LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
sgeqlf.f
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1 *> \brief \b SGEQLF
2 *
3 * =========== DOCUMENTATION ===========
4 *
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGEQLF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, LWORK, M, N
25 * ..
26 * .. Array Arguments ..
27 * REAL A( LDA, * ), TAU( * ), WORK( * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> SGEQLF 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,
60 *> if m >= n, the lower triangle of the subarray
61 *> A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
62 *> if m <= n, the elements on and below the (n-m)-th
63 *> superdiagonal contain the M-by-N lower trapezoidal matrix L;
64 *> the remaining elements, with the array TAU, represent the
65 *> orthogonal matrix Q as a product of elementary reflectors
66 *> (see Further Details).
67 *> \endverbatim
68 *>
69 *> \param[in] LDA
70 *> \verbatim
71 *> LDA is INTEGER
72 *> The leading dimension of the array A. LDA >= max(1,M).
73 *> \endverbatim
74 *>
75 *> \param[out] TAU
76 *> \verbatim
77 *> TAU is REAL array, dimension (min(M,N))
78 *> The scalar factors of the elementary reflectors (see Further
79 *> Details).
80 *> \endverbatim
81 *>
82 *> \param[out] WORK
83 *> \verbatim
84 *> WORK is REAL array, dimension (MAX(1,LWORK))
85 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
86 *> \endverbatim
87 *>
88 *> \param[in] LWORK
89 *> \verbatim
90 *> LWORK is INTEGER
91 *> The dimension of the array WORK. LWORK >= max(1,N).
92 *> For optimum performance LWORK >= N*NB, where NB is the
93 *> optimal blocksize.
94 *>
95 *> If LWORK = -1, then a workspace query is assumed; the routine
96 *> only calculates the optimal size of the WORK array, returns
97 *> this value as the first entry of the WORK array, and no error
98 *> message related to LWORK is issued by XERBLA.
99 *> \endverbatim
100 *>
101 *> \param[out] INFO
102 *> \verbatim
103 *> INFO is INTEGER
104 *> = 0: successful exit
105 *> < 0: if INFO = -i, the i-th argument had an illegal value
106 *> \endverbatim
107 *
108 * Authors:
109 * ========
110 *
111 *> \author Univ. of Tennessee
112 *> \author Univ. of California Berkeley
113 *> \author Univ. of Colorado Denver
114 *> \author NAG Ltd.
115 *
116 *> \date November 2011
117 *
118 *> \ingroup realGEcomputational
119 *
120 *> \par Further Details:
121 * =====================
122 *>
123 *> \verbatim
124 *>
125 *> The matrix Q is represented as a product of elementary reflectors
126 *>
127 *> Q = H(k) . . . H(2) H(1), where k = min(m,n).
128 *>
129 *> Each H(i) has the form
130 *>
131 *> H(i) = I - tau * v * v**T
132 *>
133 *> where tau is a real scalar, and v is a real vector with
134 *> v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
135 *> A(1:m-k+i-1,n-k+i), and tau in TAU(i).
136 *> \endverbatim
137 *>
138 * =====================================================================
139  SUBROUTINE sgeqlf( M, N, A, LDA, TAU, WORK, LWORK, INFO )
140 *
141 * -- LAPACK computational routine (version 3.4.0) --
142 * -- LAPACK is a software package provided by Univ. of Tennessee, --
143 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
144 * November 2011
145 *
146 * .. Scalar Arguments ..
147  INTEGER INFO, LDA, LWORK, M, N
148 * ..
149 * .. Array Arguments ..
150  REAL A( lda, * ), TAU( * ), WORK( * )
151 * ..
152 *
153 * =====================================================================
154 *
155 * .. Local Scalars ..
156  LOGICAL LQUERY
157  INTEGER I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
158  \$ mu, nb, nbmin, nu, nx
159 * ..
160 * .. External Subroutines ..
161  EXTERNAL sgeql2, slarfb, slarft, xerbla
162 * ..
163 * .. Intrinsic Functions ..
164  INTRINSIC max, min
165 * ..
166 * .. External Functions ..
167  INTEGER ILAENV
168  EXTERNAL ilaenv
169 * ..
170 * .. Executable Statements ..
171 *
172 * Test the input arguments
173 *
174  info = 0
175  lquery = ( lwork.EQ.-1 )
176  IF( m.LT.0 ) THEN
177  info = -1
178  ELSE IF( n.LT.0 ) THEN
179  info = -2
180  ELSE IF( lda.LT.max( 1, m ) ) THEN
181  info = -4
182  END IF
183 *
184  IF( info.EQ.0 ) THEN
185  k = min( m, n )
186  IF( k.EQ.0 ) THEN
187  lwkopt = 1
188  ELSE
189  nb = ilaenv( 1, 'SGEQLF', ' ', m, n, -1, -1 )
190  lwkopt = n*nb
191  END IF
192  work( 1 ) = lwkopt
193 *
194  IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
195  info = -7
196  END IF
197  END IF
198 *
199  IF( info.NE.0 ) THEN
200  CALL xerbla( 'SGEQLF', -info )
201  RETURN
202  ELSE IF( lquery ) THEN
203  RETURN
204  END IF
205 *
206 * Quick return if possible
207 *
208  IF( k.EQ.0 ) THEN
209  RETURN
210  END IF
211 *
212  nbmin = 2
213  nx = 1
214  iws = n
215  IF( nb.GT.1 .AND. nb.LT.k ) THEN
216 *
217 * Determine when to cross over from blocked to unblocked code.
218 *
219  nx = max( 0, ilaenv( 3, 'SGEQLF', ' ', m, n, -1, -1 ) )
220  IF( nx.LT.k ) THEN
221 *
222 * Determine if workspace is large enough for blocked code.
223 *
224  ldwork = n
225  iws = ldwork*nb
226  IF( lwork.LT.iws ) THEN
227 *
228 * Not enough workspace to use optimal NB: reduce NB and
229 * determine the minimum value of NB.
230 *
231  nb = lwork / ldwork
232  nbmin = max( 2, ilaenv( 2, 'SGEQLF', ' ', m, n, -1,
233  \$ -1 ) )
234  END IF
235  END IF
236  END IF
237 *
238  IF( nb.GE.nbmin .AND. nb.LT.k .AND. nx.LT.k ) THEN
239 *
240 * Use blocked code initially.
241 * The last kk columns are handled by the block method.
242 *
243  ki = ( ( k-nx-1 ) / nb )*nb
244  kk = min( k, ki+nb )
245 *
246  DO 10 i = k - kk + ki + 1, k - kk + 1, -nb
247  ib = min( k-i+1, nb )
248 *
249 * Compute the QL factorization of the current block
250 * A(1:m-k+i+ib-1,n-k+i:n-k+i+ib-1)
251 *
252  CALL sgeql2( m-k+i+ib-1, ib, a( 1, n-k+i ), lda, tau( i ),
253  \$ work, iinfo )
254  IF( n-k+i.GT.1 ) THEN
255 *
256 * Form the triangular factor of the block reflector
257 * H = H(i+ib-1) . . . H(i+1) H(i)
258 *
259  CALL slarft( 'Backward', 'Columnwise', m-k+i+ib-1, ib,
260  \$ a( 1, n-k+i ), lda, tau( i ), work, ldwork )
261 *
262 * Apply H**T to A(1:m-k+i+ib-1,1:n-k+i-1) from the left
263 *
264  CALL slarfb( 'Left', 'Transpose', 'Backward',
265  \$ 'Columnwise', m-k+i+ib-1, n-k+i-1, ib,
266  \$ a( 1, n-k+i ), lda, work, ldwork, a, lda,
267  \$ work( ib+1 ), ldwork )
268  END IF
269  10 CONTINUE
270  mu = m - k + i + nb - 1
271  nu = n - k + i + nb - 1
272  ELSE
273  mu = m
274  nu = n
275  END IF
276 *
277 * Use unblocked code to factor the last or only block
278 *
279  IF( mu.GT.0 .AND. nu.GT.0 )
280  \$ CALL sgeql2( mu, nu, a, lda, tau, work, iinfo )
281 *
282  work( 1 ) = iws
283  RETURN
284 *
285 * End of SGEQLF
286 *
287  END
subroutine sgeqlf(M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQLF
Definition: sgeqlf.f:140
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:125
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slarft(DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
SLARFT forms the triangular factor T of a block reflector H = I - vtvH
Definition: slarft.f:165
subroutine slarfb(SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
SLARFB applies a block reflector or its transpose to a general rectangular matrix.
Definition: slarfb.f:197