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NAME

       PDGESVX  -  use  the  LU  factorization 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 PDGESVX( FACT, TRANS, N, NRHS, A, IA, JA, DESCA, AF, IAF,  JAF,  DESCAF,  IPIV,
                           EQUED,  R,  C,  B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR,
                           WORK, LWORK, IWORK, LIWORK, INFO )

           CHARACTER       EQUED, FACT, TRANS

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

           DOUBLE          PRECISION RCOND

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

           DOUBLE          PRECISION A( * ), AF( * ), B( * ), BERR( * ), C( * ), FERR( * ), R(  *
                           ), WORK( * ), X( * )

PURPOSE

       PDGESVX  uses  the  LU  factorization  to  compute the solution to a real system of linear
       equations

       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 provided.

       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

DESCRIPTION

       In   the   following   description,   A   denotes   A(IA:IA+N-1,JA:JA+N-1),   B    denotes
       B(IB:IB+N-1,JB:JB+NRHS-1) and X denotes
       X(IX:IX+N-1,JX:JX+NRHS-1).

       The following steps are performed:

       1. If FACT = 'E', real scaling factors are computed to equilibrate
          the system:
             TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B
             TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
             TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
          or diag(C)*B (if TRANS = 'T' or 'C').

       2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
          matrix A (after equilibration if FACT = 'E') as
             A = P * L * U,
          where P is a permutation matrix, L is a unit lower triangular
          matrix, and U is upper triangular.

       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 FACT = 'E' and equilibration was used, the matrix X is
          premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if
          TRANS = 'T' or 'C') 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(IA:IA+N-1,JA:JA+N-1) is
               supplied on entry, and if not,
               whether the matrix A(IA:IA+N-1,JA:JA+N-1) should  be  equilibrated  before  it  is
               factored.  = 'F':  On entry, AF(IAF:IAF+N-1,JAF:JAF+N-1) and IPIV con-
               tain the factored form of A(IA:IA+N-1,JA:JA+N-1).  If EQUED is not 'N', the matrix
               A(IA:IA+N-1,JA:JA+N-1) has been equilibrated with scaling factors given by  R  and
               C.    A(IA:IA+N-1,JA:JA+N-1),   AF(IAF:IAF+N-1,JAF:JAF+N-1),   and  IPIV  are  not
               modified.  = 'N':  The matrix A(IA:IA+N-1,JA:JA+N-1) will be copied to
               AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored.
               = 'E':  The matrix A(IA:IA+N-1,JA:JA+N-1) will be  equili-  brated  if  necessary,
               then copied to AF(IAF:IAF+N-1,JAF:JAF+N-1) and factored.

       TRANS   (global input) CHARACTER
               Specifies the form of the system of equations:
               = 'N':  A(IA:IA+N-1,JA:JA+N-1) * X(IX:IX+N-1,JX:JX+NRHS-1)
               = B(IB:IB+N-1,JB:JB+NRHS-1)     (No transpose)
               = 'T':  A(IA:IA+N-1,JA:JA+N-1)**T * X(IX:IX+N-1,JX:JX+NRHS-1)
               = B(IB:IB+N-1,JB:JB+NRHS-1)  (Transpose)
               = 'C':  A(IA:IA+N-1,JA:JA+N-1)**H * X(IX:IX+N-1,JX:JX+NRHS-1)
               = B(IB:IB+N-1,JB:JB+NRHS-1)  (Transpose)

       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(IB:IB+N-1,JB:JB+NRHS-1) and
               X(IX:IX+N-1,JX:JX+NRHS-1).  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 N-by-N matrix A(IA:IA+N-1,JA:JA+N-1).  If FACT = 'F' and EQUED is not 'N',
               then A(IA:IA+N-1,JA:JA+N-1) must have been equilibrated by
               the scaling factors in R and/or C.  A(IA:IA+N-1,JA:JA+N-1) is not modified if FACT
               = 'F' or  'N', or if FACT = 'E' and EQUED = 'N' on exit.

               On exit, if EQUED .ne. 'N', A(IA:IA+N-1,JA:JA+N-1) is scaled as follows:
               EQUED = 'R':  A(IA:IA+N-1,JA:JA+N-1) :=
               diag(R) * A(IA:IA+N-1,JA:JA+N-1)
               EQUED = 'C':  A(IA:IA+N-1,JA:JA+N-1) :=
               A(IA:IA+N-1,JA:JA+N-1) * diag(C)
               EQUED = 'B':  A(IA:IA+N-1,JA:JA+N-1) :=
               diag(R) * A(IA:IA+N-1,JA:JA+N-1) * diag(C).

       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(IAF:IAF+N-1,JAF:JAF+N-1) is an input  argument  and  on  entry
               contains the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U
               as computed by PDGETRF.  If EQUED .ne. 'N', then AF is the factored  form  of  the
               equilibrated matrix A(IA:IA+N-1,JA:JA+N-1).

               If  FACT = 'N', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit
               returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) =  P*L*U
               of the original
               matrix A(IA:IA+N-1,JA:JA+N-1).

               If  FACT = 'E', then AF(IAF:IAF+N-1,JAF:JAF+N-1) is an output argument and on exit
               returns the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) =  P*L*U
               of the equili-
               brated matrix A(IA:IA+N-1,JA:JA+N-1) (see the description of
               A(IA:IA+N-1,JA:JA+N-1) 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.

       IPIV    (local input or local output) INTEGER array, dimension
               LOCr(M_A)+MB_A.  If  FACT  =  'F',  then  IPIV is an input argu- ment and on entry
               contains the pivot indices from the fac- torization A(IA:IA+N-1,JA:JA+N-1) = P*L*U
               as  computed  by  PDGETRF; IPIV(i) -> The global row local row i was swapped with.
               This array must be aligned with A( IA:IA+N-1, * ).

               If FACT = 'N', then IPIV is an output argument and  on  exit  contains  the  pivot
               indices  from  the  factorization  A(IA:IA+N-1,JA:JA+N-1)  = P*L*U of the original
               matrix
               A(IA:IA+N-1,JA:JA+N-1).

               If FACT = 'E', then IPIV is an output argument and  on  exit  contains  the  pivot
               indices  from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U of the equilibrated
               matrix
               A(IA:IA+N-1,JA:JA+N-1).

       EQUED   (global input or global output) CHARACTER
               Specifies the form of equilibration that  was  done.   =  'N':   No  equilibration
               (always true if FACT = 'N').
               =  'R':  Row equilibration, i.e., A(IA:IA+N-1,JA:JA+N-1) has been premultiplied by
               diag(R).  = 'C':  Column  equilibration,  i.e.,  A(IA:IA+N-1,JA:JA+N-1)  has  been
               postmultiplied by diag(C).  = 'B':  Both row and column equilibration, i.e.,
               A(IA:IA+N-1,JA:JA+N-1) has been replaced by
               diag(R)  * A(IA:IA+N-1,JA:JA+N-1) * diag(C).  EQUED is an input variable if FACT =
               'F'; otherwise, it is an output variable.

       R       (local input or local output) DOUBLE PRECISION array,
               dimension LOCr(M_A).  The row scale factors for A(IA:IA+N-1,JA:JA+N-1).
               If EQUED = 'R' or  'B',  A(IA:IA+N-1,JA:JA+N-1)  is  multiplied  on  the  left  by
               diag(R); if EQUED='N' or 'C', R is not acces- sed.  R is an input variable if FACT
               = 'F'; otherwise, R is an output variable.  If FACT = 'F' and EQUED = 'R' or  'B',
               each  element of R must be positive.  R is replicated in every process column, and
               is aligned with the distributed matrix A.

       C       (local input or local output) DOUBLE PRECISION array,
               dimension LOCc(N_A).  The column scale factors for A(IA:IA+N-1,JA:JA+N-1).
               If EQUED = 'C' or 'B',  A(IA:IA+N-1,JA:JA+N-1)  is  multiplied  on  the  right  by
               diag(C); if EQUED = 'N' or 'R', C is not accessed.  C is an input variable if FACT
               = 'F'; otherwise, C is an output variable.  If FACT = 'F' and EQUED = 'C' or C  is
               replicated in every process row, and is aligned with the distributed matrix A.

       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(IB:IB+N-1,JB:JB+NRHS-1). On exit, if
               EQUED = 'N', B(IB:IB+N-1,JB:JB+NRHS-1) is not modified; if TRANS = 'N' and EQUED =
               'R'  or 'B', B is overwritten by diag(R)*B(IB:IB+N-1,JB:JB+NRHS-1); if TRANS = 'T'
               or 'C'
               and EQUED = 'C' or 'B', B(IB:IB+N-1,JB:JB+NRHS-1) is over-
               written by diag(C)*B(IB:IB+N-1,JB:JB+NRHS-1).

       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(IX:IX+N-1,JX:JX+NRHS-1) to the original
               system of equations.  Note that A(IA:IA+N-1,JA:JA+N-1) and
               B(IB:IB+N-1,JB:JB+NRHS-1) are modified on exit if EQUED .ne. 'N', and the solution
               to the equilibrated system is inv(diag(C))*X(IX:IX+N-1,JX:JX+NRHS-1)  if  TRANS  =
               'N'  and  EQUED = 'C' or 'B', or inv(diag(R))*X(IX:IX+N-1,JX:JX+NRHS-1) if TRANS =
               'T' or 'C' and EQUED = 'R' 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(IA:IA+N-1,JA:JA+N-1)  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.

       FERR    (local output) DOUBLE PRECISION array, dimension LOCc(N_B)
               The  estimated forward error bounds for each solution vector X(j) (the j-th column
               of the solution matrix X(IX:IX+N-1,JX:JX+NRHS-1). 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  estimate  is  as  reliable  as  the
               estimate  for RCOND, and is almost always a slight overestimate of the true error.
               FERR is replicated in every process row, and is aligned with the matrices B and X.

       BERR    (local output) DOUBLE PRECISION array, dimension LOCc(N_B).
               The componentwise relative backward error of each solution vector X(j) (i.e.,  the
               smallest relative change in any entry of A(IA:IA+N-1,JA:JA+N-1) or
               B(IB:IB+N-1,JB:JB+NRHS-1)  that makes X(j) an exact solution).  BERR is replicated
               in every process row, and is aligned with the matrices B and X.

       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( PDGECON( LWORK ), PDGERFS( LWORK ) ) + LOCr( N_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.  LIWORK is local input  and  must  be  at  least
               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:  U(IA+I-1,IA+I-1) is exactly zero.  The factorization  has  been  completed,
               but  the  factor U is exactly singular, so 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 precision, and the solution
               and error bounds have not been computed.