 PZGETRI(1) compute the inverse of a distributed matrix using the LU factorization computed by PZGETRF

## SYNOPSIS

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

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

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

COMPLEX*16 A( * ), WORK( * )

## PURPOSE

PZGETRI computes the inverse of a distributed matrix using the LU factorization computed by PZGETRF. 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) COMPLEX*16 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 PZGETRF. 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) COMPLEX*16 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.