1: *> \brief \b ZLAUNHR_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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17: *
18: * Definition:
19: * ===========
20: *
21: * RECURSIVE SUBROUTINE ZLAUNHR_COL_GETRFNP2( M, N, A, LDA, D, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, M, N
25: * ..
26: * .. Array Arguments ..
27: * COMPLEX*16 A( LDA, * ), D( * )
28: * ..
29: *
30: *
31: *> \par Purpose:
32: * =============
33: *>
34: *> \verbatim
35: *>
36: *> ZLAUNHR_COL_GETRFNP2 computes the modified LU factorization without
37: *> pivoting of a complex 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 ZUNHR_COL. In ZUNHR_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: *> ZLAUNHR_COL_GETRFNP2 is called to factorize a block by the blocked
86: *> routine ZLAUNHR_COL_GETRFNP, which uses blocked code calling
87: *> Level 3 BLAS to update the submatrix. However, ZLAUNHR_COL_GETRFNP2
88: *> is self-sufficient and can be used without ZLAUNHR_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 COMPLEX*16 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 COMPLEX*16 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 be
134: *> only ( +1.0, 0.0 ) or (-1.0, 0.0 ).
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 complex16GEcomputational
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 ZLAUNHR_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: COMPLEX*16 A( LDA, * ), D( * )
178: * ..
179: *
180: * =====================================================================
181: *
182: * .. Parameters ..
183: DOUBLE PRECISION ONE
184: PARAMETER ( ONE = 1.0D+0 )
185: COMPLEX*16 CONE
186: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
187: * ..
188: * .. Local Scalars ..
189: DOUBLE PRECISION SFMIN
190: INTEGER I, IINFO, N1, N2
191: COMPLEX*16 Z
192: * ..
193: * .. External Functions ..
194: DOUBLE PRECISION DLAMCH
195: EXTERNAL DLAMCH
196: * ..
197: * .. External Subroutines ..
198: EXTERNAL ZGEMM, ZSCAL, ZTRSM, XERBLA
199: * ..
200: * .. Intrinsic Functions ..
201: INTRINSIC ABS, DBLE, DCMPLX, DIMAG, DSIGN, MAX, MIN
202: * ..
203: * .. Statement Functions ..
204: DOUBLE PRECISION CABS1
205: * ..
206: * .. Statement Function definitions ..
207: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
208: * ..
209: * .. Executable Statements ..
210: *
211: * Test the input parameters
212: *
213: INFO = 0
214: IF( M.LT.0 ) THEN
215: INFO = -1
216: ELSE IF( N.LT.0 ) THEN
217: INFO = -2
218: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
219: INFO = -4
220: END IF
221: IF( INFO.NE.0 ) THEN
222: CALL XERBLA( 'ZLAUNHR_COL_GETRFNP2', -INFO )
223: RETURN
224: END IF
225: *
226: * Quick return if possible
227: *
228: IF( MIN( M, N ).EQ.0 )
229: $ RETURN
230:
231: IF ( M.EQ.1 ) THEN
232: *
233: * One row case, (also recursion termination case),
234: * use unblocked code
235: *
236: * Transfer the sign
237: *
238: D( 1 ) = DCMPLX( -DSIGN( ONE, DBLE( A( 1, 1 ) ) ) )
239: *
240: * Construct the row of U
241: *
242: A( 1, 1 ) = A( 1, 1 ) - D( 1 )
243: *
244: ELSE IF( N.EQ.1 ) THEN
245: *
246: * One column case, (also recursion termination case),
247: * use unblocked code
248: *
249: * Transfer the sign
250: *
251: D( 1 ) = DCMPLX( -DSIGN( ONE, DBLE( A( 1, 1 ) ) ) )
252: *
253: * Construct the row of U
254: *
255: A( 1, 1 ) = A( 1, 1 ) - D( 1 )
256: *
257: * Scale the elements 2:M of the column
258: *
259: * Determine machine safe minimum
260: *
261: SFMIN = DLAMCH('S')
262: *
263: * Construct the subdiagonal elements of L
264: *
265: IF( CABS1( A( 1, 1 ) ) .GE. SFMIN ) THEN
266: CALL ZSCAL( M-1, CONE / A( 1, 1 ), A( 2, 1 ), 1 )
267: ELSE
268: DO I = 2, M
269: A( I, 1 ) = A( I, 1 ) / A( 1, 1 )
270: END DO
271: END IF
272: *
273: ELSE
274: *
275: * Divide the matrix B into four submatrices
276: *
277: N1 = MIN( M, N ) / 2
278: N2 = N-N1
279:
280: *
281: * Factor B11, recursive call
282: *
283: CALL ZLAUNHR_COL_GETRFNP2( N1, N1, A, LDA, D, IINFO )
284: *
285: * Solve for B21
286: *
287: CALL ZTRSM( 'R', 'U', 'N', 'N', M-N1, N1, CONE, A, LDA,
288: $ A( N1+1, 1 ), LDA )
289: *
290: * Solve for B12
291: *
292: CALL ZTRSM( 'L', 'L', 'N', 'U', N1, N2, CONE, A, LDA,
293: $ A( 1, N1+1 ), LDA )
294: *
295: * Update B22, i.e. compute the Schur complement
296: * B22 := B22 - B21*B12
297: *
298: CALL ZGEMM( 'N', 'N', M-N1, N2, N1, -CONE, A( N1+1, 1 ), LDA,
299: $ A( 1, N1+1 ), LDA, CONE, A( N1+1, N1+1 ), LDA )
300: *
301: * Factor B22, recursive call
302: *
303: CALL ZLAUNHR_COL_GETRFNP2( M-N1, N2, A( N1+1, N1+1 ), LDA,
304: $ D( N1+1 ), IINFO )
305: *
306: END IF
307: RETURN
308: *
309: * End of ZLAUNHR_COL_GETRFNP2
310: *
311: END
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