LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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clatrz.f
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1*> \brief \b CLATRZ factors an upper trapezoidal matrix by means of unitary transformations.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> Download CLATRZ + dependencies
9*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clatrz.f">
10*> [TGZ]</a>
11*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/clatrz.f">
12*> [ZIP]</a>
13*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/clatrz.f">
14*> [TXT]</a>
15*
16* Definition:
17* ===========
18*
19* SUBROUTINE CLATRZ( M, N, L, A, LDA, TAU, WORK )
20*
21* .. Scalar Arguments ..
22* INTEGER L, LDA, M, N
23* ..
24* .. Array Arguments ..
25* COMPLEX A( LDA, * ), TAU( * ), WORK( * )
26* ..
27*
28*
29*> \par Purpose:
30* =============
31*>
32*> \verbatim
33*>
34*> CLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix
35*> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z by means
36*> of unitary transformations, where Z is an (M+L)-by-(M+L) unitary
37*> matrix and, R and A1 are M-by-M upper triangular matrices.
38*> \endverbatim
39*
40* Arguments:
41* ==========
42*
43*> \param[in] M
44*> \verbatim
45*> M is INTEGER
46*> The number of rows of the matrix A. M >= 0.
47*> \endverbatim
48*>
49*> \param[in] N
50*> \verbatim
51*> N is INTEGER
52*> The number of columns of the matrix A. N >= 0.
53*> \endverbatim
54*>
55*> \param[in] L
56*> \verbatim
57*> L is INTEGER
58*> The number of columns of the matrix A containing the
59*> meaningful part of the Householder vectors. N-M >= L >= 0.
60*> \endverbatim
61*>
62*> \param[in,out] A
63*> \verbatim
64*> A is COMPLEX array, dimension (LDA,N)
65*> On entry, the leading M-by-N upper trapezoidal part of the
66*> array A must contain the matrix to be factorized.
67*> On exit, the leading M-by-M upper triangular part of A
68*> contains the upper triangular matrix R, and elements N-L+1 to
69*> N of the first M rows of A, with the array TAU, represent the
70*> unitary matrix Z as a product of M elementary reflectors.
71*> \endverbatim
72*>
73*> \param[in] LDA
74*> \verbatim
75*> LDA is INTEGER
76*> The leading dimension of the array A. LDA >= max(1,M).
77*> \endverbatim
78*>
79*> \param[out] TAU
80*> \verbatim
81*> TAU is COMPLEX array, dimension (M)
82*> The scalar factors of the elementary reflectors.
83*> \endverbatim
84*>
85*> \param[out] WORK
86*> \verbatim
87*> WORK is COMPLEX array, dimension (M)
88*> \endverbatim
89*
90* Authors:
91* ========
92*
93*> \author Univ. of Tennessee
94*> \author Univ. of California Berkeley
95*> \author Univ. of Colorado Denver
96*> \author NAG Ltd.
97*
98*> \ingroup latrz
99*
100*> \par Contributors:
101* ==================
102*>
103*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
104*
105*> \par Further Details:
106* =====================
107*>
108*> \verbatim
109*>
110*> The factorization is obtained by Householder's method. The kth
111*> transformation matrix, Z( k ), which is used to introduce zeros into
112*> the ( m - k + 1 )th row of A, is given in the form
113*>
114*> Z( k ) = ( I 0 ),
115*> ( 0 T( k ) )
116*>
117*> where
118*>
119*> T( k ) = I - tau*u( k )*u( k )**H, u( k ) = ( 1 ),
120*> ( 0 )
121*> ( z( k ) )
122*>
123*> tau is a scalar and z( k ) is an l element vector. tau and z( k )
124*> are chosen to annihilate the elements of the kth row of A2.
125*>
126*> The scalar tau is returned in the kth element of TAU and the vector
127*> u( k ) in the kth row of A2, such that the elements of z( k ) are
128*> in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
129*> the upper triangular part of A1.
130*>
131*> Z is given by
132*>
133*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
134*> \endverbatim
135*>
136* =====================================================================
137 SUBROUTINE clatrz( M, N, L, A, LDA, TAU, WORK )
138*
139* -- LAPACK computational routine --
140* -- LAPACK is a software package provided by Univ. of Tennessee, --
141* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
142*
143* .. Scalar Arguments ..
144 INTEGER L, LDA, M, N
145* ..
146* .. Array Arguments ..
147 COMPLEX A( LDA, * ), TAU( * ), WORK( * )
148* ..
149*
150* =====================================================================
151*
152* .. Parameters ..
153 COMPLEX ZERO
154 parameter( zero = ( 0.0e+0, 0.0e+0 ) )
155* ..
156* .. Local Scalars ..
157 INTEGER I
158 COMPLEX ALPHA
159* ..
160* .. External Subroutines ..
161 EXTERNAL clacgv, clarfg, clarz
162* ..
163* .. Intrinsic Functions ..
164 INTRINSIC conjg
165* ..
166* .. Executable Statements ..
167*
168* Quick return if possible
169*
170 IF( m.EQ.0 ) THEN
171 RETURN
172 ELSE IF( m.EQ.n ) THEN
173 DO 10 i = 1, n
174 tau( i ) = zero
175 10 CONTINUE
176 RETURN
177 END IF
178*
179 DO 20 i = m, 1, -1
180*
181* Generate elementary reflector H(i) to annihilate
182* [ A(i,i) A(i,n-l+1:n) ]
183*
184 CALL clacgv( l, a( i, n-l+1 ), lda )
185 alpha = conjg( a( i, i ) )
186 CALL clarfg( l+1, alpha, a( i, n-l+1 ), lda, tau( i ) )
187 tau( i ) = conjg( tau( i ) )
188*
189* Apply H(i) to A(1:i-1,i:n) from the right
190*
191 CALL clarz( 'Right', i-1, n-i+1, l, a( i, n-l+1 ), lda,
192 $ conjg( tau( i ) ), a( 1, i ), lda, work )
193 a( i, i ) = conjg( alpha )
194*
195 20 CONTINUE
196*
197 RETURN
198*
199* End of CLATRZ
200*
201 END
subroutine clacgv(n, x, incx)
CLACGV conjugates a complex vector.
Definition clacgv.f:72
subroutine clarfg(n, alpha, x, incx, tau)
CLARFG generates an elementary reflector (Householder matrix).
Definition clarfg.f:104
subroutine clarz(side, m, n, l, v, incv, tau, c, ldc, work)
CLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
Definition clarz.f:145
subroutine clatrz(m, n, l, a, lda, tau, work)
CLATRZ factors an upper trapezoidal matrix by means of unitary transformations.
Definition clatrz.f:138