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


SYNOPSIS

       SUBROUTINE DSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )

           CHARACTER     UPLO

           INTEGER       INFO, LDB, N, NRHS

           INTEGER       IPIV( * )

           DOUBLE        PRECISION AP( * ), B( LDB, * )


PURPOSE

       DSPSV computes the solution to a  real  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  matri­
       ces.

       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,  D  is  symmetric  and  block
       diagonal with 1-by-1 and 2-by-2 diagonal blocks.  The fac­
       tored form of A is then used to solve the system of  equa­
       tions 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)   DOUBLE   PRECISION   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.

               tipliers used to obtain the factor U or L from the
               factorization A = U*D*U**T or A = L*D*L**T as com­
               puted by DSPTRF, stored  as  a  packed  triangular
               matrix in the same storage format as A.

       IPIV    (output) INTEGER array, dimension (N)
               Details  of  the interchanges and the block struc­
               ture of D, as determined by DSPTRF.  If IPIV(k)  >
               0, then rows and columns k and IPIV(k) were inter­
               changed, 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  inter­
               changed  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 inter­
               changed and D(k:k+1,k:k+1) is  a  2-by-2  diagonal
               block.

       B         (input/output)   DOUBLE   PRECISION   array,
               dimension (LDB,NRHS)
               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, D(i,i) is exactly zero.  The
               factorization has been completed,  but  the  block
               diagonal  matrix  D  is  exactly  singular, so the
               solution could not be 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 = 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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