CPOSVX  - use the Cholesky factorization A = U**H*U or A =
       L*L**H to compute the solution to a complex system of lin­
       ear equations A * X = B,


SYNOPSIS

       SUBROUTINE CPOSVX( FACT,  UPLO, N, NRHS, A, LDA, AF, LDAF,
                          EQUED, S, B, LDB, X, LDX, RCOND,  FERR,
                          BERR, WORK, RWORK, INFO )

           CHARACTER      EQUED, FACT, UPLO

           INTEGER        INFO, LDA, LDAF, LDB, LDX, N, NRHS

           REAL           RCOND

           REAL           BERR(  * ), FERR( * ), RWORK( * ), S( *
                          )

           COMPLEX        A( LDA, * ), AF( LDAF, * ), B(  LDB,  *
                          ), WORK( * ), X( LDX, * )


PURPOSE

       CPOSVX  uses  the Cholesky factorization A = U**H*U or A =
       L*L**H to compute the solution to a complex system of lin­
       ear  equations  A  * X = B, where A is an N-by-N Hermitian
       positive definite matrix and X and B are N-by-NRHS  matri­
       ces.

       Error  bounds on the solution and a condition estimate are
       also provided.


DESCRIPTION

       The following steps are performed:

       1. If FACT = 'E', real scaling  factors  are  computed  to
       equilibrate
          the system:
             diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) *
       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(S)*A*diag(S) and B by diag(S)*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**H* U,  if UPLO = 'U', or
             A = L * L**H,  if UPLO = 'L',
          where U is an upper triangular matrix and L is a  lower

          matrix.

       3.  If  the leading i-by-i principal minor is not positive
       definite,
          then the routine returns with INFO = i. Otherwise,  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, INFO = N+1 is returned as a warning, but the
       routine
          still goes on to solve for X and compute  error  bounds
       as
          described below.

       4.  The system of equations is solved for X using the fac­
       tored 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  premulti­
       plied by
          diag(S) so that it solves the original system before
          equilibration.


ARGUMENTS

       FACT    (input) CHARACTER*1
               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  fac­
               tored 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  nec­
               essary, then copied to AF and factored.

       UPLO    (input) CHARACTER*1
               = 'U':  Upper triangle of A is stored;
               = 'L':  Lower triangle of A is stored.

       N       (input) INTEGER
               The number of linear equations, i.e., the order of
               the matrix A.  N >= 0.

               The number of right hand sides, i.e.,  the  number
               of columns of the matrices B and X.  NRHS >= 0.

       A       (input/output) COMPLEX array, dimension (LDA,N)
               On entry, the Hermitian matrix A, except if FACT =
               'F' and EQUED = 'Y', then A must contain the equi­
               librated matrix diag(S)*A*diag(S).  If UPLO = 'U',
               the leading N-by-N upper triangular part of A con­
               tains  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 tri­
               angular  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 over­
               written by diag(S)*A*diag(S).

       LDA     (input) INTEGER
               The  leading  dimension  of  the  array A.  LDA >=
               max(1,N).

       AF      (input or output) COMPLEX array, dimension
               (LDAF,N)
               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**H*U  or A =
               L*L**H, in the same storage format as A.  If EQUED
               .ne.  'N',  then  AF  is  the factored form of the
               equilibrated matrix diag(S)*A*diag(S).

               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**H*U  or  A  =
               L*L**H 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**H*U  or A =
               L*L**H of  the  equilibrated  matrix  A  (see  the
               description  of A for the form of the equilibrated
               matrix).

       LDAF    (input) INTEGER
               The leading dimension of the array  AF.   LDAF  >=
               max(1,N).

       EQUED   (input or output) CHARACTER*1
               Specifies the form of equilibration that was done.
               = 'N':  No equilibration (always true  if  FACT  =
               'N').

               replaced by diag(S) * A * diag(S).   EQUED  is  an
               input  argument if FACT = 'F'; otherwise, it is an
               output argument.

       S       (input or output) REAL array, dimension (N)
               The scale factors for A; not accessed if  EQUED  =
               'N'.  S is an input argument if FACT = 'F'; other­
               wise, S is an output argument.  If FACT = 'F'  and
               EQUED = 'Y', each element of S must be positive.

       B       (input/output) COMPLEX array, dimension (LDB,NRHS)
               On entry, the N-by-NRHS righthand side  matrix  B.
               On  exit,  if  EQUED  = 'N', B is not modified; if
               EQUED = 'Y', B is overwritten by diag(S) * B.

       LDB     (input) INTEGER
               The leading dimension of  the  array  B.   LDB  >=
               max(1,N).

       X       (output) COMPLEX array, dimension (LDX,NRHS)
               If  INFO = 0 or INFO = N+1, the N-by-NRHS solution
               matrix X to  the  original  system  of  equations.
               Note  that if EQUED = 'Y', A and B are modified on
               exit, and the solution to the equilibrated  system
               is inv(diag(S))*X.

       LDX     (input) INTEGER
               The  leading  dimension  of  the  array X.  LDX >=
               max(1,N).

       RCOND   (output) REAL
               The estimate of the reciprocal condition number of
               the  matrix  A  after equilibration (if done).  If
               RCOND is less than the machine precision (in  par­
               ticular,  if RCOND = 0), the matrix is singular to
               working precision.  This condition is indicated by
               a return code of INFO > 0.

       FERR    (output) REAL array, dimension (NRHS)
               The  estimated  forward error bound for each solu­
               tion vector X(j) (the j-th column of the  solution
               matrix  X).   If XTRUE is the true solution corre­
               sponding to X(j), FERR(j) is  an  estimated  upper
               bound  for the magnitude of the largest element in
               (X(j) - XTRUE) divided by  the  magnitude  of  the
               largest element in X(j).  The estimate is as reli­
               able as the estimate  for  RCOND,  and  is  almost
               always a slight overestimate of the true error.

       BERR    (output) REAL array, dimension (NRHS)
               The  componentwise relative backward error of each
               solution vector X(j) (i.e., the smallest  relative
               change in any element of A or B that makes X(j) an


       WORK    (workspace) COMPLEX array, dimension (2*N)

       RWORK   (workspace) REAL array, dimension (N)

       INFO    (output) INTEGER
               = 0: successful exit
               < 0: if INFO = -i, the i-th argument had an  ille­
               gal value
               > 0: if INFO = i, and i is
               <=  N:   the  leading minor of order i of A is not
               positive definite, so the factorization could  not
               be  completed,  and the solution has not been com­
               puted. RCOND = 0 is returned.  = N+1: U is nonsin­
               gular,  but  RCOND is less than machine precision,
               meaning that the matrix  is  singular  to  working
               precision.   Nevertheless,  the solution and error
               bounds are computed because there are a number  of
               situations where the computed solution can be more
               accurate than the value of RCOND would suggest.


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