LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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sgelq2.f
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1*> \brief \b SGELQ2 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*
8*> \htmlonly
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgelq2.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SGELQ2( M, N, A, LDA, TAU, WORK, INFO )
22*
23* .. Scalar Arguments ..
24* INTEGER INFO, LDA, M, N
25* ..
26* .. Array Arguments ..
27* REAL A( LDA, * ), TAU( * ), WORK( * )
28* ..
29*
30*
31*> \par Purpose:
32* =============
33*>
34*> \verbatim
35*>
36*> SGELQ2 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 REAL 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 REAL 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 REAL 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 gelq2
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 sgelq2( 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 REAL A( LDA, * ), TAU( * ), WORK( * )
139* ..
140*
141* =====================================================================
142*
143* .. Parameters ..
144 REAL ONE
145 parameter( one = 1.0e+0 )
146* ..
147* .. Local Scalars ..
148 INTEGER I, K
149 REAL AII
150* ..
151* .. External Subroutines ..
152 EXTERNAL slarf, slarfg, 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( 'SGELQ2', -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 slarfg( 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 slarf( '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 SGELQ2
196*
197 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine sgelq2(m, n, a, lda, tau, work, info)
SGELQ2 computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
Definition sgelq2.f:129
subroutine slarf(side, m, n, v, incv, tau, c, ldc, work)
SLARF applies an elementary reflector to a general rectangular matrix.
Definition slarf.f:124
subroutine slarfg(n, alpha, x, incx, tau)
SLARFG generates an elementary reflector (Householder matrix).
Definition slarfg.f:106