SSPSVX  -  use  the  diagonal  pivoting  factorization A =
       U*D*U**T or A = L*D*L**T to compute 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 matrices


SYNOPSIS

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

           CHARACTER      FACT, UPLO

           INTEGER        INFO, LDB, LDX, N, NRHS

           REAL           RCOND

           INTEGER        IPIV( * ), IWORK( * )

           REAL           AFP( * ), AP( * ), B( LDB, * ), BERR( *
                          ), FERR( * ), WORK( * ), X( LDX, * )


PURPOSE

       SSPSVX  uses  the  diagonal  pivoting  factorization  A  =
       U*D*U**T or A = L*D*L**T to compute 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 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
       precision,
          INFO = N+1 is returned as a warning,  but  the  routine

          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) REAL 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) REAL 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 SSPTRF, stored as a
               packed triangular matrix in the same storage  for­
               mat as A.

               If  FACT = 'N', then AFP is an output argument 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 SSPTRF, 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 SSPTRF.
               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 SSPTRF.

       B       (input) REAL 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) REAL 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
               (X(j) - XTRUE) divided by  the  magnitude  of  the

               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) REAL array, dimension (3*N)

       IWORK   (workspace) INTEGER 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 ]


Man(1) output converted with man2html