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
|
SUBROUTINE DLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
$ WORK )
*
* -- LAPACK auxiliary routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
INTEGER LDA, M, N, OFFSET
* ..
* .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* DLAQP2 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (min(M,N))
* The scalar factors of the elementary reflectors.
*
* VN1 (input/output) DOUBLE PRECISION array, dimension (N)
* The vector with the partial column norms.
*
* VN2 (input/output) DOUBLE PRECISION array, dimension (N)
* The vector with the exact column norms.
*
* WORK (workspace) DOUBLE PRECISION 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
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, ITEMP, J, MN, OFFPI, PVT
DOUBLE PRECISION AII, TEMP, TEMP2
* ..
* .. External Subroutines ..
EXTERNAL DLARF, DLARFG, DSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DNRM2
EXTERNAL IDAMAX, DNRM2
* ..
* .. Executable Statements ..
*
MN = MIN( M-OFFSET, N )
*
* Compute factorization.
*
DO 20 I = 1, MN
*
OFFPI = OFFSET + I
*
* Determine ith pivot column and swap if necessary.
*
PVT = ( I-1 ) + IDAMAX( N-I+1, VN1( I ), 1 )
*
IF( PVT.NE.I ) THEN
CALL DSWAP( 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 DLARFG( M-OFFPI+1, A( OFFPI, I ), A( OFFPI+1, I ), 1,
$ TAU( I ) )
ELSE
CALL DLARFG( 1, A( M, I ), A( M, I ), 1, TAU( I ) )
END IF
*
IF( I.LT.N ) THEN
*
* Apply H(i)' to A(offset+i:m,i+1:n) from the left.
*
AII = A( OFFPI, I )
A( OFFPI, I ) = ONE
CALL DLARF( '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
TEMP = ONE - ( ABS( A( OFFPI, J ) ) / VN1( J ) )**2
TEMP = MAX( TEMP, ZERO )
TEMP2 = ONE + 0.05D0*TEMP*( VN1( J ) / VN2( J ) )**2
IF( TEMP2.EQ.ONE ) THEN
IF( OFFPI.LT.M ) THEN
VN1( J ) = DNRM2( 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 DLAQP2
*
END
|
