PZPOSVX - use the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex

       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 contains
               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) COMPLEX*16 pointer into
               the local memory to an array of local dimension ( LLD_A, LOCc(JA+N-1) ).  On entry, the Hermitian
               matrix  A,  except  if  FACT  =  'F' and EQUED = 'Y', then A must contain the equilibrated 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) COMPLEX*16 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 factored 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 factorization 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  factorization 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) COMPLEX*16 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 positive.

       SC      (local input/local output) COMPLEX*16 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 positive.

       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.  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) 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 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  vector  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) 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( PZPOCON(
               LWORK ), PZPORFS( 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.

       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: 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 precision, and the solution and error bounds have not been computed.

LAPACK version 1.5                                 12 May 1997                                        PZPOSVX(l)

       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.

       PZPOSVX  -  use  the Cholesky factorization A = U**H*U or A = L*L**H 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),

       PZPOSVX  uses  the  Cholesky  factorization A = U**H*U or A = L*L**H 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.  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  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 PZPOSVX( 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, RWORK, LRWORK, INFO )

           CHARACTER       EQUED, FACT, UPLO

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

           DOUBLE          PRECISION RCOND

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

           DOUBLE          PRECISION BERR( * ), FERR( * ), SC( * ), SR( * ), RWORK( * )

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