LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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sgeql2.f
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1*> \brief \b SGEQL2 computes the QL 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 SGEQL2( 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*> SGEQL2 computes a QL factorization of a real m by n matrix A:
37*> A = Q * L.
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,out] A
56*> \verbatim
57*> A is REAL array, dimension (LDA,N)
58*> On entry, the m by n matrix A.
59*> On exit, if m >= n, the lower triangle of the subarray
60*> A(m-n+1:m,1:n) contains the n by n lower triangular matrix L;
61*> if m <= n, the elements on and below the (n-m)-th
62*> superdiagonal contain the m by n lower trapezoidal matrix L;
63*> the remaining elements, with the array TAU, represent the
64*> orthogonal matrix Q as a product of elementary reflectors
65*> (see Further Details).
66*> \endverbatim
67*>
68*> \param[in] LDA
69*> \verbatim
70*> LDA is INTEGER
71*> The leading dimension of the array A. LDA >= max(1,M).
72*> \endverbatim
73*>
74*> \param[out] TAU
75*> \verbatim
76*> TAU is REAL array, dimension (min(M,N))
77*> The scalar factors of the elementary reflectors (see Further
78*> Details).
79*> \endverbatim
80*>
81*> \param[out] WORK
82*> \verbatim
83*> WORK is REAL array, dimension (N)
84*> \endverbatim
85*>
86*> \param[out] INFO
87*> \verbatim
88*> INFO is INTEGER
89*> = 0: successful exit
90*> < 0: if INFO = -i, the i-th argument had an illegal value
91*> \endverbatim
92*
93* Authors:
94* ========
95*
96*> \author Univ. of Tennessee
97*> \author Univ. of California Berkeley
98*> \author Univ. of Colorado Denver
99*> \author NAG Ltd.
100*
101*> \ingroup geql2
102*
103*> \par Further Details:
104* =====================
105*>
106*> \verbatim
107*>
108*> The matrix Q is represented as a product of elementary reflectors
109*>
110*> Q = H(k) . . . H(2) H(1), where k = min(m,n).
111*>
112*> Each H(i) has the form
113*>
114*> H(i) = I - tau * v * v**T
115*>
116*> where tau is a real scalar, and v is a real vector with
117*> v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
118*> A(1:m-k+i-1,n-k+i), and tau in TAU(i).
119*> \endverbatim
120*>
121* =====================================================================
122 SUBROUTINE sgeql2( M, N, A, LDA, TAU, WORK, INFO )
123*
124* -- LAPACK computational routine --
125* -- LAPACK is a software package provided by Univ. of Tennessee, --
126* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
127*
128* .. Scalar Arguments ..
129 INTEGER INFO, LDA, M, N
130* ..
131* .. Array Arguments ..
132 REAL A( LDA, * ), TAU( * ), WORK( * )
133* ..
134*
135* =====================================================================
136*
137* .. Parameters ..
138 REAL ONE
139 parameter( one = 1.0e+0 )
140* ..
141* .. Local Scalars ..
142 INTEGER I, K
143 REAL AII
144* ..
145* .. External Subroutines ..
146 EXTERNAL slarf, slarfg, xerbla
147* ..
148* .. Intrinsic Functions ..
149 INTRINSIC max, min
150* ..
151* .. Executable Statements ..
152*
153* Test the input arguments
154*
155 info = 0
156 IF( m.LT.0 ) THEN
157 info = -1
158 ELSE IF( n.LT.0 ) THEN
159 info = -2
160 ELSE IF( lda.LT.max( 1, m ) ) THEN
161 info = -4
162 END IF
163 IF( info.NE.0 ) THEN
164 CALL xerbla( 'SGEQL2', -info )
165 RETURN
166 END IF
167*
168 k = min( m, n )
169*
170 DO 10 i = k, 1, -1
171*
172* Generate elementary reflector H(i) to annihilate
173* A(1:m-k+i-1,n-k+i)
174*
175 CALL slarfg( m-k+i, a( m-k+i, n-k+i ), a( 1, n-k+i ), 1,
176 \$ tau( i ) )
177*
178* Apply H(i) to A(1:m-k+i,1:n-k+i-1) from the left
179*
180 aii = a( m-k+i, n-k+i )
181 a( m-k+i, n-k+i ) = one
182 CALL slarf( 'Left', m-k+i, n-k+i-1, a( 1, n-k+i ), 1, tau( i ),
183 \$ a, lda, work )
184 a( m-k+i, n-k+i ) = aii
185 10 CONTINUE
186 RETURN
187*
188* End of SGEQL2
189*
190 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine sgeql2(m, n, a, lda, tau, work, info)
SGEQL2 computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
Definition sgeql2.f:123
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