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
|
SUBROUTINE CLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
$ WORK )
*
* -- LAPACK auxiliary routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* -- April 2011 --
*
* .. Scalar Arguments ..
INTEGER LDA, M, N, OFFSET
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
REAL VN1( * ), VN2( * )
COMPLEX A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* CLAQP2 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.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* OFFSET (input) INTEGER
* The number of rows of the matrix A that must be pivoted
* but no factorized. OFFSET >= 0.
*
* A (input/output) COMPLEX 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* JPVT (input/output) 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.
*
* TAU (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors.
*
* VN1 (input/output) REAL array, dimension (N)
* The vector with the partial column norms.
*
* VN2 (input/output) REAL array, dimension (N)
* The vector with the exact column norms.
*
* WORK (workspace) COMPLEX array, dimension (N)
*
* Further Details
* ===============
*
* Based on contributions by
* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
* X. Sun, Computer Science Dept., Duke University, USA
*
* Partial column norm updating strategy modified by
* Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
* University of Zagreb, Croatia.
* -- April 2011 --
* For more details see LAPACK Working Note 176.
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
COMPLEX CONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0,
$ CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, ITEMP, J, MN, OFFPI, PVT
REAL TEMP, TEMP2, TOL3Z
COMPLEX AII
* ..
* .. External Subroutines ..
EXTERNAL CLARF, CLARFG, CSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX, MIN, SQRT
* ..
* .. External Functions ..
INTEGER ISAMAX
REAL SCNRM2, SLAMCH
EXTERNAL ISAMAX, SCNRM2, SLAMCH
* ..
* .. 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 CSWAP( 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 CLARFG( M-OFFPI+1, A( OFFPI, I ), A( OFFPI+1, I ), 1,
$ TAU( I ) )
ELSE
CALL CLARFG( 1, A( M, I ), A( M, I ), 1, TAU( I ) )
END IF
*
IF( I.LT.N ) THEN
*
* Apply H(i)**H to A(offset+i:m,i+1:n) from the left.
*
AII = A( OFFPI, I )
A( OFFPI, I ) = CONE
CALL CLARF( 'Left', M-OFFPI+1, N-I, A( OFFPI, I ), 1,
$ CONJG( 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 ) = SCNRM2( 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 CLAQP2
*
END
|
