1*> \brief \b DGETRI
2*
3*  =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6*            http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO )
22*
23*       .. Scalar Arguments ..
24*       INTEGER            INFO, LDA, LWORK, N
25*       ..
26*       .. Array Arguments ..
27*       INTEGER            IPIV( * )
28*       DOUBLE PRECISION   A( LDA, * ), WORK( * )
29*       ..
30*
31*
32*> \par Purpose:
33*  =============
34*>
35*> \verbatim
36*>
37*> DGETRI computes the inverse of a matrix using the LU factorization
38*> computed by DGETRF.
39*>
40*> This method inverts U and then computes inv(A) by solving the system
41*> inv(A)*L = inv(U) for inv(A).
42*> \endverbatim
43*
44*  Arguments:
45*  ==========
46*
47*> \param[in] N
48*> \verbatim
49*>          N is INTEGER
50*>          The order of the matrix A.  N >= 0.
51*> \endverbatim
52*>
53*> \param[in,out] A
54*> \verbatim
55*>          A is DOUBLE PRECISION array, dimension (LDA,N)
56*>          On entry, the factors L and U from the factorization
57*>          A = P*L*U as computed by DGETRF.
58*>          On exit, if INFO = 0, the inverse of the original matrix A.
59*> \endverbatim
60*>
61*> \param[in] LDA
62*> \verbatim
63*>          LDA is INTEGER
64*>          The leading dimension of the array A.  LDA >= max(1,N).
65*> \endverbatim
66*>
67*> \param[in] IPIV
68*> \verbatim
69*>          IPIV is INTEGER array, dimension (N)
70*>          The pivot indices from DGETRF; for 1<=i<=N, row i of the
71*>          matrix was interchanged with row IPIV(i).
72*> \endverbatim
73*>
74*> \param[out] WORK
75*> \verbatim
76*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
77*>          On exit, if INFO=0, then WORK(1) returns the optimal LWORK.
78*> \endverbatim
79*>
80*> \param[in] LWORK
81*> \verbatim
82*>          LWORK is INTEGER
83*>          The dimension of the array WORK.  LWORK >= max(1,N).
84*>          For optimal performance LWORK >= N*NB, where NB is
85*>          the optimal blocksize returned by ILAENV.
86*>
87*>          If LWORK = -1, then a workspace query is assumed; the routine
88*>          only calculates the optimal size of the WORK array, returns
89*>          this value as the first entry of the WORK array, and no error
90*>          message related to LWORK is issued by XERBLA.
91*> \endverbatim
92*>
93*> \param[out] INFO
94*> \verbatim
95*>          INFO is INTEGER
96*>          = 0:  successful exit
97*>          < 0:  if INFO = -i, the i-th argument had an illegal value
98*>          > 0:  if INFO = i, U(i,i) is exactly zero; the matrix is
99*>                singular and its inverse could not be computed.
100*> \endverbatim
101*
102*  Authors:
103*  ========
104*
105*> \author Univ. of Tennessee
106*> \author Univ. of California Berkeley
107*> \author Univ. of Colorado Denver
108*> \author NAG Ltd.
109*
110*> \ingroup doubleGEcomputational
111*
112*  =====================================================================
113      SUBROUTINE DGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO )
114*
115*  -- LAPACK computational routine --
116*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
117*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
118*
119*     .. Scalar Arguments ..
120      INTEGER            INFO, LDA, LWORK, N
121*     ..
122*     .. Array Arguments ..
123      INTEGER            IPIV( * )
124      DOUBLE PRECISION   A( LDA, * ), WORK( * )
125*     ..
126*
127*  =====================================================================
128*
129*     .. Parameters ..
130      DOUBLE PRECISION   ZERO, ONE
131      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
132*     ..
133*     .. Local Scalars ..
134      LOGICAL            LQUERY
135      INTEGER            I, IWS, J, JB, JJ, JP, LDWORK, LWKOPT, NB,
136     $                   NBMIN, NN
137*     ..
138*     .. External Functions ..
139      INTEGER            ILAENV
140      EXTERNAL           ILAENV
141*     ..
142*     .. External Subroutines ..
143      EXTERNAL           DGEMM, DGEMV, DSWAP, DTRSM, DTRTRI, XERBLA
144*     ..
145*     .. Intrinsic Functions ..
146      INTRINSIC          MAX, MIN
147*     ..
148*     .. Executable Statements ..
149*
150*     Test the input parameters.
151*
152      INFO = 0
153      NB = ILAENV( 1, 'DGETRI', ' ', N, -1, -1, -1 )
154      LWKOPT = N*NB
155      WORK( 1 ) = LWKOPT
156      LQUERY = ( LWORK.EQ.-1 )
157      IF( N.LT.0 ) THEN
158         INFO = -1
159      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
160         INFO = -3
161      ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
162         INFO = -6
163      END IF
164      IF( INFO.NE.0 ) THEN
165         CALL XERBLA( 'DGETRI', -INFO )
166         RETURN
167      ELSE IF( LQUERY ) THEN
168         RETURN
169      END IF
170*
171*     Quick return if possible
172*
173      IF( N.EQ.0 )
174     $   RETURN
175*
176*     Form inv(U).  If INFO > 0 from DTRTRI, then U is singular,
177*     and the inverse is not computed.
178*
179      CALL DTRTRI( 'Upper', 'Non-unit', N, A, LDA, INFO )
180      IF( INFO.GT.0 )
181     $   RETURN
182*
183      NBMIN = 2
184      LDWORK = N
185      IF( NB.GT.1 .AND. NB.LT.N ) THEN
186         IWS = MAX( LDWORK*NB, 1 )
187         IF( LWORK.LT.IWS ) THEN
188            NB = LWORK / LDWORK
189            NBMIN = MAX( 2, ILAENV( 2, 'DGETRI', ' ', N, -1, -1, -1 ) )
190         END IF
191      ELSE
192         IWS = N
193      END IF
194*
195*     Solve the equation inv(A)*L = inv(U) for inv(A).
196*
197      IF( NB.LT.NBMIN .OR. NB.GE.N ) THEN
198*
199*        Use unblocked code.
200*
201         DO 20 J = N, 1, -1
202*
203*           Copy current column of L to WORK and replace with zeros.
204*
205            DO 10 I = J + 1, N
206               WORK( I ) = A( I, J )
207               A( I, J ) = ZERO
208   10       CONTINUE
209*
210*           Compute current column of inv(A).
211*
212            IF( J.LT.N )
213     $         CALL DGEMV( 'No transpose', N, N-J, -ONE, A( 1, J+1 ),
214     $                     LDA, WORK( J+1 ), 1, ONE, A( 1, J ), 1 )
215   20    CONTINUE
216      ELSE
217*
218*        Use blocked code.
219*
220         NN = ( ( N-1 ) / NB )*NB + 1
221         DO 50 J = NN, 1, -NB
222            JB = MIN( NB, N-J+1 )
223*
224*           Copy current block column of L to WORK and replace with
225*           zeros.
226*
227            DO 40 JJ = J, J + JB - 1
228               DO 30 I = JJ + 1, N
229                  WORK( I+( JJ-J )*LDWORK ) = A( I, JJ )
230                  A( I, JJ ) = ZERO
231   30          CONTINUE
232   40       CONTINUE
233*
234*           Compute current block column of inv(A).
235*
236            IF( J+JB.LE.N )
237     $         CALL DGEMM( 'No transpose', 'No transpose', N, JB,
238     $                     N-J-JB+1, -ONE, A( 1, J+JB ), LDA,
239     $                     WORK( J+JB ), LDWORK, ONE, A( 1, J ), LDA )
240            CALL DTRSM( 'Right', 'Lower', 'No transpose', 'Unit', N, JB,
241     $                  ONE, WORK( J ), LDWORK, A( 1, J ), LDA )
242   50    CONTINUE
243      END IF
244*
245*     Apply column interchanges.
246*
247      DO 60 J = N - 1, 1, -1
248         JP = IPIV( J )
249         IF( JP.NE.J )
250     $      CALL DSWAP( N, A( 1, J ), 1, A( 1, JP ), 1 )
251   60 CONTINUE
252*
253      WORK( 1 ) = IWS
254      RETURN
255*
256*     End of DGETRI
257*
258      END
259