CSPSVX  -  use  the  diagonal  pivoting  factorization A =
       U*D*U**T or A = L*D*L**T to compute the solution to a com­
       plex  system  of linear equations A * X = B, where A is an
       N-by-N symmetric matrix stored in packed format and X  and
       B are N-by-NRHS matrices


SYNOPSIS

       SUBROUTINE CSPSVX( FACT,  UPLO, N, NRHS, AP, AFP, IPIV, B,
                          LDB, X, LDX, RCOND, FERR,  BERR,  WORK,
                          RWORK, INFO )

           CHARACTER      FACT, UPLO

           INTEGER        INFO, LDB, LDX, N, NRHS

           REAL           RCOND

           INTEGER        IPIV( * )

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

           COMPLEX        AFP( * ), AP( * ), B( LDB, * ), WORK( *
                          ), X( LDX, * )


PURPOSE

       CSPSVX  uses  the  diagonal  pivoting  factorization  A  =
       U*D*U**T or A = L*D*L**T to compute the solution to a com­
       plex system of linear equations A * X = B, where A  is  an
       N-by-N  symmetric matrix stored in packed format 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 as
             A = U * D * U**T,  if UPLO = 'U', or
             A = L * D * L**T,  if UPLO = 'L',
          where U (or L) is a product  of  permutation  and  unit
       upper (lower)
          triangular matrices and D is symmetric and block diago­
       nal 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
       the
          reciprocal of the condition number is less than machine

          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, AFP
               and  IPIV contain the factored form of A.  AP, AFP
               and IPIV will not be modified.  = 'N':  The matrix
               A will be copied to AFP 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.

       AP      (input) COMPLEX array, dimension (N*(N+1)/2)
               The  upper  or  lower  triangle  of  the symmetric
               matrix A, packed columnwise  in  a  linear  array.
               The  j-th column of A is stored in the array AP as
               follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j)
               for  1<=i<=j;  if  UPLO  = 'L', AP(i + (j-1)*(2*n-
               j)/2) = A(i,j) for j<=i<=n.  See below for further
               details.

       AFP     (input or output) COMPLEX array, dimension
               (N*(N+1)/2)
               If FACT = 'F', then AFP 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**T  or  A  =
               L*D*L**T as computed by CSPTRF, stored as a packed
               triangular matrix in the same storage format as A.

               on  exit  contains the block diagonal matrix D and
               the multipliers used to obtain the factor U  or  L
               from  the  factorization  A  =  U*D*U**T  or  A  =
               L*D*L**T as computed by CSPTRF, stored as a packed
               triangular matrix in the same storage format as A.

       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 CSPTRF.
               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 CSPTRF.

       B       (input) COMPLEX 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 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) REAL
               The estimate of the reciprocal condition number of
               the matrix A.  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    (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

               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
               exact solution).

       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:  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
               be  more  accurate  than  the value of RCOND would
               suggest.


FURTHER DETAILS

       The packed storage scheme is illustrated by the  following
       example when N = 4, UPLO = 'U':

       Two-dimensional storage of the symmetric matrix A:

          a11 a12 a13 a14
              a22 a23 a24
                  a33 a34     (aij = aji)
                      a44

       Packed storage of the upper triangle of A:

       AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]


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