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
|
*> \brief \b SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGEQRT2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgeqrt2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgeqrt2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgeqrt2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGEQRT2( M, N, A, LDA, T, LDT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDT, M, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), T( LDT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGEQRT2 computes a QR factorization of a real M-by-N matrix A,
*> using the compact WY representation of Q.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= N.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> On entry, the real M-by-N matrix A. On exit, the elements on and
*> above the diagonal contain the N-by-N upper triangular matrix R; the
*> elements below the diagonal are the columns of V. See below for
*> further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is REAL array, dimension (LDT,N)
*> The N-by-N upper triangular factor of the block reflector.
*> The elements on and above the diagonal contain the block
*> reflector T; the elements below the diagonal are not used.
*> See below for further details.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max(1,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup realGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrix V stores the elementary reflectors H(i) in the i-th column
*> below the diagonal. For example, if M=5 and N=3, the matrix V is
*>
*> V = ( 1 )
*> ( v1 1 )
*> ( v1 v2 1 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> where the vi's represent the vectors which define H(i), which are returned
*> in the matrix A. The 1's along the diagonal of V are not stored in A. The
*> block reflector H is then given by
*>
*> H = I - V * T * V**T
*>
*> where V**T is the transpose of V.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SGEQRT2( M, N, A, LDA, T, LDT, INFO )
*
* -- LAPACK computational 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 INFO, LDA, LDT, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), T( LDT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER( ONE = 1.0, ZERO = 0.0 )
* ..
* .. Local Scalars ..
INTEGER I, K
REAL AII, ALPHA
* ..
* .. External Subroutines ..
EXTERNAL SLARFG, SGEMV, SGER, STRMV, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGEQRT2', -INFO )
RETURN
END IF
*
K = MIN( M, N )
*
DO I = 1, K
*
* Generate elem. refl. H(i) to annihilate A(i+1:m,i), tau(I) -> T(I,1)
*
CALL SLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ T( I, 1 ) )
IF( I.LT.N ) THEN
*
* Apply H(i) to A(I:M,I+1:N) from the left
*
AII = A( I, I )
A( I, I ) = ONE
*
* W(1:N-I) := A(I:M,I+1:N)^H * A(I:M,I) [W = T(:,N)]
*
CALL SGEMV( 'T',M-I+1, N-I, ONE, A( I, I+1 ), LDA,
$ A( I, I ), 1, ZERO, T( 1, N ), 1 )
*
* A(I:M,I+1:N) = A(I:m,I+1:N) + alpha*A(I:M,I)*W(1:N-1)^H
*
ALPHA = -(T( I, 1 ))
CALL SGER( M-I+1, N-I, ALPHA, A( I, I ), 1,
$ T( 1, N ), 1, A( I, I+1 ), LDA )
A( I, I ) = AII
END IF
END DO
*
DO I = 2, N
AII = A( I, I )
A( I, I ) = ONE
*
* T(1:I-1,I) := alpha * A(I:M,1:I-1)**T * A(I:M,I)
*
ALPHA = -T( I, 1 )
CALL SGEMV( 'T', M-I+1, I-1, ALPHA, A( I, 1 ), LDA,
$ A( I, I ), 1, ZERO, T( 1, I ), 1 )
A( I, I ) = AII
*
* T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
*
CALL STRMV( 'U', 'N', 'N', I-1, T, LDT, T( 1, I ), 1 )
*
* T(I,I) = tau(I)
*
T( I, I ) = T( I, 1 )
T( I, 1) = ZERO
END DO
*
* End of SGEQRT2
*
END
|
