Class for solving the equation \( f(x) = 0 \) using various numerical methods. More...

#include <Utils_zeros.hh>

Public Member Functions
	Zeros ()=default

	~Zeros ()=default

void	set_max_iterations (Integer mit)

void	set_max_fun_evaluation (Integer mfev)

void	set_tolerance (Real tol)

Real	solve_Newton (Real x_guess, Zeros_base_fun< Real > *fun)

Real	solve_Chebyshev (Real x_guess, Zeros_base_fun< Real > *fun)

Real	solve_Halley (Real x_guess, Zeros_base_fun< Real > *fun)

Real	solve_Order4 (Real x_guess, Zeros_base_fun< Real > *fun)

Real	solve_Order8 (Real x_guess, Zeros_base_fun< Real > *fun)

Real	solve_Order16 (Real x_guess, Zeros_base_fun< Real > *fun)

Real	solve_Order32 (Real x_guess, Zeros_base_fun< Real > *fun)

Integer	used_iter () const

Integer	num_fun_eval () const

Real	tolerance () const

bool	converged () const

Detailed Description

template<typename Real>
class Utils::Zeros< Real >

Class for solving the equation \( f(x) = 0 \) using various numerical methods.

This class implements multiple solvers to find the roots of a given function. The available methods include:

Newton-Raphson Method: A widely used iterative method for finding successively better approximations to the roots of a real-valued function. Learn more.
Chebyshev Method: A higher-order root-finding method that offers faster convergence compared to the Newton-Raphson method.
Halley Method: An iterative method that is a generalization of the Newton-Raphson method and provides faster convergence. Learn more.
Methods by Juan Luis Varona: A series of methods developed for enhanced convergence properties:
- Order 4 method
- Order 8 method
- Order 16 method
- Order 32 method

For a detailed exploration of these methods, refer to the paper:

An Optimal Thirty-Second-Order Iterative Method for Solving Nonlinear Equations and a Conjecture, Juan Luis Varona, Qualitative Theory of Dynamical Systems (2022). Link to the paper.

Note: This class is designed to work with user-defined functions that extend from Utils::Zeros_base_fun.

Usage Example

To use this class, first wrap your function in a derived class. For instance, for the function \( f(x) = x^2 - 2 \), you can define:

class Fun1 : public Utils::Zeros_base_fun<real_type> {
public:
    real_type eval(real_type x) const override { return x*x - 2; }
    real_type eval_D(real_type x) const override { return 2*x; }
    real_type eval_DD(real_type x) const override { return 2; }
    real_type eval_DDD(real_type x) const override { return 0; }
};

Next, instantiate the function and the solver. Then, call the desired method to find the root:

Zeros<real_type> solver;
Fun1 f;
real_type x_guess = 1.0;  // Initial guess
real_type x_solution = solver.solve_Newton(x_guess, f);

If the method converges, x_solution will contain the computed solution.

The documentation for this class was generated from the following files:

/Users/enricobertolazzi/Ricerca/PINS/PINS-submodules/UtilsLite/src/Utils_zeros.hh
/Users/enricobertolazzi/Ricerca/PINS/PINS-submodules/UtilsLite/src/Utils_zeros.cc

Public Member Functions

Detailed Description