LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ cgetrf()

 subroutine cgetrf ( integer M, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO )

CGETRF VARIANT: Crout Level 3 BLAS version of the algorithm.

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

CGETRF VARIANT: left-looking Level 3 BLAS version of the algorithm.

Purpose:

``` CGETRF 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 is the Crout Level 3 BLAS version of the algorithm.```
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 COMPLEX 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.```
Date
December 2016

Purpose:

``` CGETRF 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 is the left-looking Level 3 BLAS version of the algorithm.```
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 COMPLEX 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.```
Date
December 2016

Purpose:

``` CGETRF 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 COMPLEX 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.```
Date
December 2016

Definition at line 101 of file cgetrf.f.

102 *
103 * -- LAPACK computational routine --
104 * -- LAPACK is a software package provided by Univ. of Tennessee, --
105 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
106 *
107 * .. Scalar Arguments ..
108  INTEGER INFO, LDA, M, N
109 * ..
110 * .. Array Arguments ..
111  INTEGER IPIV( * )
112  COMPLEX A( LDA, * )
113 * ..
114 *
115 * =====================================================================
116 *
117 * .. Parameters ..
118  COMPLEX ONE
119  parameter( one = ( 1.0e+0, 0.0e+0 ) )
120 * ..
121 * .. Local Scalars ..
122  INTEGER I, IINFO, J, JB, NB
123 * ..
124 * .. External Subroutines ..
125  EXTERNAL cgemm, cgetf2, claswp, ctrsm, xerbla
126 * ..
127 * .. External Functions ..
128  INTEGER ILAENV
129  EXTERNAL ilaenv
130 * ..
131 * .. Intrinsic Functions ..
132  INTRINSIC max, min
133 * ..
134 * .. Executable Statements ..
135 *
136 * Test the input parameters.
137 *
138  info = 0
139  IF( m.LT.0 ) THEN
140  info = -1
141  ELSE IF( n.LT.0 ) THEN
142  info = -2
143  ELSE IF( lda.LT.max( 1, m ) ) THEN
144  info = -4
145  END IF
146  IF( info.NE.0 ) THEN
147  CALL xerbla( 'CGETRF', -info )
148  RETURN
149  END IF
150 *
151 * Quick return if possible
152 *
153  IF( m.EQ.0 .OR. n.EQ.0 )
154  \$ RETURN
155 *
156 * Determine the block size for this environment.
157 *
158  nb = ilaenv( 1, 'CGETRF', ' ', m, n, -1, -1 )
159  IF( nb.LE.1 .OR. nb.GE.min( m, n ) ) THEN
160 *
161 * Use unblocked code.
162 *
163  CALL cgetf2( m, n, a, lda, ipiv, info )
164  ELSE
165 *
166 * Use blocked code.
167 *
168  DO 20 j = 1, min( m, n ), nb
169  jb = min( min( m, n )-j+1, nb )
170 *
171 * Update current block.
172 *
173  CALL cgemm( 'No transpose', 'No transpose',
174  \$ m-j+1, jb, j-1, -one,
175  \$ a( j, 1 ), lda, a( 1, j ), lda, one,
176  \$ a( j, j ), lda )
177
178 *
179 * Factor diagonal and subdiagonal blocks and test for exact
180 * singularity.
181 *
182  CALL cgetf2( m-j+1, jb, a( j, j ), lda, ipiv( j ), iinfo )
183 *
184 * Adjust INFO and the pivot indices.
185 *
186  IF( info.EQ.0 .AND. iinfo.GT.0 )
187  \$ info = iinfo + j - 1
188  DO 10 i = j, min( m, j+jb-1 )
189  ipiv( i ) = j - 1 + ipiv( i )
190  10 CONTINUE
191 *
192 * Apply interchanges to column 1:J-1
193 *
194  CALL claswp( j-1, a, lda, j, j+jb-1, ipiv, 1 )
195 *
196  IF ( j+jb.LE.n ) THEN
197 *
198 * Apply interchanges to column J+JB:N
199 *
200  CALL claswp( n-j-jb+1, a( 1, j+jb ), lda, j, j+jb-1,
201  \$ ipiv, 1 )
202 *
203  CALL cgemm( 'No transpose', 'No transpose',
204  \$ jb, n-j-jb+1, j-1, -one,
205  \$ a( j, 1 ), lda, a( 1, j+jb ), lda, one,
206  \$ a( j, j+jb ), lda )
207 *
208 * Compute block row of U.
209 *
210  CALL ctrsm( 'Left', 'Lower', 'No transpose', 'Unit',
211  \$ jb, n-j-jb+1, one, a( j, j ), lda,
212  \$ a( j, j+jb ), lda )
213  END IF
214
215  20 CONTINUE
216
217  END IF
218  RETURN
219 *
220 * End of CGETRF
221 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
subroutine ctrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRSM
Definition: ctrsm.f:180
subroutine cgetf2(M, N, A, LDA, IPIV, INFO)
CGETF2 computes the LU factorization of a general m-by-n matrix using partial pivoting with row inter...
Definition: cgetf2.f:108
subroutine claswp(N, A, LDA, K1, K2, IPIV, INCX)
CLASWP performs a series of row interchanges on a general rectangular matrix.
Definition: claswp.f:115
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