PDPOSVX(3)    ScaLAPACK routine of NEC Numeric Library Collection   PDPOSVX(3)



NAME
       PDPOSVX  -  use  the Cholesky factorization A = U**T*U or A = L*L**T to
       compute  the  solution  to  a   real   system   of   linear   equations
       A(IA:IA+N-1,JA:JA+N-1) * X = B(IB:IB+N-1,JB:JB+NRHS-1),

SYNOPSIS
       SUBROUTINE PDPOSVX( FACT,  UPLO,  N,  NRHS,  A, IA, JA, DESCA, AF, IAF,
                           JAF, DESCAF, EQUED, SR, SC, B, IB,  JB,  DESCB,  X,
                           IX,  JX,  DESCX,  RCOND,  FERR,  BERR, WORK, LWORK,
                           IWORK, LIWORK, INFO )

           CHARACTER       EQUED, FACT, UPLO

           INTEGER         IA, IAF, IB, INFO, IX, JA,  JAF,  JB,  JX,  LIWORK,
                           LWORK, N, NRHS

           DOUBLE          PRECISION RCOND

           INTEGER         DESCA(  *  ),  DESCAF( * ), DESCB( * ), DESCX( * ),
                           IWORK( * )

           DOUBLE          PRECISION A( * ), AF( * ), B( * ), BERR( * ), FERR(
                           * ), SC( * ), SR( * ), WORK( * ), X( * )

PURPOSE
       PDPOSVX  uses  the  Cholesky  factorization A = U**T*U or A = L*L**T to
       compute  the  solution  to  a   real   system   of   linear   equations
       A(IA:IA+N-1,JA:JA+N-1)   *   X   =   B(IB:IB+N-1,JB:JB+NRHS-1),   where
       A(IA:IA+N-1,JA:JA+N-1)   is   an    N-by-N    matrix    and    X    and
       B(IB:IB+N-1,JB:JB+NRHS-1) are N-by-NRHS matrices.

       Error  bounds  on  the  solution and a condition estimate are also pro-
       vided.  In the following comments Y denotes Y(IY:IY+M-1,JY:JY+K-1) a M-
       by-K matrix where Y can be A, AF, B and X.


       Notes
       =====

       Each  global data object is described by an associated description vec-
       tor.  This vector stores the information required to establish the map-
       ping 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  col-
       umn.
       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


DESCRIPTION
       The following steps are performed:

       1. If FACT = 'E', real scaling factors are computed to equilibrate
          the system:
             diag(SR) * A * diag(SC) * inv(diag(SC)) * X = diag(SR) * B
          Whether or not the system will be equilibrated depends on the
          scaling of the matrix A, but if equilibration is used, A is
          overwritten by diag(SR)*A*diag(SC) and B by diag(SR)*B.

       2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
          factor the matrix A (after equilibration if FACT = 'E') as
             A = U**T* U,  if UPLO = 'U', or
             A = L * L**T,  if UPLO = 'L',
          where U is an upper triangular matrix and L is a lower triangular
          matrix.

       3. The factored form of A is used to estimate the condition number
          of the matrix A.  If the reciprocal of the condition number is
          less than machine precision, steps 4-6 are skipped.

       4. The system of equations is solved for X using the factored form
          of A.

       5. Iterative refinement is applied to improve the computed solution
          matrix and calculate error bounds and backward error estimates
          for it.

       6. If equilibration was used, the matrix X is premultiplied by
          diag(SR) so that it solves the original system before
          equilibration.


ARGUMENTS
       FACT    (global input) CHARACTER
               Specifies  whether  or not the factored form of the matrix A is
               supplied on entry, and if not, whether the matrix A  should  be
               equilibrated  before it is factored.  = 'F':  On entry, AF con-
               tains the factored form of A.  If EQUED = 'Y', the matrix A has
               been  equilibrated  with  scaling factors given by S.  A and AF
               will not be modified.  = 'N':  The matrix A will be  copied  to
               AF and factored.
               =  'E':   The  matrix A will be equilibrated if necessary, then
               copied to AF and factored.

       UPLO    (global input) CHARACTER
               = 'U':  Upper triangle of A is stored;
               = 'L':  Lower triangle of A is stored.

       N       (global input) INTEGER
               The number of rows and columns to  be  operated  on,  i.e.  the
               order  of  the distributed submatrix A(IA:IA+N-1,JA:JA+N-1).  N
               >= 0.

       NRHS    (global input) INTEGER
               The number of right hand sides, i.e., the number of columns  of
               the distributed submatrices B and X.  NRHS >= 0.

       A       (local input/local output) DOUBLE PRECISION pointer into
               the  local  memory  to  an  array  of  local dimension ( LLD_A,
               LOCc(JA+N-1) ).  On entry, the symmetric matrix  A,  except  if
               FACT  =  'F'  and  EQUED = 'Y', then A must contain the equili-
               brated matrix diag(SR)*A*diag(SC).  If UPLO = 'U', the  leading
               N-by-N upper triangular part of A contains the upper triangular
               part of the matrix A, and the strictly lower triangular part of
               A  is  not referenced.  If UPLO = 'L', the leading N-by-N lower
               triangular part of A contains the lower triangular part of  the
               matrix  A,  and  the strictly upper triangular part of A is not
               referenced.  A is not modified if FACT = 'F' or 'N', or if FACT
               = 'E' and EQUED = 'N' on exit.

               On  exit,  if  FACT  = 'E' and EQUED = 'Y', A is overwritten by
               diag(SR)*A*diag(SC).

       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.

       AF      (local input or local output) DOUBLE PRECISION pointer
               into the local memory to an array of local dimension (  LLD_AF,
               LOCc(JA+N-1)).  If FACT = 'F', then AF is an input argument and
               on entry contains  the  triangular  factor  U  or  L  from  the
               Cholesky  factorization  A  = U**T*U or A = L*L**T, in the same
               storage format as A.  If EQUED .ne. 'N', then AF  is  the  fac-
               tored form of the equilibrated matrix diag(SR)*A*diag(SC).

               If  FACT  =  'N',  then  AF  is  an output argument and on exit
               returns the triangular factor U or L from the Cholesky  factor-
               ization A = U**T*U or A = L*L**T of the original matrix A.

               If  FACT  =  'E',  then  AF  is  an output argument and on exit
               returns the triangular factor U or L from the Cholesky  factor-
               ization  A  = U**T*U or A = L*L**T of the equilibrated matrix A
               (see the description of A for  the  form  of  the  equilibrated
               matrix).

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

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

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

       EQUED   (global input/global output) CHARACTER
               Specifies  the form of equilibration that was done.  = 'N':  No
               equilibration (always true if FACT = 'N').
               = 'Y':  Equilibration was done, i.e., A has  been  replaced  by
               diag(SR)  * A * diag(SC).  EQUED is an input variable if FACT =
               'F'; otherwise, it is an output variable.

       SR      (local input/local output) DOUBLE PRECISION array,
               dimension (LLD_A) The scale factors for  A  distributed  across
               process  rows;  not  accessed  if  EQUED = 'N'.  SR is an input
               variable if FACT = 'F'; otherwise, SR is  an  output  variable.
               If FACT = 'F' and EQUED = 'Y', each element of SR must be posi-
               tive.

       SC      (local input/local output) DOUBLE PRECISION array,
               dimension (LOC(N_A)) The scale factors for A distributed across
               process  columns;  not  accessed if EQUED = 'N'. SC is an input
               variable if FACT = 'F'; otherwise, SC is  an  output  variable.
               If FACT = 'F' and EQUED = 'Y', each element of SC must be posi-
               tive.

       B       (local input/local output) DOUBLE PRECISION pointer into
               the local memory to  an  array  of  local  dimension  (  LLD_B,
               LOCc(JB+NRHS-1)  ).   On  entry,  the N-by-NRHS right-hand side
               matrix B.  On exit, if EQUED = 'N', B is not modified; if TRANS
               = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if
               TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten  by
               diag(C)*B.

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

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

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

       X       (local input/local output) DOUBLE PRECISION pointer into
               the  local  memory  to  an  array  of  local dimension ( LLD_X,
               LOCc(JX+NRHS-1) ).  If INFO = 0, the N-by-NRHS solution  matrix
               X  to  the original system of equations.  Note that A and B are
               modified on exit if EQUED .ne. 'N', and  the  solution  to  the
               equilibrated system is inv(diag(SC))*X if TRANS = 'N' and EQUED
               = 'C' or or 'B'.

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

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

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

       RCOND   (global output) DOUBLE PRECISION
               The estimate of the reciprocal condition number of the matrix A
               after  equilibration  (if  done).   If  RCOND  is less than the
               machine precision (in particular, if RCOND = 0), the matrix  is
               singular  to working precision.  This condition is indicated by
               a return code of INFO > 0, and the solution  and  error  bounds
               are not computed.

       FERR    (local output) DOUBLE PRECISION array, dimension (LOC(N_B))
               The  estimated  forward  error  bounds for each solution vector
               X(j) (the j-th column of the solution matrix X).  If  XTRUE  is
               the  true solution, FERR(j) bounds the magnitude of the largest
               entry in (X(j) - XTRUE) divided by the magnitude of the largest
               entry  in  X(j).  The quality of the error bound depends on the
               quality of the estimate of norm(inv(A)) computed in  the  code;
               if the estimate of norm(inv(A)) is accurate, the error bound is
               guaranteed.

       BERR    (local output) DOUBLE PRECISION array, dimension (LOC(N_B))
               The componentwise relative backward error of each solution vec-
               tor  X(j) (i.e., the smallest relative change in any entry of A
               or B that makes X(j) an exact solution).

       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 = MAX( PDPOCON( LWORK ), PDPORFS( LWORK ) ) +
               LOCr( N_A ).  LWORK = 3*DESCA( LLD_ )

               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 = DESCA( LLD_ ) LIWORK = LOCr(N_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 INFO = -i, the i-th argument had an illegal value
               > 0: if INFO = i, and i is
               <=  N:  if  INFO  = i, the leading minor of order i of A is not
               positive definite, so the factorization could not be completed,
               and  the  solution  and  error bounds could not be computed.  =
               N+1: RCOND is less than machine precision.   The  factorization
               has  been completed, but the matrix is singular to working pre-
               cision, and  the  solution  and  error  bounds  have  not  been
               computed.



ScaLAPACK routine               31 October 2017                     PDPOSVX(3)