LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
cgetrf.f
Go to the documentation of this file.
1 C> \brief \b CGETRF VARIANT: iterative version of Sivan Toledo's recursive LU algorithm
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 * Definition:
9 * ===========
10 *
11 * SUBROUTINE CGETRF( M, N, A, LDA, IPIV, INFO )
12 *
13 * .. Scalar Arguments ..
14 * INTEGER INFO, LDA, M, N
15 * ..
16 * .. Array Arguments ..
17 * INTEGER IPIV( * )
18 * COMPLEX A( LDA, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 C>\details \b Purpose:
25 C>\verbatim
26 C>
27 C> CGETRF computes an LU factorization of a general M-by-N matrix A
28 C> using partial pivoting with row interchanges.
29 C>
30 C> The factorization has the form
31 C> A = P * L * U
32 C> where P is a permutation matrix, L is lower triangular with unit
33 C> diagonal elements (lower trapezoidal if m > n), and U is upper
34 C> triangular (upper trapezoidal if m < n).
35 C>
36 C> This code implements an iterative version of Sivan Toledo's recursive
37 C> LU algorithm[1]. For square matrices, this iterative versions should
38 C> be within a factor of two of the optimum number of memory transfers.
39 C>
40 C> The pattern is as follows, with the large blocks of U being updated
41 C> in one call to DTRSM, and the dotted lines denoting sections that
42 C> have had all pending permutations applied:
43 C>
44 C> 1 2 3 4 5 6 7 8
45 C> +-+-+---+-------+------
46 C> | |1| | |
47 C> |.+-+ 2 | |
48 C> | | | | |
49 C> |.|.+-+-+ 4 |
50 C> | | | |1| |
51 C> | | |.+-+ |
52 C> | | | | | |
53 C> |.|.|.|.+-+-+---+ 8
54 C> | | | | | |1| |
55 C> | | | | |.+-+ 2 |
56 C> | | | | | | | |
57 C> | | | | |.|.+-+-+
58 C> | | | | | | | |1|
59 C> | | | | | | |.+-+
60 C> | | | | | | | | |
61 C> |.|.|.|.|.|.|.|.+-----
62 C> | | | | | | | | |
63 C>
64 C> The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
65 C> the binary expansion of the current column. Each Schur update is
66 C> applied as soon as the necessary portion of U is available.
67 C>
68 C> [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
69 C> Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
70 C> 1065-1081. http://dx.doi.org/10.1137/S0895479896297744
71 C>
72 C>\endverbatim
73 *
74 * Arguments:
75 * ==========
76 *
77 C> \param[in] M
78 C> \verbatim
79 C> M is INTEGER
80 C> The number of rows of the matrix A. M >= 0.
81 C> \endverbatim
82 C>
83 C> \param[in] N
84 C> \verbatim
85 C> N is INTEGER
86 C> The number of columns of the matrix A. N >= 0.
87 C> \endverbatim
88 C>
89 C> \param[in,out] A
90 C> \verbatim
91 C> A is COMPLEX array, dimension (LDA,N)
92 C> On entry, the M-by-N matrix to be factored.
93 C> On exit, the factors L and U from the factorization
94 C> A = P*L*U; the unit diagonal elements of L are not stored.
95 C> \endverbatim
96 C>
97 C> \param[in] LDA
98 C> \verbatim
99 C> LDA is INTEGER
100 C> The leading dimension of the array A. LDA >= max(1,M).
101 C> \endverbatim
102 C>
103 C> \param[out] IPIV
104 C> \verbatim
105 C> IPIV is INTEGER array, dimension (min(M,N))
106 C> The pivot indices; for 1 <= i <= min(M,N), row i of the
107 C> matrix was interchanged with row IPIV(i).
108 C> \endverbatim
109 C>
110 C> \param[out] INFO
111 C> \verbatim
112 C> INFO is INTEGER
113 C> = 0: successful exit
114 C> < 0: if INFO = -i, the i-th argument had an illegal value
115 C> > 0: if INFO = i, U(i,i) is exactly zero. The factorization
116 C> has been completed, but the factor U is exactly
117 C> singular, and division by zero will occur if it is used
118 C> to solve a system of equations.
119 C> \endverbatim
120 C>
121 *
122 * Authors:
123 * ========
124 *
125 C> \author Univ. of Tennessee
126 C> \author Univ. of California Berkeley
127 C> \author Univ. of Colorado Denver
128 C> \author NAG Ltd.
129 *
130 C> \date November 2011
131 *
132 C> \ingroup variantsGEcomputational
133 *
134 * =====================================================================
135  SUBROUTINE cgetrf( M, N, A, LDA, IPIV, INFO )
136 *
137 * -- LAPACK computational routine (version 3.X) --
138 * -- LAPACK is a software package provided by Univ. of Tennessee, --
139 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
140 * November 2011
141 *
142 * .. Scalar Arguments ..
143  INTEGER info, lda, m, n
144 * ..
145 * .. Array Arguments ..
146  INTEGER ipiv( * )
147  COMPLEX a( lda, * )
148 * ..
149 *
150 * =====================================================================
151 *
152 * .. Parameters ..
153  COMPLEX one, negone
154  REAL zero
155  parameter( one = (1.0e+0, 0.0e+0) )
156  parameter( negone = (-1.0e+0, 0.0e+0) )
157  parameter( zero = 0.0e+0 )
158 * ..
159 * .. Local Scalars ..
160  REAL sfmin, pivmag
161  COMPLEX tmp
162  INTEGER i, j, jp, nstep, ntopiv, npived, kahead
163  INTEGER kstart, ipivstart, jpivstart, kcols
164 * ..
165 * .. External Functions ..
166  REAL slamch
167  INTEGER icamax
168  LOGICAL sisnan
169  EXTERNAL slamch, icamax, sisnan
170 * ..
171 * .. External Subroutines ..
172  EXTERNAL ctrsm, cscal, xerbla, claswp
173 * ..
174 * .. Intrinsic Functions ..
175  INTRINSIC max, min, iand, abs
176 * ..
177 * .. Executable Statements ..
178 *
179 * Test the input parameters.
180 *
181  info = 0
182  IF( m.LT.0 ) THEN
183  info = -1
184  ELSE IF( n.LT.0 ) THEN
185  info = -2
186  ELSE IF( lda.LT.max( 1, m ) ) THEN
187  info = -4
188  END IF
189  IF( info.NE.0 ) THEN
190  CALL xerbla( 'CGETRF', -info )
191  return
192  END IF
193 *
194 * Quick return if possible
195 *
196  IF( m.EQ.0 .OR. n.EQ.0 )
197  \$ return
198 *
199 * Compute machine safe minimum
200 *
201  sfmin = slamch( 'S' )
202 *
203  nstep = min( m, n )
204  DO j = 1, nstep
205  kahead = iand( j, -j )
206  kstart = j + 1 - kahead
207  kcols = min( kahead, m-j )
208 *
209 * Find pivot.
210 *
211  jp = j - 1 + icamax( m-j+1, a( j, j ), 1 )
212  ipiv( j ) = jp
213
214 * Permute just this column.
215  IF (jp .NE. j) THEN
216  tmp = a( j, j )
217  a( j, j ) = a( jp, j )
218  a( jp, j ) = tmp
219  END IF
220
221 * Apply pending permutations to L
222  ntopiv = 1
223  ipivstart = j
224  jpivstart = j - ntopiv
225  DO WHILE ( ntopiv .LT. kahead )
226  CALL claswp( ntopiv, a( 1, jpivstart ), lda, ipivstart, j,
227  \$ ipiv, 1 )
228  ipivstart = ipivstart - ntopiv;
229  ntopiv = ntopiv * 2;
230  jpivstart = jpivstart - ntopiv;
231  END DO
232
233 * Permute U block to match L
234  CALL claswp( kcols, a( 1,j+1 ), lda, kstart, j, ipiv, 1 )
235
236 * Factor the current column
237  pivmag = abs( a( j, j ) )
238  IF( pivmag.NE.zero .AND. .NOT.sisnan( pivmag ) ) THEN
239  IF( pivmag .GE. sfmin ) THEN
240  CALL cscal( m-j, one / a( j, j ), a( j+1, j ), 1 )
241  ELSE
242  DO i = 1, m-j
243  a( j+i, j ) = a( j+i, j ) / a( j, j )
244  END DO
245  END IF
246  ELSE IF( pivmag .EQ. zero .AND. info .EQ. 0 ) THEN
247  info = j
248  END IF
249
250 * Solve for U block.
251  CALL ctrsm( 'Left', 'Lower', 'No transpose', 'Unit', kahead,
252  \$ kcols, one, a( kstart, kstart ), lda,
253  \$ a( kstart, j+1 ), lda )
254 * Schur complement.
255  CALL cgemm( 'No transpose', 'No transpose', m-j,
256  \$ kcols, kahead, negone, a( j+1, kstart ), lda,
257  \$ a( kstart, j+1 ), lda, one, a( j+1, j+1 ), lda )
258  END DO
259
260 * Handle pivot permutations on the way out of the recursion
261  npived = iand( nstep, -nstep )
262  j = nstep - npived
263  DO WHILE ( j .GT. 0 )
264  ntopiv = iand( j, -j )
265  CALL claswp( ntopiv, a( 1, j-ntopiv+1 ), lda, j+1, nstep,
266  \$ ipiv, 1 )
267  j = j - ntopiv
268  END DO
269
270 * If short and wide, handle the rest of the columns.
271  IF ( m .LT. n ) THEN
272  CALL claswp( n-m, a( 1, m+kcols+1 ), lda, 1, m, ipiv, 1 )
273  CALL ctrsm( 'Left', 'Lower', 'No transpose', 'Unit', m,
274  \$ n-m, one, a, lda, a( 1,m+kcols+1 ), lda )
275  END IF
276
277  return
278 *
279 * End of CGETRF
280 *
281  END