1*> \brief \b DGETRF2
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*  Definition:
9*  ===========
10*
11*       RECURSIVE SUBROUTINE DGETRF2( M, N, A, LDA, IPIV, INFO )
12*
13*       .. Scalar Arguments ..
14*       INTEGER            INFO, LDA, M, N
15*       ..
16*       .. Array Arguments ..
17*       INTEGER            IPIV( * )
18*       DOUBLE PRECISION   A( LDA, * )
19*       ..
20*
21*
22*> \par Purpose:
23*  =============
24*>
25*> \verbatim
26*>
27*> DGETRF2 computes an LU factorization of a general M-by-N matrix A
28*> using partial pivoting with row interchanges.
29*>
30*> The factorization has the form
31*>    A = P * L * U
32*> where P is a permutation matrix, L is lower triangular with unit
33*> diagonal elements (lower trapezoidal if m > n), and U is upper
34*> triangular (upper trapezoidal if m < n).
35*>
36*> This is the recursive version of the algorithm. It divides
37*> the matrix into four submatrices:
38*>
39*>        [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
40*>    A = [ -----|----- ]  with n1 = min(m,n)
41*>        [  A21 | A22  ]       n2 = n-n1
42*>
43*>                                       [ A11 ]
44*> The subroutine calls itself to factor [ --- ],
45*>                                       [ A12 ]
46*>                 [ A12 ]
47*> do the swaps on [ --- ], solve A12, update A22,
48*>                 [ A22 ]
49*>
50*> then calls itself to factor A22 and do the swaps on A21.
51*>
52*> \endverbatim
53*
54*  Arguments:
55*  ==========
56*
57*> \param[in] M
58*> \verbatim
59*>          M is INTEGER
60*>          The number of rows of the matrix A.  M >= 0.
61*> \endverbatim
62*>
63*> \param[in] N
64*> \verbatim
65*>          N is INTEGER
66*>          The number of columns of the matrix A.  N >= 0.
67*> \endverbatim
68*>
69*> \param[in,out] A
70*> \verbatim
71*>          A is DOUBLE PRECISION array, dimension (LDA,N)
72*>          On entry, the M-by-N matrix to be factored.
73*>          On exit, the factors L and U from the factorization
74*>          A = P*L*U; the unit diagonal elements of L are not stored.
75*> \endverbatim
76*>
77*> \param[in] LDA
78*> \verbatim
79*>          LDA is INTEGER
80*>          The leading dimension of the array A.  LDA >= max(1,M).
81*> \endverbatim
82*>
83*> \param[out] IPIV
84*> \verbatim
85*>          IPIV is INTEGER array, dimension (min(M,N))
86*>          The pivot indices; for 1 <= i <= min(M,N), row i of the
87*>          matrix was interchanged with row IPIV(i).
88*> \endverbatim
89*>
90*> \param[out] INFO
91*> \verbatim
92*>          INFO is INTEGER
93*>          = 0:  successful exit
94*>          < 0:  if INFO = -i, the i-th argument had an illegal value
95*>          > 0:  if INFO = i, U(i,i) is exactly zero. The factorization
96*>                has been completed, but the factor U is exactly
97*>                singular, and division by zero will occur if it is used
98*>                to solve a system of equations.
99*> \endverbatim
100*
101*  Authors:
102*  ========
103*
104*> \author Univ. of Tennessee
105*> \author Univ. of California Berkeley
106*> \author Univ. of Colorado Denver
107*> \author NAG Ltd.
108*
109*> \date November 2015
110*
111*> \ingroup doubleGEcomputational
112*
113*  =====================================================================
114      RECURSIVE SUBROUTINE DGETRF2( M, N, A, LDA, IPIV, INFO )
115*
116*  -- LAPACK computational routine (version 3.6.0) --
117*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
118*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
119*     November 2015
120*
121*     .. Scalar Arguments ..
122      INTEGER            INFO, LDA, M, N
123*     ..
124*     .. Array Arguments ..
125      INTEGER            IPIV( * )
126      DOUBLE PRECISION   A( LDA, * )
127*     ..
128*
129*  =====================================================================
130*
131*     .. Parameters ..
132      DOUBLE PRECISION   ONE, ZERO
133      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
134*     ..
135*     .. Local Scalars ..
136      DOUBLE PRECISION   SFMIN, TEMP
137      INTEGER            I, IINFO, N1, N2
138*     ..
139*     .. External Functions ..
140      DOUBLE PRECISION   DLAMCH
141      INTEGER            IDAMAX
142      EXTERNAL           DLAMCH, IDAMAX
143*     ..
144*     .. External Subroutines ..
145      EXTERNAL           DGEMM, DSCAL, DLASWP, DTRSM, XERBLA
146*     ..
147*     .. Intrinsic Functions ..
148      INTRINSIC          MAX, MIN
149*     ..
150*     .. Executable Statements ..
151*
152*     Test the input parameters
153*
154      INFO = 0
155      IF( M.LT.0 ) THEN
156         INFO = -1
157      ELSE IF( N.LT.0 ) THEN
158         INFO = -2
159      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
160         INFO = -4
161      END IF
162      IF( INFO.NE.0 ) THEN
163         CALL XERBLA( 'DGETRF2', -INFO )
164         RETURN
165      END IF
166*
167*     Quick return if possible
168*
169      IF( M.EQ.0 .OR. N.EQ.0 )
170     $   RETURN
171
172      IF ( M.EQ.1 ) THEN
173*
174*        Use unblocked code for one row case
175*        Just need to handle IPIV and INFO
176*
177         IPIV( 1 ) = 1
178         IF ( A(1,1).EQ.ZERO )
179     $      INFO = 1
180*
181      ELSE IF( N.EQ.1 ) THEN
182*
183*        Use unblocked code for one column case
184*
185*
186*        Compute machine safe minimum
187*
188         SFMIN = DLAMCH('S')
189*
190*        Find pivot and test for singularity
191*
192         I = IDAMAX( M, A( 1, 1 ), 1 )
193         IPIV( 1 ) = I
194         IF( A( I, 1 ).NE.ZERO ) THEN
195*
196*           Apply the interchange
197*
198            IF( I.NE.1 ) THEN
199               TEMP = A( 1, 1 )
200               A( 1, 1 ) = A( I, 1 )
201               A( I, 1 ) = TEMP
202            END IF
203*
204*           Compute elements 2:M of the column
205*
206            IF( ABS(A( 1, 1 )) .GE. SFMIN ) THEN
207               CALL DSCAL( M-1, ONE / A( 1, 1 ), A( 2, 1 ), 1 )
208            ELSE
209               DO 10 I = 1, M-1
210                  A( 1+I, 1 ) = A( 1+I, 1 ) / A( 1, 1 )
211   10          CONTINUE
212            END IF
213*
214         ELSE
215            INFO = 1
216         END IF
217*
218      ELSE
219*
220*        Use recursive code
221*
222         N1 = MIN( M, N ) / 2
223         N2 = N-N1
224*
225*               [ A11 ]
226*        Factor [ --- ]
227*               [ A21 ]
228*
229         CALL DGETRF2( M, N1, A, LDA, IPIV, IINFO )
230
231         IF ( INFO.EQ.0 .AND. IINFO.GT.0 )
232     $      INFO = IINFO
233*
234*                              [ A12 ]
235*        Apply interchanges to [ --- ]
236*                              [ A22 ]
237*
238         CALL DLASWP( N2, A( 1, N1+1 ), LDA, 1, N1, IPIV, 1 )
239*
240*        Solve A12
241*
242         CALL DTRSM( 'L', 'L', 'N', 'U', N1, N2, ONE, A, LDA,
243     $               A( 1, N1+1 ), LDA )
244*
245*        Update A22
246*
247         CALL DGEMM( 'N', 'N', M-N1, N2, N1, -ONE, A( N1+1, 1 ), LDA,
248     $               A( 1, N1+1 ), LDA, ONE, A( N1+1, N1+1 ), LDA )
249*
250*        Factor A22
251*
252         CALL DGETRF2( M-N1, N2, A( N1+1, N1+1 ), LDA, IPIV( N1+1 ),
253     $                 IINFO )
254*
255*        Adjust INFO and the pivot indices
256*
257         IF ( INFO.EQ.0 .AND. IINFO.GT.0 )
258     $      INFO = IINFO + N1
259         DO 20 I = N1+1, MIN( M, N )
260            IPIV( I ) = IPIV( I ) + N1
261   20    CONTINUE
262*
263*        Apply interchanges to A21
264*
265         CALL DLASWP( N1, A( 1, 1 ), LDA, N1+1, MIN( M, N), IPIV, 1 )
266*
267      END IF
268      RETURN
269*
270*     End of DGETRF2
271*
272      END
273