to the identity matrix if desired
*
IF( LSAME( INIT, 'I' ) )
$ CALL CLASET( 'Full', M, N, CZERO, CONE, A, LDA )
*
* If no rotation possible, still multiply by
* a random complex number from the circle |x| = 1
*
* 2) Compute Rotation by computing Householder
* Transformations H(2), H(3), ..., H(n). Note that the
* order in which they are computed is irrelevant.
*
DO 40 J = 1, NXFRM
X( J ) = CZERO
40 CONTINUE
*
DO 60 IXFRM = 2, NXFRM
KBEG = NXFRM - IXFRM + 1
*
* Generate independent normal( 0, 1 ) random numbers
*
DO 50 J = KBEG, NXFRM
X( J ) = CLARND( 3, ISEED )
50 CONTINUE
*
* Generate a Householder transformation from the random vector X
*
XNORM = SCNRM2( IXFRM, X( KBEG ), 1 )
XABS = ABS( X( KBEG ) )
IF( XABS.NE.CZERO ) THEN
CSIGN = X( KBEG ) / XABS
ELSE
CSIGN = CONE
END IF
XNORMS = CSIGN*XNORM
X( NXFRM+KBEG ) = -CSIGN
FACTOR = XNORM*( XNORM+XABS )
IF( ABS( FACTOR ).LT.TOOSML ) THEN
INFO = 1
CALL XERBLA( 'CLAROR', -INFO )
RETURN
ELSE
FACTOR = ONE / FACTOR
END IF
X( KBEG ) = X( KBEG ) + XNORMS
*
* Apply Householder transformation to A
*
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.3 .OR. ITYPE.EQ.4 ) THEN
*
* Apply H(k) on the left of A
*
CALL CGEMV( 'C', IXFRM, N, CONE, A( KBEG, 1 ), LDA,
$ X( KBEG ), 1, CZERO, X( 2*NXFRM+1 ), 1 )
CALL CGERC( IXFRM, N, -CMPLX( FACTOR ), X( KBEG ), 1,
$ X( 2*NXFRM+1 ), 1, A( KBEG, 1 ), LDA )
*
END IF
*
IF( ITYPE.GE.2 .AND. ITYPE.LE.4 ) THEN
*
* Apply H(k)* (or H(k)') on the right of A
*
IF( ITYPE.EQ.4 ) THEN
CALL CLACGV( IXFRM, X( KBEG ), 1 )
END IF
*
CALL CGEMV( 'N', M, IXFRM, CONE, A( 1, KBEG ), LDA,
$ X( KBEG ), 1, CZERO, X( 2*NXFRM+1 ), 1 )
CALL CGERC( M, IXFRM, -CMPLX( FACTOR ), X( 2*NXFRM+1 ), 1,
$ X( KBEG ), 1, A( 1, KBEG ), LDA )
*
END IF
60 CONTINUE
*
X( 1 ) = CLARND( 3, ISEED )
XABS = ABS( X( 1 ) )
IF( XABS.NE.ZERO ) THEN
CSIGN = X( 1 ) / XABS
ELSE
CSIGN = CONE
END IF
X( 2*NXFRM ) = CSIGN
*
* Scale the matrix A by D.
*
IF( ITYPE.EQ.1 .OR. ITYPE.EQ.3 .OR. ITYPE.EQ.4 ) THEN
DO 70 IROW = 1, M
CALL CSCAL( N, CONJG( X( NXFRM+IROW ) ), A( IROW, 1 ), LDA )
70 CONTINUE
END IF
*
IF( ITYPE.EQ.2 .OR. ITYPE.EQ.3 ) THEN
DO 80 JCOL = 1, N
CALL CSCAL( M, X( NXFRM+JCOL ), A( 1, JCOL ), 1 )
80 CONTINUE
END IF
*
IF( ITYPE.EQ.4 ) THEN
DO 90 JCOL = 1, N
CALL CSCAL( M, CONJG( X( NXFRM+JCOL ) ), A( 1, JCOL ), 1 )
90 CONTINUE
END IF
RETURN
*
* End of CLAROR
*
END
|
