LAPACK  3.8.0 LAPACK: Linear Algebra PACKage
dlatrz.f
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1 *> \brief \b DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER L, LDA, M, N
25 * ..
26 * .. Array Arguments ..
27 * DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
37 *> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means
38 *> of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal
39 *> matrix and, R and A1 are M-by-M upper triangular matrices.
40 *> \endverbatim
41 *
42 * Arguments:
43 * ==========
44 *
45 *> \param[in] M
46 *> \verbatim
47 *> M is INTEGER
48 *> The number of rows of the matrix A. M >= 0.
49 *> \endverbatim
50 *>
51 *> \param[in] N
52 *> \verbatim
53 *> N is INTEGER
54 *> The number of columns of the matrix A. N >= 0.
55 *> \endverbatim
56 *>
57 *> \param[in] L
58 *> \verbatim
59 *> L is INTEGER
60 *> The number of columns of the matrix A containing the
61 *> meaningful part of the Householder vectors. N-M >= L >= 0.
62 *> \endverbatim
63 *>
64 *> \param[in,out] A
65 *> \verbatim
66 *> A is DOUBLE PRECISION array, dimension (LDA,N)
67 *> On entry, the leading M-by-N upper trapezoidal part of the
68 *> array A must contain the matrix to be factorized.
69 *> On exit, the leading M-by-M upper triangular part of A
70 *> contains the upper triangular matrix R, and elements N-L+1 to
71 *> N of the first M rows of A, with the array TAU, represent the
72 *> orthogonal matrix Z as a product of M elementary reflectors.
73 *> \endverbatim
74 *>
75 *> \param[in] LDA
76 *> \verbatim
77 *> LDA is INTEGER
78 *> The leading dimension of the array A. LDA >= max(1,M).
79 *> \endverbatim
80 *>
81 *> \param[out] TAU
82 *> \verbatim
83 *> TAU is DOUBLE PRECISION array, dimension (M)
84 *> The scalar factors of the elementary reflectors.
85 *> \endverbatim
86 *>
87 *> \param[out] WORK
88 *> \verbatim
89 *> WORK is DOUBLE PRECISION array, dimension (M)
90 *> \endverbatim
91 *
92 * Authors:
93 * ========
94 *
95 *> \author Univ. of Tennessee
96 *> \author Univ. of California Berkeley
97 *> \author Univ. of Colorado Denver
98 *> \author NAG Ltd.
99 *
100 *> \date December 2016
101 *
102 *> \ingroup doubleOTHERcomputational
103 *
104 *> \par Contributors:
105 * ==================
106 *>
107 *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
108 *
109 *> \par Further Details:
110 * =====================
111 *>
112 *> \verbatim
113 *>
114 *> The factorization is obtained by Householder's method. The kth
115 *> transformation matrix, Z( k ), which is used to introduce zeros into
116 *> the ( m - k + 1 )th row of A, is given in the form
117 *>
118 *> Z( k ) = ( I 0 ),
119 *> ( 0 T( k ) )
120 *>
121 *> where
122 *>
123 *> T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ),
124 *> ( 0 )
125 *> ( z( k ) )
126 *>
127 *> tau is a scalar and z( k ) is an l element vector. tau and z( k )
128 *> are chosen to annihilate the elements of the kth row of A2.
129 *>
130 *> The scalar tau is returned in the kth element of TAU and the vector
131 *> u( k ) in the kth row of A2, such that the elements of z( k ) are
132 *> in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
133 *> the upper triangular part of A1.
134 *>
135 *> Z is given by
136 *>
137 *> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
138 *> \endverbatim
139 *>
140 * =====================================================================
141  SUBROUTINE dlatrz( M, N, L, A, LDA, TAU, WORK )
142 *
143 * -- LAPACK computational routine (version 3.7.0) --
144 * -- LAPACK is a software package provided by Univ. of Tennessee, --
145 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146 * December 2016
147 *
148 * .. Scalar Arguments ..
149  INTEGER L, LDA, M, N
150 * ..
151 * .. Array Arguments ..
152  DOUBLE PRECISION A( lda, * ), TAU( * ), WORK( * )
153 * ..
154 *
155 * =====================================================================
156 *
157 * .. Parameters ..
158  DOUBLE PRECISION ZERO
159  parameter( zero = 0.0d+0 )
160 * ..
161 * .. Local Scalars ..
162  INTEGER I
163 * ..
164 * .. External Subroutines ..
165  EXTERNAL dlarfg, dlarz
166 * ..
167 * .. Executable Statements ..
168 *
169 * Test the input arguments
170 *
171 * Quick return if possible
172 *
173  IF( m.EQ.0 ) THEN
174  RETURN
175  ELSE IF( m.EQ.n ) THEN
176  DO 10 i = 1, n
177  tau( i ) = zero
178  10 CONTINUE
179  RETURN
180  END IF
181 *
182  DO 20 i = m, 1, -1
183 *
184 * Generate elementary reflector H(i) to annihilate
185 * [ A(i,i) A(i,n-l+1:n) ]
186 *
187  CALL dlarfg( l+1, a( i, i ), a( i, n-l+1 ), lda, tau( i ) )
188 *
189 * Apply H(i) to A(1:i-1,i:n) from the right
190 *
191  CALL dlarz( 'Right', i-1, n-i+1, l, a( i, n-l+1 ), lda,
192  \$ tau( i ), a( 1, i ), lda, work )
193 *
194  20 CONTINUE
195 *
196  RETURN
197 *
198 * End of DLATRZ
199 *
200  END
subroutine dlatrz(M, N, L, A, LDA, TAU, WORK)
DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
Definition: dlatrz.f:142
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:147
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:108