SUBROUTINE PDGETRI( N, A, IA, JA, DESCA, IPIV, WORK, LWORK,
$ IWORK, LIWORK, INFO )
*
* -- ScaLAPACK routine (version 1.7.4) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* v1.7.4: May 10, 2006
* v1.7: May 1, 1997
*
* .. Scalar Arguments ..
INTEGER IA, INFO, JA, LIWORK, LWORK, N
* ..
* .. Array Arguments ..
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.
*
* =====================================================================
*
* .. Parameters ..
INTEGER BLOCK_CYCLIC_2D, CSRC_, CTXT_, DLEN_, DTYPE_,
$ LLD_, MB_, M_, NB_, N_, RSRC_
PARAMETER ( BLOCK_CYCLIC_2D = 1, DLEN_ = 9, DTYPE_ = 1,
$ CTXT_ = 2, M_ = 3, N_ = 4, MB_ = 5, NB_ = 6,
$ RSRC_ = 7, CSRC_ = 8, LLD_ = 9 )
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IACOL, IAROW, ICOFF, ICTXT, IROFF, IW, J,
$ JB, JN, LCM, LIWMIN, LWMIN, MP, MYCOL, MYROW,
$ NN, NP, NPCOL, NPROW, NQ
* ..
* .. Local Arrays ..
INTEGER DESCW( DLEN_ ), IDUM1( 2 ), IDUM2( 2 )
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, PCHK1MAT,
$ PDGEMM, PDLACPY, PDLASET, PDLAPIV,
$ PDTRSM, PDTRTRI, PXERBLA
* ..
* .. External Functions ..
INTEGER ICEIL, ILCM, INDXG2P, NUMROC
EXTERNAL ICEIL, ILCM, INDXG2P, NUMROC
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, MAX, MIN, MOD
* ..
* .. Executable Statements ..
*
* Get grid parameters
*
ICTXT = DESCA( CTXT_ )
CALL BLACS_GRIDINFO( ICTXT, NPROW, NPCOL, MYROW, MYCOL )
*
* Test the input parameters
*
INFO = 0
IF( NPROW.EQ.-1 ) THEN
INFO = -(500+CTXT_)
ELSE
CALL CHK1MAT( N, 1, N, 1, IA, JA, DESCA, 5, INFO )
IF( INFO.EQ.0 ) THEN
IROFF = MOD( IA-1, DESCA( MB_ ) )
ICOFF = MOD( JA-1, DESCA( NB_ ) )
IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ),
$ NPROW )
NP = NUMROC( N+IROFF, DESCA( MB_ ), MYROW, IAROW, NPROW )
LWMIN = NP * DESCA( NB_ )
*
MP = NUMROC( DESCA( M_ ), DESCA( MB_ ), MYROW,
$ DESCA( RSRC_ ), NPROW )
NQ = NUMROC( DESCA( N_ ), DESCA( NB_ ), MYCOL,
$ DESCA( CSRC_ ), NPCOL )
IF( NPROW.EQ.NPCOL ) THEN
LIWMIN = NQ + DESCA( NB_ )
ELSE
*
* Use the formula for the workspace given in PxLAPIV
* to compute the minimum size LIWORK for IWORK
*
* The formula in PxLAPIV is
* LDW = LOCc( M_P + MOD(IP-1, MB_P) ) +
* MB_P * CEIL( CEIL(LOCr(M_P)/MB_P) / (LCM/NPROW) )
*
* where
* M_P is the global length of the pivot vector
* MP = DESCA( M_ ) + DESCA( MB_ ) * NPROW
* I_P is IA
* I_P = IA
* MB_P is the block size use for the block cyclic distribution of the
* pivot vector
* MB_P = DESCA (MB_ )
* LOCc ( . )
* NUMROC ( . , DESCA ( NB_ ), MYCOL, DESCA ( CSRC_ ), NPCOL )
* LOCr ( . )
* NUMROC ( . , DESCA ( MB_ ), MYROW, DESCA ( RSRC_ ), NPROW )
* CEIL ( X / Y )
* ICEIL( X, Y )
* LCM
* LCM = ILCM( NPROW, NPCOL )
*
LCM = ILCM( NPROW, NPCOL )
LIWMIN = NUMROC( DESCA( M_ ) + DESCA( MB_ ) * NPROW
$ + MOD ( IA - 1, DESCA( MB_ ) ), DESCA ( NB_ ),
$ MYCOL, DESCA( CSRC_ ), NPCOL ) +
$ MAX ( DESCA( MB_ ) * ICEIL ( ICEIL(
$ NUMROC( DESCA( M_ ) + DESCA( MB_ ) * NPROW,
$ DESCA( MB_ ), MYROW, DESCA( RSRC_ ), NPROW ),
$ DESCA( MB_ ) ), LCM / NPROW ), DESCA( NB_ ) )
*
END IF
*
WORK( 1 ) = DBLE( LWMIN )
IWORK( 1 ) = LIWMIN
LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
IF( IROFF.NE.ICOFF .OR. IROFF.NE.0 ) THEN
INFO = -4
ELSE IF( DESCA( MB_ ).NE.DESCA( NB_ ) ) THEN
INFO = -(500+NB_)
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -8
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
END IF
IF( LWORK.EQ.-1 ) THEN
IDUM1( 1 ) = -1
ELSE
IDUM1( 1 ) = 1
END IF
IDUM2( 1 ) = 8
IF( LIWORK.EQ.-1 ) THEN
IDUM1( 2 ) = -1
ELSE
IDUM1( 2 ) = 1
END IF
IDUM2( 2 ) = 10
CALL PCHK1MAT( N, 1, N, 1, IA, JA, DESCA, 5, 2, IDUM1, IDUM2,
$ INFO )
END IF
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PDGETRI', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Form inv(U). If INFO > 0 from PDTRTRI, then U is singular,
* and the inverse is not computed.
*
CALL PDTRTRI( 'Upper', 'Non-unit', N, A, IA, JA, DESCA, INFO )
IF( INFO.GT.0 )
$ RETURN
*
* Define array descriptor for working array WORK
*
JN = MIN( ICEIL( JA, DESCA( NB_ ) ) * DESCA( NB_ ), JA+N-1 )
NN = ( ( JA+N-2 ) / DESCA( NB_ ) ) * DESCA( NB_ ) + 1
IACOL = INDXG2P( NN, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ), NPCOL )
CALL DESCSET( DESCW, N+IROFF, DESCA( NB_ ), DESCA( MB_ ),
$ DESCA( NB_ ), IAROW, IACOL, ICTXT, MAX( 1, NP ) )
IW = IROFF + 1
*
* Solve the equation inv(A)*L=inv(U) for inv(A) using blocked code.
*
DO 10 J = NN, JN+1, -DESCA( NB_ )
JB = MIN( DESCA( NB_ ), JA+N-J )
I = IA + J - JA
*
* Copy current block column of L to WORK and replace with zeros.
*
CALL PDLACPY( 'Lower', JA+N-1-J, JB, A, I+1, J, DESCA,
$ WORK, IW+J-JA+1, 1, DESCW )
CALL PDLASET( 'Lower', JA+N-1-J, JB, ZERO, ZERO, A, I+1, J,
$ DESCA )
*
* Compute current block column of inv(A).
*
IF( J+JB.LE.JA+N-1 )
$ CALL PDGEMM( 'No transpose', 'No transpose', N, JB,
$ JA+N-J-JB, -ONE, A, IA, J+JB, DESCA, WORK,
$ IW+J+JB-JA, 1, DESCW, ONE, A, IA, J, DESCA )
CALL PDTRSM( 'Right', 'Lower', 'No transpose', 'Unit', N, JB,
$ ONE, WORK, IW+J-JA, 1, DESCW, A, IA, J, DESCA )
DESCW( CSRC_ ) = MOD( DESCW( CSRC_ ) + NPCOL - 1, NPCOL )
*
10 CONTINUE
*
* Handle the last block of columns separately
*
JB = JN-JA+1
*
* Copy current block column of L to WORK and replace with zeros.
*
CALL PDLACPY( 'Lower', N-1, JB, A, IA+1, JA, DESCA, WORK, IW+1,
$ 1, DESCW )
CALL PDLASET( 'Lower', N-1, JB, ZERO, ZERO, A, IA+1, JA, DESCA )
*
* Compute current block column of inv(A).
*
IF( JA+JB.LE.JA+N-1 )
$ CALL PDGEMM( 'No transpose', 'No transpose', N, JB,
$ N-JB, -ONE, A, IA, JA+JB, DESCA, WORK, IW+JB, 1,
$ DESCW, ONE, A, IA, JA, DESCA )
CALL PDTRSM( 'Right', 'Lower', 'No transpose', 'Unit', N, JB,
$ ONE, WORK, IW, 1, DESCW, A, IA, JA, DESCA )
*
* Use the row pivots and apply them to the columns of the global
* matrix.
*
*
* JL: I do not get why the size of the PIVOT vector is DESCA( M_ ) + DESCA( MB_ )*NPROW
* should be DESCA( M_ ) + DESCA( MB_ ) no?
*
CALL DESCSET( DESCW, DESCA( M_ ) + DESCA( MB_ )*NPROW, 1,
$ DESCA( MB_ ), 1, DESCA( RSRC_ ), MYCOL, ICTXT,
$ MP+DESCA( MB_ ) )
CALL PDLAPIV( 'Backward', 'Columns', 'Column', N, N, A, IA,
$ JA, DESCA, IPIV, IA, 1, DESCW, IWORK )
*
WORK( 1 ) = DBLE( LWMIN )
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of PDGETRI
*
END