1: *> \brief \b DGELQF
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DGELQF + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelqf.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelqf.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, LWORK, 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: *> DGELQF 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 (MAX(1,LWORK))
90: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
91: *> \endverbatim
92: *>
93: *> \param[in] LWORK
94: *> \verbatim
95: *> LWORK is INTEGER
96: *> The dimension of the array WORK. LWORK >= max(1,M).
97: *> For optimum performance LWORK >= M*NB, where NB is the
98: *> optimal blocksize.
99: *>
100: *> If LWORK = -1, then a workspace query is assumed; the routine
101: *> only calculates the optimal size of the WORK array, returns
102: *> this value as the first entry of the WORK array, and no error
103: *> message related to LWORK is issued by XERBLA.
104: *> \endverbatim
105: *>
106: *> \param[out] INFO
107: *> \verbatim
108: *> INFO is INTEGER
109: *> = 0: successful exit
110: *> < 0: if INFO = -i, the i-th argument had an illegal value
111: *> \endverbatim
112: *
113: * Authors:
114: * ========
115: *
116: *> \author Univ. of Tennessee
117: *> \author Univ. of California Berkeley
118: *> \author Univ. of Colorado Denver
119: *> \author NAG Ltd.
120: *
121: *> \ingroup doubleGEcomputational
122: *
123: *> \par Further Details:
124: * =====================
125: *>
126: *> \verbatim
127: *>
128: *> The matrix Q is represented as a product of elementary reflectors
129: *>
130: *> Q = H(k) . . . H(2) H(1), where k = min(m,n).
131: *>
132: *> Each H(i) has the form
133: *>
134: *> H(i) = I - tau * v * v**T
135: *>
136: *> where tau is a real scalar, and v is a real vector with
137: *> v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
138: *> and tau in TAU(i).
139: *> \endverbatim
140: *>
141: * =====================================================================
142: SUBROUTINE DGELQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
143: *
144: * -- LAPACK computational routine --
145: * -- LAPACK is a software package provided by Univ. of Tennessee, --
146: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
147: *
148: * .. Scalar Arguments ..
149: INTEGER INFO, LDA, LWORK, M, N
150: * ..
151: * .. Array Arguments ..
152: DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
153: * ..
154: *
155: * =====================================================================
156: *
157: * .. Local Scalars ..
158: LOGICAL LQUERY
159: INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
160: $ NBMIN, NX
161: * ..
162: * .. External Subroutines ..
163: EXTERNAL DGELQ2, DLARFB, DLARFT, XERBLA
164: * ..
165: * .. Intrinsic Functions ..
166: INTRINSIC MAX, MIN
167: * ..
168: * .. External Functions ..
169: INTEGER ILAENV
170: EXTERNAL ILAENV
171: * ..
172: * .. Executable Statements ..
173: *
174: * Test the input arguments
175: *
176: INFO = 0
177: NB = ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
178: LWKOPT = M*NB
179: WORK( 1 ) = LWKOPT
180: LQUERY = ( LWORK.EQ.-1 )
181: IF( M.LT.0 ) THEN
182: INFO = -1
183: ELSE IF( N.LT.0 ) THEN
184: INFO = -2
185: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
186: INFO = -4
187: ELSE IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
188: INFO = -7
189: END IF
190: IF( INFO.NE.0 ) THEN
191: CALL XERBLA( 'DGELQF', -INFO )
192: RETURN
193: ELSE IF( LQUERY ) THEN
194: RETURN
195: END IF
196: *
197: * Quick return if possible
198: *
199: K = MIN( M, N )
200: IF( K.EQ.0 ) THEN
201: WORK( 1 ) = 1
202: RETURN
203: END IF
204: *
205: NBMIN = 2
206: NX = 0
207: IWS = M
208: IF( NB.GT.1 .AND. NB.LT.K ) THEN
209: *
210: * Determine when to cross over from blocked to unblocked code.
211: *
212: NX = MAX( 0, ILAENV( 3, 'DGELQF', ' ', M, N, -1, -1 ) )
213: IF( NX.LT.K ) THEN
214: *
215: * Determine if workspace is large enough for blocked code.
216: *
217: LDWORK = M
218: IWS = LDWORK*NB
219: IF( LWORK.LT.IWS ) THEN
220: *
221: * Not enough workspace to use optimal NB: reduce NB and
222: * determine the minimum value of NB.
223: *
224: NB = LWORK / LDWORK
225: NBMIN = MAX( 2, ILAENV( 2, 'DGELQF', ' ', M, N, -1,
226: $ -1 ) )
227: END IF
228: END IF
229: END IF
230: *
231: IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
232: *
233: * Use blocked code initially
234: *
235: DO 10 I = 1, K - NX, NB
236: IB = MIN( K-I+1, NB )
237: *
238: * Compute the LQ factorization of the current block
239: * A(i:i+ib-1,i:n)
240: *
241: CALL DGELQ2( IB, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
242: $ IINFO )
243: IF( I+IB.LE.M ) THEN
244: *
245: * Form the triangular factor of the block reflector
246: * H = H(i) H(i+1) . . . H(i+ib-1)
247: *
248: CALL DLARFT( 'Forward', 'Rowwise', N-I+1, IB, A( I, I ),
249: $ LDA, TAU( I ), WORK, LDWORK )
250: *
251: * Apply H to A(i+ib:m,i:n) from the right
252: *
253: CALL DLARFB( 'Right', 'No transpose', 'Forward',
254: $ 'Rowwise', M-I-IB+1, N-I+1, IB, A( I, I ),
255: $ LDA, WORK, LDWORK, A( I+IB, I ), LDA,
256: $ WORK( IB+1 ), LDWORK )
257: END IF
258: 10 CONTINUE
259: ELSE
260: I = 1
261: END IF
262: *
263: * Use unblocked code to factor the last or only block.
264: *
265: IF( I.LE.K )
266: $ CALL DGELQ2( M-I+1, N-I+1, A( I, I ), LDA, TAU( I ), WORK,
267: $ IINFO )
268: *
269: WORK( 1 ) = IWS
270: RETURN
271: *
272: * End of DGELQF
273: *
274: END
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