SUBROUTINE PDPOCON( UPLO, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, $ LWORK, IWORK, LIWORK, INFO ) * * -- ScaLAPACK routine (version 1.7) -- * University of Tennessee, Knoxville, Oak Ridge National Laboratory, * and University of California, Berkeley. * May 25, 2001 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER IA, INFO, JA, LIWORK, LWORK, N DOUBLE PRECISION ANORM, RCOND * .. * .. Array Arguments .. INTEGER DESCA( * ), IWORK( * ) DOUBLE PRECISION A( * ), WORK( * ) * .. * * Purpose * ======= * * PDPOCON estimates the reciprocal of the condition number (in the * 1-norm) of a real symmetric positive definite distributed matrix * using the Cholesky factorization A = U**T*U or A = L*L**T computed by * PDPOTRF. * * 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) DOUBLE PRECISION 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 PDPOTRF. * * 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) DOUBLE PRECISION * The 1-norm (or infinity-norm) of the symmetric distributed * matrix A(IA:IA+N-1,JA:JA+N-1). * * RCOND (global output) DOUBLE PRECISION * 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) 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 >= 2*LOCr(N+MOD(IA-1,MB_A)) + 2*LOCc(N+MOD(JA-1,NB_A))+ * MAX( 2, MAX(NB_A*CEIL(NPROW-1,NPCOL),LOCc(N+MOD(JA-1,NB_A)) + * NB_A*CEIL(NPCOL-1,NPROW)) ). * * 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. * LIWORK is local input and must be at least * LIWORK >= LOCr(N+MOD(IA-1,MB_A)). * * 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. * * ===================================================================== * * .. 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 ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+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, LIWMIN, LWMIN, MYCOL, MYROW, NP, $ NPCOL, NPROW, NPMOD, NQ, NQMOD DOUBLE PRECISION AINVNM, SCALE, SL, SU, SMLNUM DOUBLE PRECISION WMAX * .. * .. Local Arrays .. INTEGER DESCV( DLEN_ ), DESCX( DLEN_ ), IDUM1( 3 ), $ IDUM2( 3 ) * .. * .. External Subroutines .. EXTERNAL BLACS_GRIDINFO, CHK1MAT, DESCSET, DGEBR2D, $ DGEBS2D, INFOG2L, PCHK1MAT, PDAMAX, $ PDLATRS, PDLACON, PDRSCL, PB_TOPGET, $ PB_TOPSET, PXERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ICEIL, INDXG2P, NUMROC DOUBLE PRECISION PDLAMCH EXTERNAL ICEIL, INDXG2P, LSAME, NUMROC, PDLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, ICHAR, MAX, 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 = -(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 + 2*NQMOD + $ MAX( 2, MAX( DESCA( NB_ )* $ MAX( 1, ICEIL( NPROW-1, NPCOL ) ), NQMOD + $ DESCA( NB_ )* $ MAX( 1, ICEIL( NPCOL-1, NPROW ) ) ) ) WORK( 1 ) = DBLE( LWMIN ) LIWMIN = NPMOD IWORK( 1 ) = LIWMIN LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.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( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN IWORK( 1 ) = LIWMIN 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( LIWORK.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, 'PDPOCON', -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 PB_TOPGET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) CALL PB_TOPGET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) CALL PB_TOPSET( ICTXT, 'Combine', 'Columnwise', '1-tree' ) CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', '1-tree' ) * SMLNUM = PDLAMCH( 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 IPNL = IPV + NP IPNU = IPNL + NQ IPW = IPNU + 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 PDLACON( N, WORK( IPV ), IV, JV, DESCV, WORK( IPX ), IX, JX, $ DESCX, IWORK, AINVNM, KASE ) IF( KASE.NE.0 ) THEN IF( UPPER ) THEN * * Multiply by inv(U'). * DESCX( CSRC_ ) = IACOL CALL PDLATRS( 'Upper', 'Transpose', 'Non-unit', NORMIN, $ N, A, IA, JA, DESCA, WORK( IPX ), IX, JX, $ DESCX, SL, WORK( IPNL ), WORK( IPW ) ) DESCX( CSRC_ ) = MYCOL NORMIN = 'Y' * * Multiply by inv(U). * DESCX( CSRC_ ) = IACOL CALL PDLATRS( 'Upper', 'No transpose', 'Non-unit', NORMIN, $ N, A, IA, JA, DESCA, WORK( IPX ), IX, JX, $ DESCX, SU, WORK( IPNU ), WORK( IPW ) ) DESCX( CSRC_ ) = MYCOL ELSE * * Multiply by inv(L). * DESCX( CSRC_ ) = IACOL CALL PDLATRS( 'Lower', 'No transpose', 'Non-unit', NORMIN, $ N, A, IA, JA, DESCA, WORK( IPX ), IX, JX, $ DESCX, SL, WORK( IPNL ), WORK( IPW ) ) DESCX( CSRC_ ) = MYCOL NORMIN = 'Y' * * Multiply by inv(L'). * DESCX( CSRC_ ) = IACOL CALL PDLATRS( 'Lower', 'Transpose', 'Non-unit', NORMIN, $ N, A, IA, JA, DESCA, WORK( IPX ), IX, JX, $ DESCX, SU, WORK( 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 PDAMAX( N, WMAX, IXX, WORK( IPX ), IX, JX, DESCX, 1 ) IF( DESCX( M_ ).EQ.1 .AND. N.EQ.1 ) THEN CALL PB_TOPGET( ICTXT, 'Broadcast', 'Columnwise', CBTOP ) IF( MYROW.EQ.IAROW ) THEN CALL DGEBS2D( ICTXT, 'Column', CBTOP, 1, 1, WMAX, 1 ) ELSE CALL DGEBR2D( ICTXT, 'Column', CBTOP, 1, 1, WMAX, 1, $ IAROW, MYCOL ) END IF END IF IF( SCALE.LT.ABS( WMAX )*SMLNUM .OR. SCALE.EQ.ZERO ) $ GO TO 20 CALL PDRSCL( 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 PB_TOPSET( ICTXT, 'Combine', 'Columnwise', COLCTOP ) CALL PB_TOPSET( ICTXT, 'Combine', 'Rowwise', ROWCTOP ) * RETURN * * End of PDPOCON * END