SPPSV  -  compute  the solution to a real system of linear
       equations A * X = B,


SYNOPSIS

       SUBROUTINE SPPSV( UPLO, N, NRHS, AP, B, LDB, INFO )

           CHARACTER     UPLO

           INTEGER       INFO, LDB, N, NRHS

           REAL          AP( * ), B( LDB, * )


PURPOSE

       SPPSV computes the solution to a  real  system  of  linear
       equations  A * X = B, where A is an N-by-N symmetric posi­
       tive definite matrix stored in packed format and X  and  B
       are N-by-NRHS matrices.

       The Cholesky decomposition is used to factor A 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.  The factored form of A is then used to
       solve the system of equations A * X = B.


ARGUMENTS

       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 matrix B.  NRHS >= 0.

       AP      (input/output) REAL array, dimension (N*(N+1)/2)
               On  entry, the upper or lower triangle of the sym­
               metric 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)*(2n-j)/2) = A(i,j) for j<=i<=n.   See
               below for further details.

               On  exit,  if INFO = 0, the factor U or L from the
               Cholesky factorization A = U**T*U or A  =  L*L**T,
               in the same storage format as A.

               On  entry, the N-by-NRHS right hand side matrix B.
               On exit, if  INFO  =  0,  the  N-by-NRHS  solution
               matrix X.

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

       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille­
               gal value
               > 0:  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 has not
               been computed.


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 = conjg(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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