SYNOPSIS
 SUBROUTINE SPOSVX(
 FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
 CHARACTER EQUED, FACT, UPLO
 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
 REAL RCOND
 INTEGER IWORK( * )
 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), BERR( * ), FERR( * ), S( * ), WORK( * ), X( LDX, * )
PURPOSE
SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equationsA * X = B, where A is an NbyN symmetric positive definite matrix and X and B are NbyNRHS matrices.
Error bounds on the solution and a condition estimate are also provided.
DESCRIPTION
The following steps are performed:1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') 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.
3. If the leading ibyi principal minor is not positive definite,
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
still goes on to solve for X and compute error bounds as
described below.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(S) so that it solves the original system before
equilibration.
ARGUMENTS
 FACT (input) CHARACTER*1

Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 'F': On entry, AF contains the factored form of A.
If EQUED = 'Y', the matrix A has been equilibrated
with scaling factors given by S. A and AF will not
be modified.
= 'N': The matrix A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then 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/output) REAL array, dimension (LDA,N)
 On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). If UPLO = 'U', the leading NbyN upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading NbyN lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,N).
 AF (input or output) REAL array, dimension (LDAF,N)
 If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S). If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A. If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).
 LDAF (input) INTEGER
 The leading dimension of the array AF. LDAF >= max(1,N).
 EQUED (input or output) CHARACTER*1

Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.  S (input or output) REAL array, dimension (N)
 The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive.
 B (input/output) REAL array, dimension (LDB,NRHS)
 On entry, the NbyNRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * 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 NbyNRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*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 after equilibration (if done). 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 solution vector X(j) (the jth column of the solution matrix X). If XTRUE is the true solution corresponding 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 reliable 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 ith argument had an illegal value
> 0: if INFO = i, and i is
<= N: 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. RCOND = 0 is returned. = N+1: U 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 number of situations where the computed solution can be more accurate than the value of RCOND would suggest.