LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
dgelqt3.f
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1 *> \brief \b DGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * RECURSIVE SUBROUTINE DGELQT3( M, N, A, LDA, T, LDT, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, M, N, LDT
25 * ..
26 * .. Array Arguments ..
27 * DOUBLE PRECISION A( LDA, * ), T( LDT, * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> DGELQT3 recursively computes a LQ factorization of a real M-by-N
37 *> matrix A, using the compact WY representation of Q.
38 *>
39 *> Based on the algorithm of Elmroth and Gustavson,
40 *> IBM J. Res. Develop. Vol 44 No. 4 July 2000.
41 *> \endverbatim
42 *
43 * Arguments:
44 * ==========
45 *
46 *> \param[in] M
47 *> \verbatim
48 *> M is INTEGER
49 *> The number of rows of the matrix A. M =< N.
50 *> \endverbatim
51 *>
52 *> \param[in] N
53 *> \verbatim
54 *> N is INTEGER
55 *> The number of columns of the matrix A. N >= 0.
56 *> \endverbatim
57 *>
58 *> \param[in,out] A
59 *> \verbatim
60 *> A is DOUBLE PRECISION array, dimension (LDA,N)
61 *> On entry, the real M-by-N matrix A. On exit, the elements on and
62 *> below the diagonal contain the N-by-N lower triangular matrix L; the
63 *> elements above the diagonal are the rows of V. See below for
64 *> further details.
65 *> \endverbatim
66 *>
67 *> \param[in] LDA
68 *> \verbatim
69 *> LDA is INTEGER
70 *> The leading dimension of the array A. LDA >= max(1,M).
71 *> \endverbatim
72 *>
73 *> \param[out] T
74 *> \verbatim
75 *> T is DOUBLE PRECISION array, dimension (LDT,N)
76 *> The N-by-N upper triangular factor of the block reflector.
77 *> The elements on and above the diagonal contain the block
78 *> reflector T; the elements below the diagonal are not used.
79 *> See below for further details.
80 *> \endverbatim
81 *>
82 *> \param[in] LDT
83 *> \verbatim
84 *> LDT is INTEGER
85 *> The leading dimension of the array T. LDT >= max(1,N).
86 *> \endverbatim
87 *>
88 *> \param[out] INFO
89 *> \verbatim
90 *> INFO is INTEGER
91 *> = 0: successful exit
92 *> < 0: if INFO = -i, the i-th argument had an illegal value
93 *> \endverbatim
94 *
95 * Authors:
96 * ========
97 *
98 *> \author Univ. of Tennessee
99 *> \author Univ. of California Berkeley
100 *> \author Univ. of Colorado Denver
101 *> \author NAG Ltd.
102 *
103 *> \date November 2017
104 *
105 *> \ingroup doubleGEcomputational
106 *
107 *> \par Further Details:
108 * =====================
109 *>
110 *> \verbatim
111 *>
112 *> The matrix V stores the elementary reflectors H(i) in the i-th row
113 *> above the diagonal. For example, if M=5 and N=3, the matrix V is
114 *>
115 *> V = ( 1 v1 v1 v1 v1 )
116 *> ( 1 v2 v2 v2 )
117 *> ( 1 v3 v3 v3 )
118 *>
119 *>
120 *> where the vi's represent the vectors which define H(i), which are returned
121 *> in the matrix A. The 1's along the diagonal of V are not stored in A. The
122 *> block reflector H is then given by
123 *>
124 *> H = I - V * T * V**T
125 *>
126 *> where V**T is the transpose of V.
127 *>
128 *> For details of the algorithm, see Elmroth and Gustavson (cited above).
129 *> \endverbatim
130 *>
131 * =====================================================================
132  RECURSIVE SUBROUTINE dgelqt3( M, N, A, LDA, T, LDT, INFO )
133 *
134 * -- LAPACK computational routine (version 3.8.0) --
135 * -- LAPACK is a software package provided by Univ. of Tennessee, --
136 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
137 * November 2017
138 *
139 * .. Scalar Arguments ..
140  INTEGER INFO, LDA, M, N, LDT
141 * ..
142 * .. Array Arguments ..
143  DOUBLE PRECISION A( lda, * ), T( ldt, * )
144 * ..
145 *
146 * =====================================================================
147 *
148 * .. Parameters ..
149  DOUBLE PRECISION ONE
150  parameter( one = 1.0d+00 )
151 * ..
152 * .. Local Scalars ..
153  INTEGER I, I1, J, J1, M1, M2, IINFO
154 * ..
155 * .. External Subroutines ..
156  EXTERNAL dlarfg, dtrmm, dgemm, xerbla
157 * ..
158 * .. Executable Statements ..
159 *
160  info = 0
161  IF( m .LT. 0 ) THEN
162  info = -1
163  ELSE IF( n .LT. m ) THEN
164  info = -2
165  ELSE IF( lda .LT. max( 1, m ) ) THEN
166  info = -4
167  ELSE IF( ldt .LT. max( 1, m ) ) THEN
168  info = -6
169  END IF
170  IF( info.NE.0 ) THEN
171  CALL xerbla( 'DGELQT3', -info )
172  RETURN
173  END IF
174 *
175  IF( m.EQ.1 ) THEN
176 *
177 * Compute Householder transform when N=1
178 *
179  CALL dlarfg( n, a, a( 1, min( 2, n ) ), lda, t )
180 *
181  ELSE
182 *
183 * Otherwise, split A into blocks...
184 *
185  m1 = m/2
186  m2 = m-m1
187  i1 = min( m1+1, m )
188  j1 = min( m+1, n )
189 *
190 * Compute A(1:M1,1:N) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
191 *
192  CALL dgelqt3( m1, n, a, lda, t, ldt, iinfo )
193 *
194 * Compute A(J1:M,1:N) = Q1^H A(J1:M,1:N) [workspace: T(1:N1,J1:N)]
195 *
196  DO i=1,m2
197  DO j=1,m1
198  t( i+m1, j ) = a( i+m1, j )
199  END DO
200  END DO
201  CALL dtrmm( 'R', 'U', 'T', 'U', m2, m1, one,
202  & a, lda, t( i1, 1 ), ldt )
203 *
204  CALL dgemm( 'N', 'T', m2, m1, n-m1, one, a( i1, i1 ), lda,
205  & a( 1, i1 ), lda, one, t( i1, 1 ), ldt)
206 *
207  CALL dtrmm( 'R', 'U', 'N', 'N', m2, m1, one,
208  & t, ldt, t( i1, 1 ), ldt )
209 *
210  CALL dgemm( 'N', 'N', m2, n-m1, m1, -one, t( i1, 1 ), ldt,
211  & a( 1, i1 ), lda, one, a( i1, i1 ), lda )
212 *
213  CALL dtrmm( 'R', 'U', 'N', 'U', m2, m1 , one,
214  & a, lda, t( i1, 1 ), ldt )
215 *
216  DO i=1,m2
217  DO j=1,m1
218  a( i+m1, j ) = a( i+m1, j ) - t( i+m1, j )
219  t( i+m1, j )=0
220  END DO
221  END DO
222 *
223 * Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
224 *
225  CALL dgelqt3( m2, n-m1, a( i1, i1 ), lda,
226  & t( i1, i1 ), ldt, iinfo )
227 *
228 * Compute T3 = T(J1:N1,1:N) = -T1 Y1^H Y2 T2
229 *
230  DO i=1,m2
231  DO j=1,m1
232  t( j, i+m1 ) = (a( j, i+m1 ))
233  END DO
234  END DO
235 *
236  CALL dtrmm( 'R', 'U', 'T', 'U', m1, m2, one,
237  & a( i1, i1 ), lda, t( 1, i1 ), ldt )
238 *
239  CALL dgemm( 'N', 'T', m1, m2, n-m, one, a( 1, j1 ), lda,
240  & a( i1, j1 ), lda, one, t( 1, i1 ), ldt )
241 *
242  CALL dtrmm( 'L', 'U', 'N', 'N', m1, m2, -one, t, ldt,
243  & t( 1, i1 ), ldt )
244 *
245  CALL dtrmm( 'R', 'U', 'N', 'N', m1, m2, one,
246  & t( i1, i1 ), ldt, t( 1, i1 ), ldt )
247 *
248 *
249 *
250 * Y = (Y1,Y2); L = [ L1 0 ]; T = [T1 T3]
251 * [ A(1:N1,J1:N) L2 ] [ 0 T2]
252 *
253  END IF
254 *
255  RETURN
256 *
257 * End of DGELQT3
258 *
259  END
subroutine dtrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRMM
Definition: dtrmm.f:179
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:189
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:108
recursive subroutine dgelqt3(M, N, A, LDA, T, LDT, INFO)
DGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact...
Definition: dgelqt3.f:133