PZGESVX - use the LU factorization to compute the solution to a complex system of linear equations

       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*16 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*16 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 PZGETRF.  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 PZGETRF; 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) COMPLEX*16 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*16 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) COMPLEX*16 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( PZGECON(
               LWORK ), PZGERFS( 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) DOUBLE PRECISION 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.

LAPACK version 1.5                                 12 May 1997                                        PZGESVX(l)

       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.

       PZGESVX  -  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),

       PZGESVX 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

       SUBROUTINE PZGESVX( 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

           DOUBLE          PRECISION RCOND

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

           DOUBLE          PRECISION BERR( * ), C( * ), FERR( * ), R( * ), RWORK( * )

           COMPLEX*16      A( * ), AF( * ), B( * ), WORK( * ), X( * )