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 equations
   A * X = B, where A is an N-by-N symmetric positive definite matrix 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 = '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 i-by-i 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 N-by-N 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 N-by-N 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 N-by-NRHS 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 N-by-NRHS 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 j-th 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 i-th 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.