LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
subroutine dgetrf ( integer  M,
integer  N,
double precision, dimension( lda, * )  A,
integer  LDA,
integer, dimension( * )  IPIV,
integer  INFO 
)

DGETRF VARIANT: iterative version of Sivan Toledo's recursive LU algorithm

Purpose: 

 DGETRF computes an LU factorization of a general M-by-N matrix A
 using partial pivoting with row interchanges.

 The factorization has the form
    A = P * L * U
 where P is a permutation matrix, L is lower triangular with unit
 diagonal elements (lower trapezoidal if m > n), and U is upper
 triangular (upper trapezoidal if m < n).

 This code implements an iterative version of Sivan Toledo's recursive
 LU algorithm[1].  For square matrices, this iterative versions should
 be within a factor of two of the optimum number of memory transfers.

 The pattern is as follows, with the large blocks of U being updated
 in one call to DTRSM, and the dotted lines denoting sections that
 have had all pending permutations applied:

  1 2 3 4 5 6 7 8
 +-+-+---+-------+------
 | |1|   |       |
 |.+-+ 2 |       |
 | | |   |       |
 |.|.+-+-+   4   |
 | | | |1|       |
 | | |.+-+       |
 | | | | |       |
 |.|.|.|.+-+-+---+  8
 | | | | | |1|   |
 | | | | |.+-+ 2 |
 | | | | | | |   |
 | | | | |.|.+-+-+
 | | | | | | | |1|
 | | | | | | |.+-+
 | | | | | | | | |
 |.|.|.|.|.|.|.|.+-----
 | | | | | | | | |

 The 1-2-1-4-1-2-1-8-... pattern is the position of the last 1 bit in
 the binary expansion of the current column.  Each Schur update is
 applied as soon as the necessary portion of U is available.

 [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with
 Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
 1065-1081. http://dx.doi.org/10.1137/S0895479896297744
Parameters
[in]M
          M is INTEGER
          The number of rows of the matrix A.  M >= 0.
[in]N
          N is INTEGER
          The number of columns of the matrix A.  N >= 0.
[in,out]A
          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix to be factored.
          On exit, the factors L and U from the factorization
          A = P*L*U; the unit diagonal elements of L are not stored.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).
[out]IPIV
          IPIV is INTEGER array, dimension (min(M,N))
          The pivot indices; for 1 <= i <= min(M,N), row i of the
          matrix was interchanged with row IPIV(i).
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
                has been completed, but the factor U is exactly
                singular, and division by zero will occur if it is used
                to solve a system of equations.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
November 2011

Definition at line 136 of file dgetrf.f.

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  DOUBLE PRECISION a( lda, * )
148 * ..
149 *
150 * =====================================================================
151 *
152 * .. Parameters ..
153  DOUBLE PRECISION one, zero, negone
154  parameter ( one = 1.0d+0, zero = 0.0d+0 )
155  parameter ( negone = -1.0d+0 )
156 * ..
157 * .. Local Scalars ..
158  DOUBLE PRECISION sfmin, tmp
159  INTEGER i, j, jp, nstep, ntopiv, npived, kahead
160  INTEGER kstart, ipivstart, jpivstart, kcols
161 * ..
162 * .. External Functions ..
163  DOUBLE PRECISION dlamch
164  INTEGER idamax
165  LOGICAL disnan
166  EXTERNAL dlamch, idamax, disnan
167 * ..
168 * .. External Subroutines ..
169  EXTERNAL dtrsm, dscal, xerbla, dlaswp
170 * ..
171 * .. Intrinsic Functions ..
172  INTRINSIC max, min, iand
173 * ..
174 * .. Executable Statements ..
175 *
176 * Test the input parameters.
177 *
178  info = 0
179  IF( m.LT.0 ) THEN
180  info = -1
181  ELSE IF( n.LT.0 ) THEN
182  info = -2
183  ELSE IF( lda.LT.max( 1, m ) ) THEN
184  info = -4
185  END IF
186  IF( info.NE.0 ) THEN
187  CALL xerbla( 'DGETRF', -info )
188  RETURN
189  END IF
190 *
191 * Quick return if possible
192 *
193  IF( m.EQ.0 .OR. n.EQ.0 )
194  $ RETURN
195 *
196 * Compute machine safe minimum
197 *
198  sfmin = dlamch( 'S' )
199 *
200  nstep = min( m, n )
201  DO j = 1, nstep
202  kahead = iand( j, -j )
203  kstart = j + 1 - kahead
204  kcols = min( kahead, m-j )
205 *
206 * Find pivot.
207 *
208  jp = j - 1 + idamax( m-j+1, a( j, j ), 1 )
209  ipiv( j ) = jp
210 
211 * Permute just this column.
212  IF (jp .NE. j) THEN
213  tmp = a( j, j )
214  a( j, j ) = a( jp, j )
215  a( jp, j ) = tmp
216  END IF
217 
218 * Apply pending permutations to L
219  ntopiv = 1
220  ipivstart = j
221  jpivstart = j - ntopiv
222  DO WHILE ( ntopiv .LT. kahead )
223  CALL dlaswp( ntopiv, a( 1, jpivstart ), lda, ipivstart, j,
224  $ ipiv, 1 )
225  ipivstart = ipivstart - ntopiv;
226  ntopiv = ntopiv * 2;
227  jpivstart = jpivstart - ntopiv;
228  END DO
229 
230 * Permute U block to match L
231  CALL dlaswp( kcols, a( 1,j+1 ), lda, kstart, j, ipiv, 1 )
232 
233 * Factor the current column
234  IF( a( j, j ).NE.zero .AND. .NOT.disnan( a( j, j ) ) ) THEN
235  IF( abs(a( j, j )) .GE. sfmin ) THEN
236  CALL dscal( m-j, one / a( j, j ), a( j+1, j ), 1 )
237  ELSE
238  DO i = 1, m-j
239  a( j+i, j ) = a( j+i, j ) / a( j, j )
240  END DO
241  END IF
242  ELSE IF( a( j,j ) .EQ. zero .AND. info .EQ. 0 ) THEN
243  info = j
244  END IF
245 
246 * Solve for U block.
247  CALL dtrsm( 'Left', 'Lower', 'No transpose', 'Unit', kahead,
248  $ kcols, one, a( kstart, kstart ), lda,
249  $ a( kstart, j+1 ), lda )
250 * Schur complement.
251  CALL dgemm( 'No transpose', 'No transpose', m-j,
252  $ kcols, kahead, negone, a( j+1, kstart ), lda,
253  $ a( kstart, j+1 ), lda, one, a( j+1, j+1 ), lda )
254  END DO
255 
256 * Handle pivot permutations on the way out of the recursion
257  npived = iand( nstep, -nstep )
258  j = nstep - npived
259  DO WHILE ( j .GT. 0 )
260  ntopiv = iand( j, -j )
261  CALL dlaswp( ntopiv, a( 1, j-ntopiv+1 ), lda, j+1, nstep,
262  $ ipiv, 1 )
263  j = j - ntopiv
264  END DO
265 
266 * If short and wide, handle the rest of the columns.
267  IF ( m .LT. n ) THEN
268  CALL dlaswp( n-m, a( 1, m+kcols+1 ), lda, 1, m, ipiv, 1 )
269  CALL dtrsm( 'Left', 'Lower', 'No transpose', 'Unit', m,
270  $ n-m, one, a, lda, a( 1,m+kcols+1 ), lda )
271  END IF
272 
273  RETURN
274 *
275 * End of DGETRF
276 *
disnan
logical function disnan(DIN)
DISNAN tests input for NaN.
Definition: disnan.f:61
dtrsm
subroutine dtrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRSM
Definition: dtrsm.f:183
idamax
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:53
dlamch
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
dgemm
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:189
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
dlaswp
subroutine dlaswp(N, A, LDA, K1, K2, IPIV, INCX)
DLASWP performs a series of row interchanges on a general rectangular matrix.
Definition: dlaswp.f:116
dscal
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:55

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