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
|
*> \brief \b SLAQP2 computes a QR factorization with column pivoting of the matrix block.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLAQP2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaqp2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaqp2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaqp2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
* WORK )
*
* .. Scalar Arguments ..
* INTEGER LDA, M, N, OFFSET
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* REAL A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAQP2 computes a QR factorization with column pivoting of
*> the block A(OFFSET+1:M,1:N).
*> The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
*> \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] OFFSET
*> \verbatim
*> OFFSET is INTEGER
*> The number of rows of the matrix A that must be pivoted
*> but no factorized. OFFSET >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
*> the triangular factor obtained; the elements in block
*> A(OFFSET+1:M,1:N) below the diagonal, together with the
*> array TAU, represent the orthogonal matrix Q as a product of
*> elementary reflectors. Block A(1:OFFSET,1:N) has been
*> accordingly pivoted, but no factorized.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*> JPVT is INTEGER array, dimension (N)
*> On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
*> to the front of A*P (a leading column); if JPVT(i) = 0,
*> the i-th column of A is a free column.
*> On exit, if JPVT(i) = k, then the i-th column of A*P
*> was the k-th column of A.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is REAL array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[in,out] VN1
*> \verbatim
*> VN1 is REAL array, dimension (N)
*> The vector with the partial column norms.
*> \endverbatim
*>
*> \param[in,out] VN2
*> \verbatim
*> VN2 is REAL array, dimension (N)
*> The vector with the exact column norms.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (N)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup realOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
*> X. Sun, Computer Science Dept., Duke University, USA
*> \n
*> Partial column norm updating strategy modified on April 2011
*> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
*> University of Zagreb, Croatia.
*
*> \par References:
* ================
*>
*> LAPACK Working Note 176
*
*> \htmlonly
*> <a href="http://www.netlib.org/lapack/lawnspdf/lawn176.pdf">[PDF]</a>
*> \endhtmlonly
*
* =====================================================================
SUBROUTINE SLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
$ WORK )
*
* -- 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 ..
INTEGER LDA, M, N, OFFSET
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
REAL A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, ITEMP, J, MN, OFFPI, PVT
REAL AII, TEMP, TEMP2, TOL3Z
* ..
* .. External Subroutines ..
EXTERNAL SLARF, SLARFG, SSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. External Functions ..
INTEGER ISAMAX
REAL SLAMCH, SNRM2
EXTERNAL ISAMAX, SLAMCH, SNRM2
* ..
* .. Executable Statements ..
*
MN = MIN( M-OFFSET, N )
TOL3Z = SQRT(SLAMCH('Epsilon'))
*
* Compute factorization.
*
DO 20 I = 1, MN
*
OFFPI = OFFSET + I
*
* Determine ith pivot column and swap if necessary.
*
PVT = ( I-1 ) + ISAMAX( N-I+1, VN1( I ), 1 )
*
IF( PVT.NE.I ) THEN
CALL SSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
ITEMP = JPVT( PVT )
JPVT( PVT ) = JPVT( I )
JPVT( I ) = ITEMP
VN1( PVT ) = VN1( I )
VN2( PVT ) = VN2( I )
END IF
*
* Generate elementary reflector H(i).
*
IF( OFFPI.LT.M ) THEN
CALL SLARFG( M-OFFPI+1, A( OFFPI, I ), A( OFFPI+1, I ), 1,
$ TAU( I ) )
ELSE
CALL SLARFG( 1, A( M, I ), A( M, I ), 1, TAU( I ) )
END IF
*
IF( I.LT.N ) THEN
*
* Apply H(i)**T to A(offset+i:m,i+1:n) from the left.
*
AII = A( OFFPI, I )
A( OFFPI, I ) = ONE
CALL SLARF( 'Left', M-OFFPI+1, N-I, A( OFFPI, I ), 1,
$ TAU( I ), A( OFFPI, I+1 ), LDA, WORK( 1 ) )
A( OFFPI, I ) = AII
END IF
*
* Update partial column norms.
*
DO 10 J = I + 1, N
IF( VN1( J ).NE.ZERO ) THEN
*
* NOTE: The following 4 lines follow from the analysis in
* Lapack Working Note 176.
*
TEMP = ONE - ( ABS( A( OFFPI, J ) ) / VN1( J ) )**2
TEMP = MAX( TEMP, ZERO )
TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
IF( TEMP2 .LE. TOL3Z ) THEN
IF( OFFPI.LT.M ) THEN
VN1( J ) = SNRM2( M-OFFPI, A( OFFPI+1, J ), 1 )
VN2( J ) = VN1( J )
ELSE
VN1( J ) = ZERO
VN2( J ) = ZERO
END IF
ELSE
VN1( J ) = VN1( J )*SQRT( TEMP )
END IF
END IF
10 CONTINUE
*
20 CONTINUE
*
RETURN
*
* End of SLAQP2
*
END
|
