newton(4rheolef) [debian man page]

newton(4rheolef)						    rheolef-6.1 						  newton(4rheolef)

NAME

       newton -- Newton nonlinear algorithm

DESCRIPTION

       Nonlinear Newton algorithm for the resolution of the following problem:

	      F(u) = 0

       A simple call to the algorithm writes:

	   my_problem P;
	   field uh (Vh);
	   newton (P, uh, tol, max_iter);

       The my_problem class may contains methods for the evaluation of F (aka residue) and its derivative:

	   class my_problem {
	   public:
	     my_problem();
	     field residue (const field& uh) const;
	     void update_derivative (const field& uh) const;
	     field derivative_solve (const field& mrh) const;
	     Float norm (const field& uh) const;
	     Float dual_norm (const field& Muh) const;
	   };

       See the example p-laplacian.h in the user's documentation for more.

IMPLEMENTATION

       template <class Problem, class Field>
       int newton (Problem P, Field& uh, Float& tol, size_t& max_iter, odiststream *p_derr = 0) {
	   if (p_derr) *p_derr << "# Newton: n r" << std::endl;
	   for (size_t n = 0; true; n++) {
	     Field rh = P.residue(uh);
	     Float r = P.dual_norm(rh);
	     if (p_derr) *p_derr << n << " " << r << std::endl;
	     if (r <= tol) { tol = r; max_iter = n; return 0; }
	     if (n == max_iter) { tol = r; return 1; }
	     P.update_derivative (uh);
	     Field delta_uh = P.derivative_solve (-rh);
	     uh += delta_uh;
	   }
       }

rheolef-6.1							    rheolef-6.1 						  newton(4rheolef)

pcg(4rheolef)							    rheolef-6.1 						     pcg(4rheolef)

NAME

       pcg -- conjugate gradient algorithm.

SYNOPSIS

	   template <class Matrix, class Vector, class Preconditioner, class Real>
	   int pcg (const Matrix &A, Vector &x, const Vector &b,
	     const Preconditioner &M, int &max_iter, Real &tol, odiststream *p_derr=0);

EXAMPLE

       The simplest call to 'pcg' has the folling form:

	   size_t max_iter = 100;
	   double tol = 1e-7;
	   int status = pcg(a, x, b, EYE, max_iter, tol, &derr);

DESCRIPTION

       pcg solves the symmetric positive definite linear system Ax=b using the Conjugate Gradient method.

       The return value indicates convergence within max_iter (input) iterations(0), or no convergence within max_iter iterations(1).  Upon suc-
       cessful return, output arguments have the following values:

       x      approximate solution to Ax = b

       max_iter
	      the number of iterations performed before the tolerance was reached

       tol    the residual after the final iteration

NOTE

       pcg is an iterative template routine.

       pcg follows the algorithm described on p. 15 in

       @quotation Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, R. Barrett, M.  Berry,  T.	F.
       Chan,  J.  Demmel,  J.  Donato,	J.  Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. Van der Vorst, SIAM, 1994, ftp.netlib.org/templates/tem-
       plates.ps.  @end quotation

       The present implementation is inspired from IML++ 1.2 iterative method library, http://math.nist.gov/iml++.

IMPLEMENTATION

       template <class Matrix, class Vector, class Vector2, class Preconditioner, class Real, class Size>
       int pcg(const Matrix &A, Vector &x, const Vector2 &Mb, const Preconditioner &M,
	       Size &max_iter, Real &tol, odiststream *p_derr = 0, std::string label = "cg")
       {
	   Vector b = M.solve(Mb);
	   Real norm2_b = dot(Mb,b);
	   if (norm2_b == Real(0)) norm2_b = 1;
	   Vector Mr = Mb - A*x;
	   Real  norm2_r = 0;
	   if (p_derr) (*p_derr) << "[" << label << "] #iteration residue" << std::endl;
	   Vector p;
	   for (Size n = 0; n <= max_iter; n++) {
	       Vector r = M.solve(Mr);
	       Real prev_norm2_r = norm2_r;
	       norm2_r = dot(Mr, r);
	       if (p_derr) (*p_derr) << "[" << label << "] " << n << " " << ::sqrt(norm2_r/norm2_b) << std::endl;
	       if (norm2_r <= sqr(tol)*norm2_b) {
		 tol = ::sqrt(norm2_r/norm2_b);
		 max_iter = n;
		 return 0;
	       }
	       if (n == 0) {
		 p = r;
	       } else {
		 Real beta = norm2_r/prev_norm2_r;
		 p = r + beta*p;
	       }
	       Vector Mq = A*p;
	       Real alpha = norm2_r/dot(Mq, p);
	       x  += alpha*p;
	       Mr -= alpha*Mq;
	   }
	   tol = ::sqrt(norm2_r/norm2_b);
	   return 1;
       }

rheolef-6.1							    rheolef-6.1 						     pcg(4rheolef)