LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
dgelq2.f
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1 *> \brief \b DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
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 * SUBROUTINE DGELQ2( M, N, A, LDA, TAU, WORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, 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 *> DGELQ2 computes an LQ factorization of a real m-by-n matrix A:
37 *>
38 *> A = ( L 0 ) * Q
39 *>
40 *> where:
41 *>
42 *> Q is a n-by-n orthogonal matrix;
43 *> L is a lower-triangular m-by-m matrix;
44 *> 0 is a m-by-(n-m) zero matrix, if m < n.
45 *>
46 *> \endverbatim
47 *
48 * Arguments:
49 * ==========
50 *
51 *> \param[in] M
52 *> \verbatim
53 *> M is INTEGER
54 *> The number of rows of the matrix A. M >= 0.
55 *> \endverbatim
56 *>
57 *> \param[in] N
58 *> \verbatim
59 *> N is INTEGER
60 *> The number of columns of the matrix A. N >= 0.
61 *> \endverbatim
62 *>
63 *> \param[in,out] A
64 *> \verbatim
65 *> A is DOUBLE PRECISION array, dimension (LDA,N)
66 *> On entry, the m by n matrix A.
67 *> On exit, the elements on and below the diagonal of the array
68 *> contain the m by min(m,n) lower trapezoidal matrix L (L is
69 *> lower triangular if m <= n); the elements above the diagonal,
70 *> with the array TAU, represent the orthogonal matrix Q as a
71 *> product of elementary reflectors (see Further Details).
72 *> \endverbatim
73 *>
74 *> \param[in] LDA
75 *> \verbatim
76 *> LDA is INTEGER
77 *> The leading dimension of the array A. LDA >= max(1,M).
78 *> \endverbatim
79 *>
80 *> \param[out] TAU
81 *> \verbatim
82 *> TAU is DOUBLE PRECISION array, dimension (min(M,N))
83 *> The scalar factors of the elementary reflectors (see Further
84 *> Details).
85 *> \endverbatim
86 *>
87 *> \param[out] WORK
88 *> \verbatim
89 *> WORK is DOUBLE PRECISION array, dimension (M)
90 *> \endverbatim
91 *>
92 *> \param[out] INFO
93 *> \verbatim
94 *> INFO is INTEGER
95 *> = 0: successful exit
96 *> < 0: if INFO = -i, the i-th argument had an illegal value
97 *> \endverbatim
98 *
99 * Authors:
100 * ========
101 *
102 *> \author Univ. of Tennessee
103 *> \author Univ. of California Berkeley
104 *> \author Univ. of Colorado Denver
105 *> \author NAG Ltd.
106 *
107 *> \ingroup doubleGEcomputational
108 *
109 *> \par Further Details:
110 * =====================
111 *>
112 *> \verbatim
113 *>
114 *> The matrix Q is represented as a product of elementary reflectors
115 *>
116 *> Q = H(k) . . . H(2) H(1), where k = min(m,n).
117 *>
118 *> Each H(i) has the form
119 *>
120 *> H(i) = I - tau * v * v**T
121 *>
122 *> where tau is a real scalar, and v is a real vector with
123 *> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
124 *> and tau in TAU(i).
125 *> \endverbatim
126 *>
127 * =====================================================================
128  SUBROUTINE dgelq2( M, N, A, LDA, TAU, WORK, INFO )
129 *
130 * -- LAPACK computational routine --
131 * -- LAPACK is a software package provided by Univ. of Tennessee, --
132 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
133 *
134 * .. Scalar Arguments ..
135  INTEGER INFO, LDA, M, N
136 * ..
137 * .. Array Arguments ..
138  DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
139 * ..
140 *
141 * =====================================================================
142 *
143 * .. Parameters ..
144  DOUBLE PRECISION ONE
145  parameter( one = 1.0d+0 )
146 * ..
147 * .. Local Scalars ..
148  INTEGER I, K
149  DOUBLE PRECISION AII
150 * ..
151 * .. External Subroutines ..
152  EXTERNAL dlarf, dlarfg, xerbla
153 * ..
154 * .. Intrinsic Functions ..
155  INTRINSIC max, min
156 * ..
157 * .. Executable Statements ..
158 *
159 * Test the input arguments
160 *
161  info = 0
162  IF( m.LT.0 ) THEN
163  info = -1
164  ELSE IF( n.LT.0 ) THEN
165  info = -2
166  ELSE IF( lda.LT.max( 1, m ) ) THEN
167  info = -4
168  END IF
169  IF( info.NE.0 ) THEN
170  CALL xerbla( 'DGELQ2', -info )
171  RETURN
172  END IF
173 *
174  k = min( m, n )
175 *
176  DO 10 i = 1, k
177 *
178 * Generate elementary reflector H(i) to annihilate A(i,i+1:n)
179 *
180  CALL dlarfg( n-i+1, a( i, i ), a( i, min( i+1, n ) ), lda,
181  $ tau( i ) )
182  IF( i.LT.m ) THEN
183 *
184 * Apply H(i) to A(i+1:m,i:n) from the right
185 *
186  aii = a( i, i )
187  a( i, i ) = one
188  CALL dlarf( 'Right', m-i, n-i+1, a( i, i ), lda, tau( i ),
189  $ a( i+1, i ), lda, work )
190  a( i, i ) = aii
191  END IF
192  10 CONTINUE
193  RETURN
194 *
195 * End of DGELQ2
196 *
197  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dgelq2(M, N, A, LDA, TAU, WORK, INFO)
DGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
Definition: dgelq2.f:129
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:124
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106