1: *> \brief \b DLAORHR_COL_GETRFNP2
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * RECURSIVE SUBROUTINE DLAORHR_COL_GETRFNP2( M, N, A, LDA, D, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, M, N
25: * ..
26: * .. Array Arguments ..
27: * DOUBLE PRECISION A( LDA, * ), D( * )
28: * ..
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> DLAORHR_COL_GETRFNP2 computes the modified LU factorization without
37: *> pivoting of a real general M-by-N matrix A. The factorization has
38: *> the form:
39: *>
40: *> A - S = L * U,
41: *>
42: *> where:
43: *> S is a m-by-n diagonal sign matrix with the diagonal D, so that
44: *> D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
45: *> as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
46: *> i-1 steps of Gaussian elimination. This means that the diagonal
47: *> element at each step of "modified" Gaussian elimination is at
48: *> least one in absolute value (so that division-by-zero not
49: *> possible during the division by the diagonal element);
50: *>
51: *> L is a M-by-N lower triangular matrix with unit diagonal elements
52: *> (lower trapezoidal if M > N);
53: *>
54: *> and U is a M-by-N upper triangular matrix
55: *> (upper trapezoidal if M < N).
56: *>
57: *> This routine is an auxiliary routine used in the Householder
58: *> reconstruction routine DORHR_COL. In DORHR_COL, this routine is
59: *> applied to an M-by-N matrix A with orthonormal columns, where each
60: *> element is bounded by one in absolute value. With the choice of
61: *> the matrix S above, one can show that the diagonal element at each
62: *> step of Gaussian elimination is the largest (in absolute value) in
63: *> the column on or below the diagonal, so that no pivoting is required
64: *> for numerical stability [1].
65: *>
66: *> For more details on the Householder reconstruction algorithm,
67: *> including the modified LU factorization, see [1].
68: *>
69: *> This is the recursive version of the LU factorization algorithm.
70: *> Denote A - S by B. The algorithm divides the matrix B into four
71: *> submatrices:
72: *>
73: *> [ B11 | B12 ] where B11 is n1 by n1,
74: *> B = [ -----|----- ] B21 is (m-n1) by n1,
75: *> [ B21 | B22 ] B12 is n1 by n2,
76: *> B22 is (m-n1) by n2,
77: *> with n1 = min(m,n)/2, n2 = n-n1.
78: *>
79: *>
80: *> The subroutine calls itself to factor B11, solves for B21,
81: *> solves for B12, updates B22, then calls itself to factor B22.
82: *>
83: *> For more details on the recursive LU algorithm, see [2].
84: *>
85: *> DLAORHR_COL_GETRFNP2 is called to factorize a block by the blocked
86: *> routine DLAORHR_COL_GETRFNP, which uses blocked code calling
87: *> Level 3 BLAS to update the submatrix. However, DLAORHR_COL_GETRFNP2
88: *> is self-sufficient and can be used without DLAORHR_COL_GETRFNP.
89: *>
90: *> [1] "Reconstructing Householder vectors from tall-skinny QR",
91: *> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
92: *> E. Solomonik, J. Parallel Distrib. Comput.,
93: *> vol. 85, pp. 3-31, 2015.
94: *>
95: *> [2] "Recursion leads to automatic variable blocking for dense linear
96: *> algebra algorithms", F. Gustavson, IBM J. of Res. and Dev.,
97: *> vol. 41, no. 6, pp. 737-755, 1997.
98: *> \endverbatim
99: *
100: * Arguments:
101: * ==========
102: *
103: *> \param[in] M
104: *> \verbatim
105: *> M is INTEGER
106: *> The number of rows of the matrix A. M >= 0.
107: *> \endverbatim
108: *>
109: *> \param[in] N
110: *> \verbatim
111: *> N is INTEGER
112: *> The number of columns of the matrix A. N >= 0.
113: *> \endverbatim
114: *>
115: *> \param[in,out] A
116: *> \verbatim
117: *> A is DOUBLE PRECISION array, dimension (LDA,N)
118: *> On entry, the M-by-N matrix to be factored.
119: *> On exit, the factors L and U from the factorization
120: *> A-S=L*U; the unit diagonal elements of L are not stored.
121: *> \endverbatim
122: *>
123: *> \param[in] LDA
124: *> \verbatim
125: *> LDA is INTEGER
126: *> The leading dimension of the array A. LDA >= max(1,M).
127: *> \endverbatim
128: *>
129: *> \param[out] D
130: *> \verbatim
131: *> D is DOUBLE PRECISION array, dimension min(M,N)
132: *> The diagonal elements of the diagonal M-by-N sign matrix S,
133: *> D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
134: *> be only plus or minus one.
135: *> \endverbatim
136: *>
137: *> \param[out] INFO
138: *> \verbatim
139: *> INFO is INTEGER
140: *> = 0: successful exit
141: *> < 0: if INFO = -i, the i-th argument had an illegal value
142: *> \endverbatim
143: *>
144: * Authors:
145: * ========
146: *
147: *> \author Univ. of Tennessee
148: *> \author Univ. of California Berkeley
149: *> \author Univ. of Colorado Denver
150: *> \author NAG Ltd.
151: *
152: *> \ingroup doubleGEcomputational
153: *
154: *> \par Contributors:
155: * ==================
156: *>
157: *> \verbatim
158: *>
159: *> November 2019, Igor Kozachenko,
160: *> Computer Science Division,
161: *> University of California, Berkeley
162: *>
163: *> \endverbatim
164: *
165: * =====================================================================
166: RECURSIVE SUBROUTINE DLAORHR_COL_GETRFNP2( M, N, A, LDA, D, INFO )
167: IMPLICIT NONE
168: *
169: * -- LAPACK computational routine --
170: * -- LAPACK is a software package provided by Univ. of Tennessee, --
171: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
172: *
173: * .. Scalar Arguments ..
174: INTEGER INFO, LDA, M, N
175: * ..
176: * .. Array Arguments ..
177: DOUBLE PRECISION A( LDA, * ), D( * )
178: * ..
179: *
180: * =====================================================================
181: *
182: * .. Parameters ..
183: DOUBLE PRECISION ONE
184: PARAMETER ( ONE = 1.0D+0 )
185: * ..
186: * .. Local Scalars ..
187: DOUBLE PRECISION SFMIN
188: INTEGER I, IINFO, N1, N2
189: * ..
190: * .. External Functions ..
191: DOUBLE PRECISION DLAMCH
192: EXTERNAL DLAMCH
193: * ..
194: * .. External Subroutines ..
195: EXTERNAL DGEMM, DSCAL, DTRSM, XERBLA
196: * ..
197: * .. Intrinsic Functions ..
198: INTRINSIC ABS, DSIGN, MAX, MIN
199: * ..
200: * .. Executable Statements ..
201: *
202: * Test the input parameters
203: *
204: INFO = 0
205: IF( M.LT.0 ) THEN
206: INFO = -1
207: ELSE IF( N.LT.0 ) THEN
208: INFO = -2
209: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
210: INFO = -4
211: END IF
212: IF( INFO.NE.0 ) THEN
213: CALL XERBLA( 'DLAORHR_COL_GETRFNP2', -INFO )
214: RETURN
215: END IF
216: *
217: * Quick return if possible
218: *
219: IF( MIN( M, N ).EQ.0 )
220: $ RETURN
221:
222: IF ( M.EQ.1 ) THEN
223: *
224: * One row case, (also recursion termination case),
225: * use unblocked code
226: *
227: * Transfer the sign
228: *
229: D( 1 ) = -DSIGN( ONE, A( 1, 1 ) )
230: *
231: * Construct the row of U
232: *
233: A( 1, 1 ) = A( 1, 1 ) - D( 1 )
234: *
235: ELSE IF( N.EQ.1 ) THEN
236: *
237: * One column case, (also recursion termination case),
238: * use unblocked code
239: *
240: * Transfer the sign
241: *
242: D( 1 ) = -DSIGN( ONE, A( 1, 1 ) )
243: *
244: * Construct the row of U
245: *
246: A( 1, 1 ) = A( 1, 1 ) - D( 1 )
247: *
248: * Scale the elements 2:M of the column
249: *
250: * Determine machine safe minimum
251: *
252: SFMIN = DLAMCH('S')
253: *
254: * Construct the subdiagonal elements of L
255: *
256: IF( ABS( A( 1, 1 ) ) .GE. SFMIN ) THEN
257: CALL DSCAL( M-1, ONE / A( 1, 1 ), A( 2, 1 ), 1 )
258: ELSE
259: DO I = 2, M
260: A( I, 1 ) = A( I, 1 ) / A( 1, 1 )
261: END DO
262: END IF
263: *
264: ELSE
265: *
266: * Divide the matrix B into four submatrices
267: *
268: N1 = MIN( M, N ) / 2
269: N2 = N-N1
270:
271: *
272: * Factor B11, recursive call
273: *
274: CALL DLAORHR_COL_GETRFNP2( N1, N1, A, LDA, D, IINFO )
275: *
276: * Solve for B21
277: *
278: CALL DTRSM( 'R', 'U', 'N', 'N', M-N1, N1, ONE, A, LDA,
279: $ A( N1+1, 1 ), LDA )
280: *
281: * Solve for B12
282: *
283: CALL DTRSM( 'L', 'L', 'N', 'U', N1, N2, ONE, A, LDA,
284: $ A( 1, N1+1 ), LDA )
285: *
286: * Update B22, i.e. compute the Schur complement
287: * B22 := B22 - B21*B12
288: *
289: CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( N1+1, 1 ), LDA,
290: $ A( 1, N1+1 ), LDA, ONE, A( N1+1, N1+1 ), LDA )
291: *
292: * Factor B22, recursive call
293: *
294: CALL DLAORHR_COL_GETRFNP2( M-N1, N2, A( N1+1, N1+1 ), LDA,
295: $ D( N1+1 ), IINFO )
296: *
297: END IF
298: RETURN
299: *
300: * End of DLAORHR_COL_GETRFNP2
301: *
302: END
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