DPOSVX(l)								 )								 DPOSVX(l)

NAME

       DPOSVX - use 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,

SYNOPSIS

       SUBROUTINE DPOSVX( 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

	   DOUBLE	  PRECISION RCOND

	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), BERR( * ), FERR( * ), S( * ), WORK( * ), X( LDX, * )

PURPOSE

       DPOSVX  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 equili-
	       brated before it is factored.  = 'F':  On entry, AF contains the factored form of A.  If EQUED = 'Y', the matrix A has been equili-
	       brated 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) DOUBLE PRECISION 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 modi-
	       fied 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) DOUBLE PRECISION 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'; oth-
	       erwise, it is an output argument.

       S       (input or output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
	       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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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.

LAPACK version 3.0						   15 June 2000 							 DPOSVX(l)