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
|
*> \brief \b CTZRQF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CTZRQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ctzrqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ctzrqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ctzrqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CTZRQF( M, N, A, LDA, TAU, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* COMPLEX A( LDA, * ), TAU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine CTZRZF.
*>
*> CTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
*> to upper triangular form by means of unitary transformations.
*>
*> The upper trapezoidal matrix A is factored as
*>
*> A = ( R 0 ) * Z,
*>
*> where Z is an N-by-N unitary matrix and R is an M-by-M upper
*> triangular matrix.
*> \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 >= M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> On entry, the leading M-by-N upper trapezoidal part of the
*> array A must contain the matrix to be factorized.
*> On exit, the leading M-by-M upper triangular part of A
*> contains the upper triangular matrix R, and elements M+1 to
*> N of the first M rows of A, with the array TAU, represent the
*> unitary matrix Z as a product of M elementary reflectors.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is COMPLEX array, dimension (M)
*> The scalar factors of the elementary reflectors.
*> \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 November 2011
*
*> \ingroup complexOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The factorization is obtained by Householder's method. The kth
*> transformation matrix, Z( k ), whose conjugate transpose is used to
*> introduce zeros into the (m - k + 1)th row of A, is given in the form
*>
*> Z( k ) = ( I 0 ),
*> ( 0 T( k ) )
*>
*> where
*>
*> T( k ) = I - tau*u( k )*u( k )**H, u( k ) = ( 1 ),
*> ( 0 )
*> ( z( k ) )
*>
*> tau is a scalar and z( k ) is an ( n - m ) element vector.
*> tau and z( k ) are chosen to annihilate the elements of the kth row
*> of X.
*>
*> The scalar tau is returned in the kth element of TAU and the vector
*> u( k ) in the kth row of A, such that the elements of z( k ) are
*> in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
*> the upper triangular part of A.
*>
*> Z is given by
*>
*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CTZRQF( M, N, A, LDA, TAU, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), TAU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX CONE, CZERO
PARAMETER ( CONE = ( 1.0E+0, 0.0E+0 ),
$ CZERO = ( 0.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, K, M1
COMPLEX ALPHA
* ..
* .. Intrinsic Functions ..
INTRINSIC CONJG, MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL CAXPY, CCOPY, CGEMV, CGERC, CLACGV, CLARFG,
$ XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CTZRQF', -INFO )
RETURN
END IF
*
* Perform the factorization.
*
IF( M.EQ.0 )
$ RETURN
IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = CZERO
10 CONTINUE
ELSE
M1 = MIN( M+1, N )
DO 20 K = M, 1, -1
*
* Use a Householder reflection to zero the kth row of A.
* First set up the reflection.
*
A( K, K ) = CONJG( A( K, K ) )
CALL CLACGV( N-M, A( K, M1 ), LDA )
ALPHA = A( K, K )
CALL CLARFG( N-M+1, ALPHA, A( K, M1 ), LDA, TAU( K ) )
A( K, K ) = ALPHA
TAU( K ) = CONJG( TAU( K ) )
*
IF( TAU( K ).NE.CZERO .AND. K.GT.1 ) THEN
*
* We now perform the operation A := A*P( k )**H.
*
* Use the first ( k - 1 ) elements of TAU to store a( k ),
* where a( k ) consists of the first ( k - 1 ) elements of
* the kth column of A. Also let B denote the first
* ( k - 1 ) rows of the last ( n - m ) columns of A.
*
CALL CCOPY( K-1, A( 1, K ), 1, TAU, 1 )
*
* Form w = a( k ) + B*z( k ) in TAU.
*
CALL CGEMV( 'No transpose', K-1, N-M, CONE, A( 1, M1 ),
$ LDA, A( K, M1 ), LDA, CONE, TAU, 1 )
*
* Now form a( k ) := a( k ) - conjg(tau)*w
* and B := B - conjg(tau)*w*z( k )**H.
*
CALL CAXPY( K-1, -CONJG( TAU( K ) ), TAU, 1, A( 1, K ),
$ 1 )
CALL CGERC( K-1, N-M, -CONJG( TAU( K ) ), TAU, 1,
$ A( K, M1 ), LDA, A( 1, M1 ), LDA )
END IF
20 CONTINUE
END IF
*
RETURN
*
* End of CTZRQF
*
END
|
