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NAME

       PCGESVX  -  use the LU factorization to compute the solution to a complex 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 PCGESVX( 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, RWORK, LRWORK, INFO )

           CHARACTER       EQUED, FACT, TRANS

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

           REAL            RCOND

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

           REAL            BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * )

           COMPLEX         A( * ), AF( * ), B( * ), WORK( * ), X( * )

PURPOSE

       PCGESVX uses the LU factorization to compute the solution to a complex  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)  (Conjugate 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) COMPLEX 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) COMPLEX 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  PCGETRF.  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 PCGETRF; 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) REAL 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) REAL 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) COMPLEX 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) COMPLEX 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) REAL
               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) REAL 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) REAL 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) 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
               = MAX( PCGECON( LWORK ), PCGERFS( 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.

       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_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 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.