SUBROUTINE PCPOCON( UPLO, N, A, IA, JA, DESCA, ANORM, RCOND, WORK,
$ LWORK, RWORK, LRWORK, INFO )
*
* -- ScaLAPACK routine (version 1.5) --
* University of Tennessee, Knoxville, Oak Ridge National Laboratory,
* and University of California, Berkeley.
* May 1, 1997
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER IA, INFO, JA, LRWORK, LWORK, N
REAL ANORM, RCOND
* ..
* .. Array Arguments ..
INTEGER DESCA( * )
REAL RWORK( * )
COMPLEX A( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* PCPOCON estimates the reciprocal of the condition number (in the
* 1-norm) of a complex Hermitian positive definite distributed matrix
* using the Cholesky factorization A = U**H*U or A = L*L**H computed by
* PCPOTRF.
*
* An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and
* the reciprocal of the condition number is computed as
* RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) *
* norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).
*
* 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
* =========
*
* UPLO (global input) CHARACTER
* Specifies whether the factor stored in
* A(IA:IA+N-1,JA:JA+N-1) is upper or lower triangular.
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (global input) INTEGER
* The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1).
* N >= 0.
*
* A (local input) COMPLEX pointer into the local memory to
* an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this
* array contains the local pieces of the factors L or U from
* the Cholesky factorization A(IA:IA+N-1,JA:JA+N-1) = U'*U or
* L*L', as computed by PCPOTRF.
*
* 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.
*
* ANORM (global input) REAL
* The 1-norm (or infinity-norm) of the hermitian distributed
* matrix A(IA:IA+N-1,JA:JA+N-1).
*
* RCOND (global output) REAL
* The reciprocal of the condition number of the distributed
* matrix A(IA:IA+N-1,JA:JA+N-1), computed as
* RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) *
* norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).
*
* WORK (local workspace/local output) COMPLEX 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 >= 2*LOCr(N+MOD(IA-1,MB_A)) +
* MAX( 2, MAX(NB_A*MAX(1,CEIL(P-1,Q)),LOCc(N+MOD(JA-1,NB_A)) +
* NB_A*MAX(1,CEIL(Q-1,P))) ).
*
* 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.
*
* RWORK (local workspace/local output) REAL array,
* dimension (LRWORK)
* On exit, RWORK(1) returns the minimal and optimal LRWORK.
*
* LRWORK (local or global input) INTEGER
* The dimension of the array RWORK.
* LRWORK is local input and must be at least
* LRWORK >= 2*LOCc(N+MOD(JA-1,NB_A)).
*
* If LRWORK = -1, then LRWORK 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.
*
* =====================================================================
*
* .. 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 )
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
CHARACTER CBTOP, COLCTOP, NORMIN, ROWCTOP
INTEGER IACOL, IAROW, ICOFF, ICTXT, IIA, IPNL, IPNU,
$ IPV, IPW, IPX, IROFF, IV, IX, IXX, JJA, JV,
$ JX, KASE, LRWMIN, LWMIN, MYCOL, MYROW, NP,
$ NPCOL, NPROW, NPMOD, NQ, NQMOD
REAL AINVNM, SCALE, SL, SU, SMLNUM
COMPLEX WMAX, ZDUM
* ..
* .. Local Arrays ..
INTEGER DESCV( DLEN_ ), DESCX( DLEN_ ), IDUM1( 3 ),
$ IDUM2( 3 )
* ..
* .. External Subroutines ..
EXTERNAL BLACS_GRIDINFO, CGEBR2D, CGEBS2D, CHK1MAT,
$ DESCSET, INFOG2L, PCAMAX, PCHK1MAT, PCLATRS,
$ PCLACON, PCSRSCL, PTOPGET, PTOPSET,
$ PXERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ICEIL, INDXG2P, NUMROC
REAL PSLAMCH
EXTERNAL ICEIL, INDXG2P, LSAME, NUMROC, PSLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, ICHAR, MAX, MOD, REAL
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
* ..
* .. 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 = -(600+CTXT_)
ELSE
CALL CHK1MAT( N, 2, N, 2, IA, JA, DESCA, 6, INFO )
IF( INFO.EQ.0 ) THEN
UPPER = LSAME( UPLO, 'U' )
IAROW = INDXG2P( IA, DESCA( MB_ ), MYROW, DESCA( RSRC_ ),
$ NPROW )
IACOL = INDXG2P( JA, DESCA( NB_ ), MYCOL, DESCA( CSRC_ ),
$ NPCOL )
NPMOD = NUMROC( N + MOD( IA-1, DESCA( MB_ ) ), DESCA( MB_ ),
$ MYROW, IAROW, NPROW )
NQMOD = NUMROC( N + MOD( JA-1, DESCA( NB_ ) ), DESCA( NB_ ),
$ MYCOL, IACOL, NPCOL )
LWMIN = 2*NPMOD +
$ MAX( 2, MAX( DESCA( NB_ )*
$ MAX( 1, ICEIL( NPROW-1, NPCOL ) ), NQMOD +
$ DESCA( NB_ )*
$ MAX( 1, ICEIL( NPCOL-1, NPROW ) ) ) )
WORK( 1 ) = REAL( LWMIN )
LRWMIN = 2*NQMOD
RWORK( 1 ) = REAL( LRWMIN )
LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 )
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -7
ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -10
ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( UPPER ) THEN
IDUM1( 1 ) = ICHAR( 'U' )
ELSE
IDUM1( 1 ) = ICHAR( 'L' )
END IF
IDUM2( 1 ) = 1
IF( LWORK.EQ.-1 ) THEN
IDUM1( 2 ) = -1
ELSE
IDUM1( 2 ) = 1
END IF
IDUM2( 2 ) = 10
IF( LRWORK.EQ.-1 ) THEN
IDUM1( 3 ) = -1
ELSE
IDUM1( 3 ) = 1
END IF
IDUM2( 3 ) = 12
CALL PCHK1MAT( N, 2, N, 2, IA, JA, DESCA, 6, 3, IDUM1, IDUM2,
$ INFO )
END IF
*
IF( INFO.NE.0 ) THEN
CALL PXERBLA( ICTXT, 'PCPOCON', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
ELSE IF( N.EQ.1 ) THEN
RCOND = ONE
RETURN
END IF
*
CALL PTOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
CALL PTOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
CALL PTOPSET( ICTXT, 'Combine', 'Columnwise', '1-tree' )
CALL PTOPSET( ICTXT, 'Combine', 'Rowwise', '1-tree' )
*
SMLNUM = PSLAMCH( ICTXT, 'Safe minimum' )
IROFF = MOD( IA-1, DESCA( MB_ ) )
ICOFF = MOD( JA-1, DESCA( NB_ ) )
CALL INFOG2L( IA, JA, DESCA, NPROW, NPCOL, MYROW, MYCOL, IIA, JJA,
$ IAROW, IACOL )
NP = NUMROC( N+IROFF, DESCA( MB_ ), MYROW, IAROW, NPROW )
NQ = NUMROC( N+ICOFF, DESCA( NB_ ), MYCOL, IACOL, NPCOL )
IV = IROFF + 1
IX = IV
JV = ICOFF + 1
JX = JV
*
IPX = 1
IPV = IPX + NP
IPW = IPV + NP
IPNL = 1
IPNU = IPNL + NQ
*
CALL DESCSET( DESCV, N+IROFF, 1, DESCA( MB_ ), 1, IAROW, MYCOL,
$ ICTXT, MAX( 1, NP ) )
CALL DESCSET( DESCX, N+IROFF, 1, DESCA( MB_ ), 1, IAROW, MYCOL,
$ ICTXT, MAX( 1, NP ) )
*
* Estimate the 1-norm (or I-norm) of inv(A).
*
AINVNM = ZERO
KASE = 0
NORMIN = 'N'
*
10 CONTINUE
CALL PCLACON( N, WORK( IPV ), IV, JV, DESCV, WORK( IPX ), IX, JX,
$ DESCX, AINVNM, KASE )
IF( KASE.NE.0 ) THEN
IF( UPPER ) THEN
*
* Multiply by inv(U').
*
DESCX( CSRC_ ) = IACOL
CALL PCLATRS( 'Upper', 'Conjugate transpose', 'Non-unit',
$ NORMIN, N, A, IA, JA, DESCA, WORK( IPX ),
$ IX, JX, DESCX, SL, RWORK( IPNL ),
$ WORK( IPW ) )
DESCX( CSRC_ ) = MYCOL
NORMIN = 'Y'
*
* Multiply by inv(U).
*
DESCX( CSRC_ ) = IACOL
CALL PCLATRS( 'Upper', 'No transpose', 'Non-unit', NORMIN,
$ N, A, IA, JA, DESCA, WORK( IPX ), IX, JX,
$ DESCX, SU, RWORK( IPNU ), WORK( IPW ) )
DESCX( CSRC_ ) = MYCOL
ELSE
*
* Multiply by inv(L).
*
DESCX( CSRC_ ) = IACOL
CALL PCLATRS( 'Lower', 'No transpose', 'Non-unit', NORMIN,
$ N, A, IA, JA, DESCA, WORK( IPX ), IX, JX,
$ DESCX, SL, RWORK( IPNL ), WORK( IPW ) )
DESCX( CSRC_ ) = MYCOL
NORMIN = 'Y'
*
* Multiply by inv(L').
*
DESCX( CSRC_ ) = IACOL
CALL PCLATRS( 'Lower', 'Conjugate transpose', 'Non-unit',
$ NORMIN, N, A, IA, JA, DESCA, WORK( IPX ),
$ IX, JX, DESCX, SU, RWORK( IPNU ),
$ WORK( IPW ) )
DESCX( CSRC_ ) = MYCOL
END IF
*
* Multiply by 1/SCALE if doing so will not cause overflow.
*
SCALE = SL*SU
IF( SCALE.NE.ONE ) THEN
CALL PCAMAX( N, WMAX, IXX, WORK( IPX ), IX, JX, DESCX, 1 )
IF( DESCX( M_ ).EQ.1 .AND. N.EQ.1 ) THEN
CALL PTOPGET( ICTXT, 'Broadcast', 'Columnwise', CBTOP )
IF( MYROW.EQ.IAROW ) THEN
CALL CGEBS2D( ICTXT, 'Column', CBTOP, 1, 1, WMAX, 1 )
ELSE
CALL CGEBR2D( ICTXT, 'Column', CBTOP, 1, 1, WMAX, 1,
$ IAROW, MYCOL )
END IF
END IF
IF( SCALE.LT.CABS1( WMAX )*SMLNUM .OR. SCALE.EQ.ZERO )
$ GO TO 20
CALL PCSRSCL( N, SCALE, WORK( IPX ), IX, JX, DESCX, 1 )
END IF
GO TO 10
END IF
*
* Compute the estimate of the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
20 CONTINUE
*
CALL PTOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP )
CALL PTOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP )
*
RETURN
*
* End of PCPOCON
*
END