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NAME

       PDGETRI  - compute the inverse of a distributed matrix using the LU factorization computed
       by PDGETRF

SYNOPSIS

       SUBROUTINE PDGETRI( N, A, IA, JA, DESCA, IPIV, WORK, LWORK, IWORK, LIWORK, INFO )

           INTEGER         IA, INFO, JA, LIWORK, LWORK, N

           INTEGER         DESCA( * ), IPIV( * ), IWORK( * )

           DOUBLE          PRECISION A( * ), WORK( * )

PURPOSE

       PDGETRI computes the inverse of a distributed matrix using the LU  factorization  computed
       by  PDGETRF.  This  method  inverts  U  and  then  computes  the  inverse  of  sub(  A ) =
       A(IA:IA+N-1,JA:JA+N-1) denoted InvA by solving the system InvA*L = inv(U) for InvA.

       Notes
       =====

       Each global data object is described by an associated  description  vector.   This  vector
       stores the information required to establish the mapping between an object element and its
       corresponding process and memory location.

       Let A be a generic term for any 2D block cyclicly distributed array.  Such a global  array
       has  an  associated  description vector DESCA.  In the following comments, the character _
       should be read as "of the global array".

       NOTATION        STORED IN      EXPLANATION
       ---------------  --------------   --------------------------------------   DTYPE_A(global)
       DESCA( DTYPE_ )The descriptor type.  In this case,
                                      DTYPE_A = 1.
       CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                                      the BLACS process grid A is distribu-
                                      ted over. The context itself is glo-
                                      bal, but the handle (the integer
                                      value) may vary.
       M_A    (global) DESCA( M_ )    The number of rows in the global
                                      array A.
       N_A    (global) DESCA( N_ )    The number of columns in the global
                                      array A.
       MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                                      the rows of the array.
       NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                                      the columns of the array.
       RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                                      row  of the array A is distributed.  CSRC_A (global) DESCA(
       CSRC_ ) The process column over which the
                                      first column of the array A is
                                      distributed.
       LLD_A  (local)  DESCA( LLD_ )  The leading dimension of the local
                                      array.  LLD_A >= MAX(1,LOCr(M_A)).

       Let K be the number of rows or columns of  a  distributed  matrix,  and  assume  that  its
       process grid has dimension p x q.
       LOCr(  K  )  denotes  the  number  of elements of K that a process would receive if K were
       distributed over the p processes of its process column.
       Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K
       were distributed over the q processes of its process row.
       The  values  of  LOCr()  and  LOCc()  may  be  determined via a call to the ScaLAPACK tool
       function, NUMROC:
               LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
               LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ).  An  upper  bound  for  these
       quantities may be computed by:
               LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
               LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

ARGUMENTS

       N       (global input) INTEGER
               The  number  of  rows  and  columns  to  be  operated  on,  i.e.  the order of the
               distributed submatrix sub( A ). N >= 0.

       A       (local input/local output) DOUBLE PRECISION pointer into the
               local memory to an array of dimension (LLD_A,LOCc(JA+N-1)).  On entry,  the  local
               pieces  of  the L and U obtained by the factorization sub( A ) = P*L*U computed by
               PDGETRF. On exit, if INFO = 0, sub( A )  contains  the  inverse  of  the  original
               distributed matrix sub( A ).

       IA      (global input) INTEGER
               The row index in the global array A indicating the first row of sub( A ).

       JA      (global input) INTEGER
               The column index in the global array A indicating the first column of sub( A ).

       DESCA   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix A.

       IPIV    (local input) INTEGER array, dimension LOCr(M_A)+MB_A
               keeps track of the pivoting information. IPIV(i) is the global row index the local
               row i was swapped with.  This array is tied to the distributed matrix A.

       WORK    (local workspace/local output) DOUBLE PRECISION array,
               dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK.

       LWORK   (local or global input) INTEGER
               The dimension of the array WORK.  LWORK is local input and must be at least  LWORK
               =  LOCr(N+MOD(IA-1,MB_A))*NB_A.  WORK  is used to keep a copy of at most an entire
               column block of sub( A ).

               If LWORK = -1, then LWORK is global input and a workspace query  is  assumed;  the
               routine  only calculates the minimum and optimal size for all work arrays. Each of
               these values is returned in the first entry of the corresponding work  array,  and
               no error message is issued by PXERBLA.

       IWORK   (local workspace/local output) INTEGER array,
               dimension (LIWORK) On exit, IWORK(1) returns the minimal and optimal LIWORK.

       LIWORK  (local or global input) INTEGER
               The  dimension of the array IWORK used as workspace for physically transposing the
               pivots.  LIWORK is local input and must be at least if NPROW == NPCOL then  LIWORK
               = LOCc( N_A + MOD(JA-1, NB_A) ) + NB_A, else LIWORK =  LOCc( N_A + MOD(JA-1, NB_A)
               ) + MAX( CEIL(CEIL(LOCr(M_A)/MB_A)/(LCM/NPROW)), NB_A ) where  LCM  is  the  least
               common multiple of process rows and columns (NPROW and NPCOL).  end if

               If  LIWORK = -1, then LIWORK is global input and a workspace query is assumed; the
               routine only calculates the minimum and optimal size for all work arrays. Each  of
               these  values  is returned in the first entry of the corresponding work array, and
               no error message is issued by PXERBLA.

       INFO    (global output) INTEGER
               = 0:  successful exit
               < 0:  If the i-th argument is an array and the j-entry had an illegal value,  then
               INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then
               INFO = -i.  > 0:  If INFO = K, U(IA+K-1,IA+K-1) is exactly  zero;  the  matrix  is
               singular and its inverse could not be computed.