1: SUBROUTINE DLAQP2( M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2,
2: $ WORK )
3: *
4: * -- LAPACK auxiliary routine (version 3.2.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * June 2010
8: *
9: * .. Scalar Arguments ..
10: INTEGER LDA, M, N, OFFSET
11: * ..
12: * .. Array Arguments ..
13: INTEGER JPVT( * )
14: DOUBLE PRECISION A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
15: $ WORK( * )
16: * ..
17: *
18: * Purpose
19: * =======
20: *
21: * DLAQP2 computes a QR factorization with column pivoting of
22: * the block A(OFFSET+1:M,1:N).
23: * The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
24: *
25: * Arguments
26: * =========
27: *
28: * M (input) INTEGER
29: * The number of rows of the matrix A. M >= 0.
30: *
31: * N (input) INTEGER
32: * The number of columns of the matrix A. N >= 0.
33: *
34: * OFFSET (input) INTEGER
35: * The number of rows of the matrix A that must be pivoted
36: * but no factorized. OFFSET >= 0.
37: *
38: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
39: * On entry, the M-by-N matrix A.
40: * On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
41: * the triangular factor obtained; the elements in block
42: * A(OFFSET+1:M,1:N) below the diagonal, together with the
43: * array TAU, represent the orthogonal matrix Q as a product of
44: * elementary reflectors. Block A(1:OFFSET,1:N) has been
45: * accordingly pivoted, but no factorized.
46: *
47: * LDA (input) INTEGER
48: * The leading dimension of the array A. LDA >= max(1,M).
49: *
50: * JPVT (input/output) INTEGER array, dimension (N)
51: * On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
52: * to the front of A*P (a leading column); if JPVT(i) = 0,
53: * the i-th column of A is a free column.
54: * On exit, if JPVT(i) = k, then the i-th column of A*P
55: * was the k-th column of A.
56: *
57: * TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
58: * The scalar factors of the elementary reflectors.
59: *
60: * VN1 (input/output) DOUBLE PRECISION array, dimension (N)
61: * The vector with the partial column norms.
62: *
63: * VN2 (input/output) DOUBLE PRECISION array, dimension (N)
64: * The vector with the exact column norms.
65: *
66: * WORK (workspace) DOUBLE PRECISION array, dimension (N)
67: *
68: * Further Details
69: * ===============
70: *
71: * Based on contributions by
72: * G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
73: * X. Sun, Computer Science Dept., Duke University, USA
74: *
75: * Partial column norm updating strategy modified by
76: * Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
77: * University of Zagreb, Croatia.
78: * June 2010
79: * For more details see LAPACK Working Note 176.
80: * =====================================================================
81: *
82: * .. Parameters ..
83: DOUBLE PRECISION ZERO, ONE
84: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
85: * ..
86: * .. Local Scalars ..
87: INTEGER I, ITEMP, J, MN, OFFPI, PVT
88: DOUBLE PRECISION AII, TEMP, TEMP2, TOL3Z
89: * ..
90: * .. External Subroutines ..
91: EXTERNAL DLARF, DLARFG, DSWAP
92: * ..
93: * .. Intrinsic Functions ..
94: INTRINSIC ABS, MAX, MIN, SQRT
95: * ..
96: * .. External Functions ..
97: INTEGER IDAMAX
98: DOUBLE PRECISION DLAMCH, DNRM2
99: EXTERNAL IDAMAX, DLAMCH, DNRM2
100: * ..
101: * .. Executable Statements ..
102: *
103: MN = MIN( M-OFFSET, N )
104: TOL3Z = SQRT(DLAMCH('Epsilon'))
105: *
106: * Compute factorization.
107: *
108: DO 20 I = 1, MN
109: *
110: OFFPI = OFFSET + I
111: *
112: * Determine ith pivot column and swap if necessary.
113: *
114: PVT = ( I-1 ) + IDAMAX( N-I+1, VN1( I ), 1 )
115: *
116: IF( PVT.NE.I ) THEN
117: CALL DSWAP( M, A( 1, PVT ), 1, A( 1, I ), 1 )
118: ITEMP = JPVT( PVT )
119: JPVT( PVT ) = JPVT( I )
120: JPVT( I ) = ITEMP
121: VN1( PVT ) = VN1( I )
122: VN2( PVT ) = VN2( I )
123: END IF
124: *
125: * Generate elementary reflector H(i).
126: *
127: IF( OFFPI.LT.M ) THEN
128: CALL DLARFG( M-OFFPI+1, A( OFFPI, I ), A( OFFPI+1, I ), 1,
129: $ TAU( I ) )
130: ELSE
131: CALL DLARFG( 1, A( M, I ), A( M, I ), 1, TAU( I ) )
132: END IF
133: *
134: IF( I.LE.N ) THEN
135: *
136: * Apply H(i)' to A(offset+i:m,i+1:n) from the left.
137: *
138: AII = A( OFFPI, I )
139: A( OFFPI, I ) = ONE
140: CALL DLARF( 'Left', M-OFFPI+1, N-I, A( OFFPI, I ), 1,
141: $ TAU( I ), A( OFFPI, I+1 ), LDA, WORK( 1 ) )
142: A( OFFPI, I ) = AII
143: END IF
144: *
145: * Update partial column norms.
146: *
147: DO 10 J = I + 1, N
148: IF( VN1( J ).NE.ZERO ) THEN
149: *
150: * NOTE: The following 4 lines follow from the analysis in
151: * Lapack Working Note 176.
152: *
153: TEMP = ONE - ( ABS( A( OFFPI, J ) ) / VN1( J ) )**2
154: TEMP = MAX( TEMP, ZERO )
155: TEMP2 = TEMP*( VN1( J ) / VN2( J ) )**2
156: IF( TEMP2 .LE. TOL3Z ) THEN
157: IF( OFFPI.LT.M ) THEN
158: VN1( J ) = DNRM2( M-OFFPI, A( OFFPI+1, J ), 1 )
159: VN2( J ) = VN1( J )
160: ELSE
161: VN1( J ) = ZERO
162: VN2( J ) = ZERO
163: END IF
164: ELSE
165: VN1( J ) = VN1( J )*SQRT( TEMP )
166: END IF
167: END IF
168: 10 CONTINUE
169: *
170: 20 CONTINUE
171: *
172: RETURN
173: *
174: * End of DLAQP2
175: *
176: END
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