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dppsvx.f(3)				      LAPACK				      dppsvx.f(3)

NAME
       dppsvx.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dppsvx (FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX, RCOND, FERR,
	   BERR, WORK, IWORK, INFO)
	    DPPSVX computes the solution to system of linear equations A * X = B for OTHER
	   matrices

Function/Subroutine Documentation
   subroutine dppsvx (characterFACT, characterUPLO, integerN, integerNRHS, double precision,
       dimension( * )AP, double precision, dimension( * )AFP, characterEQUED, double precision,
       dimension( * )S, double precision, dimension( ldb, * )B, integerLDB, double precision,
       dimension( ldx, * )X, integerLDX, double precisionRCOND, double precision, dimension( *
       )FERR, double precision, dimension( * )BERR, double precision, dimension( * )WORK,
       integer, dimension( * )IWORK, integerINFO)
	DPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

       Purpose:

	    DPPSVX 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 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 = '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.

       Parameters:
	   FACT

		     FACT is 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, AFP contains the factored form of A.
			     If EQUED = 'Y', the matrix A has been equilibrated
			     with scaling factors given by S.  AP and AFP will not
			     be modified.
		     = 'N':  The matrix A will be copied to AFP and factored.
		     = 'E':  The matrix A will be equilibrated if necessary, then
			     copied to AFP and factored.

	   UPLO

		     UPLO is CHARACTER*1
		     = 'U':  Upper triangle of A is stored;
		     = 'L':  Lower triangle of A is stored.

	   N

		     N is INTEGER
		     The number of linear equations, i.e., the order of the
		     matrix A.	N >= 0.

	   NRHS

		     NRHS is INTEGER
		     The number of right hand sides, i.e., the number of columns
		     of the matrices B and X.  NRHS >= 0.

	   AP

		     AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
		     On entry, the upper or lower triangle of the symmetric matrix
		     A, packed columnwise in a linear array, except if FACT = 'F'
		     and EQUED = 'Y', then A must contain the equilibrated matrix
		     diag(S)*A*diag(S).  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.  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).

	   AFP

		     AFP is DOUBLE PRECISION array, dimension
				       (N*(N+1)/2)
		     If FACT = 'F', then AFP 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 AFP is the factored
		     form of the equilibrated matrix A.

		     If FACT = 'N', then AFP 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 AFP 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 AP for the form of the
		     equilibrated matrix).

	   EQUED

		     EQUED is 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

		     S is 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

		     B is 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

		     LDB is INTEGER
		     The leading dimension of the array B.  LDB >= max(1,N).

	   X

		     X is 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

		     LDX is INTEGER
		     The leading dimension of the array X.  LDX >= max(1,N).

	   RCOND

		     RCOND is 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

		     FERR is 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

		     BERR is 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

		     WORK is DOUBLE PRECISION array, dimension (3*N)

	   IWORK

		     IWORK is INTEGER array, dimension (N)

	   INFO

		     INFO is 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.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   April 2012

       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 = conjg(aji))
			    a44

	     Packed storage of the upper triangle of A:

	     AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

       Definition at line 312 of file dppsvx.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dppsvx.f(3)
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