ZHESVX  -  use the diagonal pivoting factorization to com­
       pute the solution to a complex system of linear  equations
       A * X = B,


SYNOPSIS

       SUBROUTINE ZHESVX( FACT,  UPLO, N, NRHS, A, LDA, AF, LDAF,
                          IPIV, B,  LDB,  X,  LDX,  RCOND,  FERR,
                          BERR, WORK, LWORK, RWORK, INFO )

           CHARACTER      FACT, UPLO

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

           DOUBLE         PRECISION RCOND

           INTEGER        IPIV( * )

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

           COMPLEX*16     A(  LDA,  * ), AF( LDAF, * ), B( LDB, *
                          ), WORK( * ), X( LDX, * )


PURPOSE

       ZHESVX uses the diagonal pivoting factorization to compute
       the solution to a complex system of linear equations A * X
       = B, where A is an N-by-N Hermitian matrix and X and B are
       N-by-NRHS matrices.

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


DESCRIPTION

       The following steps are performed:

       1. If FACT = 'N', the diagonal pivoting method is used  to
       factor A.
          The form of the factorization is
             A = U * D * U**H,  if UPLO = 'U', or
             A = L * D * L**H,  if UPLO = 'L',
          where  U  (or  L)  is a product of permutation and unit
       upper (lower)
          triangular matrices, and D is Hermitian and block diag­
       onal with
          1-by-1 and 2-by-2 diagonal blocks.

       2.  If  some D(i,i)=0, so that D is exactly singular, 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

          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.

       3.  The system of equations is solved for X using the fac­
       tored form
          of A.

       4. Iterative refinement is applied to improve the computed
       solution
          matrix  and  calculate  error bounds and backward error
       estimates
          for it.


ARGUMENTS

       FACT    (input) CHARACTER*1
               Specifies whether or not the factored  form  of  A
               has  been supplied on entry.  = 'F':  On entry, AF
               and IPIV contain the factored form of  A.   A,  AF
               and IPIV will not be modified.  = 'N':  The matrix
               A will be 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.

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

       A       (input) COMPLEX*16 array, dimension (LDA,N)
               The Hermitian matrix A.  If UPLO = 'U', the  lead­
               ing 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 refer­
               enced.  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  tri­
               angular part of A is not referenced.

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

               (LDAF,N)
               If FACT = 'F', then AF is an input argument and on
               entry contains the block diagonal matrix D and the
               multipliers  used to obtain the factor U or L from
               the factorization A = U*D*U**H or A = L*D*L**H  as
               computed by ZHETRF.

               If  FACT  = 'N', then AF is an output argument and
               on exit returns the block diagonal  matrix  D  and
               the  multipliers  used to obtain the factor U or L
               from  the  factorization  A  =  U*D*U**H  or  A  =
               L*D*L**H.

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

       IPIV    (input or output) INTEGER array, dimension (N)
               If FACT = 'F', then IPIV is an input argument  and
               on  entry contains details of the interchanges and
               the block structure of D, as determined by ZHETRF.
               If  IPIV(k)  >  0,  then  rows  and  columns k and
               IPIV(k) were interchanged and D(k,k) is  a  1-by-1
               diagonal  block.   If  UPLO  =  'U'  and IPIV(k) =
               IPIV(k-1) < 0,  then  rows  and  columns  k-1  and
               -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a
               2-by-2 diagonal block.  If UPLO = 'L' and  IPIV(k)
               =  IPIV(k+1)  <  0,  then rows and columns k+1 and
               -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a
               2-by-2 diagonal block.

               If FACT = 'N', then IPIV is an output argument and
               on exit contains details of the  interchanges  and
               the block structure of D, as determined by ZHETRF.

       B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
               The N-by-NRHS right hand side matrix B.

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

       X       (output) COMPLEX*16 array, dimension (LDX,NRHS)
               If  INFO = 0 or INFO = N+1, the N-by-NRHS solution
               matrix X.

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

       RCOND   (output) DOUBLE PRECISION
               The estimate of the reciprocal condition number of
               the matrix A.  If RCOND is less than  the  machine

               matrix is singular  to  working  precision.   This
               condition  is indicated by a return code of INFO >
               0.

       FERR    (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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
               exact solution).

       WORK    (workspace/output) COMPLEX*16 array, dimension
               (LWORK)
               On exit, if INFO = 0, WORK(1) returns the  optimal
               LWORK.

       LWORK   (input) INTEGER
               The  length  of  WORK.  LWORK >= 2*N, and for best
               performance LWORK >= N*NB, where NB is the optimal
               blocksize for ZHETRF.

               If  LWORK = -1, then a workspace query is assumed;
               the routine only calculates the  optimal  size  of
               the  WORK  array,  returns this value as the first
               entry of the WORK  array,  and  no  error  message
               related to LWORK is issued by XERBLA.

       RWORK   (workspace) DOUBLE PRECISION 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:  D(i,i) is exactly zero.  The  factorization
               has  been  completed  but  the factor D is exactly
               singular, so the solution and error  bounds  could
               not  be computed. RCOND = 0 is returned.  = N+1: D
               is nonsingular, 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 num­
               ber of situations where the computed solution  can

               suggest.


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