LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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dlatrz.f
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1*> \brief \b DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> Download DLATRZ + dependencies
9*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrz.f">
10*> [TGZ]</a>
11*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrz.f">
12*> [ZIP]</a>
13*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrz.f">
14*> [TXT]</a>
15*
16* Definition:
17* ===========
18*
19* SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
20*
21* .. Scalar Arguments ..
22* INTEGER L, LDA, M, N
23* ..
24* .. Array Arguments ..
25* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
26* ..
27*
28*
29*> \par Purpose:
30* =============
31*>
32*> \verbatim
33*>
34*> DLATRZ factors the M-by-(M+L) real 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 orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
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 DOUBLE PRECISION 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*> orthogonal 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 DOUBLE PRECISION array, dimension (M)
82*> The scalar factors of the elementary reflectors.
83*> \endverbatim
84*>
85*> \param[out] WORK
86*> \verbatim
87*> WORK is DOUBLE PRECISION 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 )**T, 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 dlatrz( 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 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
148* ..
149*
150* =====================================================================
151*
152* .. Parameters ..
153 DOUBLE PRECISION ZERO
154 parameter( zero = 0.0d+0 )
155* ..
156* .. Local Scalars ..
157 INTEGER I
158* ..
159* .. External Subroutines ..
160 EXTERNAL dlarfg, dlarz
161* ..
162* .. Executable Statements ..
163*
164* Test the input arguments
165*
166* Quick return if possible
167*
168 IF( m.EQ.0 ) THEN
169 RETURN
170 ELSE IF( m.EQ.n ) THEN
171 DO 10 i = 1, n
172 tau( i ) = zero
173 10 CONTINUE
174 RETURN
175 END IF
176*
177 DO 20 i = m, 1, -1
178*
179* Generate elementary reflector H(i) to annihilate
180* [ A(i,i) A(i,n-l+1:n) ]
181*
182 CALL dlarfg( l+1, a( i, i ), a( i, n-l+1 ), lda, tau( i ) )
183*
184* Apply H(i) to A(1:i-1,i:n) from the right
185*
186 CALL dlarz( 'Right', i-1, n-i+1, l, a( i, n-l+1 ), lda,
187 $ tau( i ), a( 1, i ), lda, work )
188*
189 20 CONTINUE
190*
191 RETURN
192*
193* End of DLATRZ
194*
195 END
subroutine dlarfg(n, alpha, x, incx, tau)
DLARFG generates an elementary reflector (Householder matrix).
Definition dlarfg.f:104
subroutine dlarz(side, m, n, l, v, incv, tau, c, ldc, work)
DLARZ applies an elementary reflector (as returned by stzrzf) to a general matrix.
Definition dlarz.f:143
subroutine dlatrz(m, n, l, a, lda, tau, work)
DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
Definition dlatrz.f:138