Methods Based on First-Order Approximation and Fixed-Point Methods
I. Methods Based on First-Order Approximation
Consider the function f : R → R defined by f(x) = 0.25 cos(2x)^2 − x^2.
Implement the methods of computing a root of the equation f(x) = 0 based on the first-order approximation of the function. In each of the methods below, take the parameter values ωa = 0.9, ωf = 0.1, tol = 10−15 and maxit = 100 in the stopping criterion of the corresponding algorithms.
Put the C++ implementations of each algorithm in the rootscalar.hpp file. Use the struct
cs117::root::scalar::RootScalarResult
as the return value of each function associated to the algorithm. Also, use the struct
cs117::root::scalar::param
for the input parameters. You may add additional members to this struct as needed. For the implementation to the specific function given by (fun), use the root_scalar_firstorderex.cpp as the filename, that is, the file where the main function is located. Print the results of each method by invoking the member function print of RootScalarResult.
In the following list, use the initial point x_0 = 0.5 and the second point x_1 = 0 in the case of the secant method. 1 Secant Method
2 Newton Method
3 Steffensen Method
4 Inexact Newton Method (Apply the backward, forward and centered finite difference approximations of the derivative and use the step size h = √eps.)
II. Fixed-Point Methods
A point x ∈ R is called a fixed point of g : R → R if g(x) = x. Observe that a fixed point of g(x) = 0.5 cos(2x) is a root of the equation f(x) = 0 in Item 1. Repeat the tasks in the previous item to the following methods. Use the filename fixpointex.cpp.
1 Fixed-Point Method
2 Aitken Acceleration Method
