Hi AGCM,
I understand that you are trying to utilize both Newton's method and the secant method in MATLAB to find the zeros of several functions, such as polynomials and differential equations, by narrowing down intervals until you are close enough to an approximation or the actual value of zero. You aim to start with a broader interval and make it smaller using these methods to find the roots more accurately.
Here is a concise MATLAB approach to achieve your goal:
function root = findRootCombined(f, df, x0, x1)
% Use the secant method to get an initial approximation of the root
[x_approx, iter_secant] = sekant(f, x0, x1);
disp(['Secant method used ', num2str(iter_secant), ' iterations. Approximation: ', num2str(x_approx)]);
% Refine the approximation using Newton's method
[root, iter_newton] = newton(f, df, x_approx);
if iter_newton == 1000
disp('Newton method reached maximum iterations. Result may be inaccurate.');
else
disp(['Newton method refined the root in ', num2str(iter_newton), ' iterations. Final Root: ', num2str(root)]);
end
end
% Secant method function
function [xout, iter] = sekant(f, x, x1)
iter = 0;
maxiter = 1000;
while abs(x1-x) > 1e-6 && iter < maxiter
iter = iter + 1;
d = (f(x1) - f(x)) / (x1-x);
xout = x1 - f(x1) / d;
x = x1;
x1 = xout;
end
end
% Newton's method function
function [x, iter] = newton(f, df, x)
maxiter = 1000;
iter = 0;
d = 100;
while abs(d) > sqrt(eps) * abs(x) && iter < maxiter
iter = iter + 1;
d = -f(x) / df(x);
x = x + d;
end
end
This combined approach starts with the secant method to find an initial approximation of the root within a given interval (x0 to x1). It then uses Newton's method, starting from this approximation, to refine the search and find the root more accurately. This strategy leverages the strengths of both methods: the secant method's ability to work without the derivative and its broader convergence characteristics, followed by Newton's method's rapid convergence near the root.
Make sure to replace f and df with your specific function and its derivative when calling findRootCombined. The initial guesses x0 and x1 should be chosen based on your knowledge of the function's behavior or through preliminary analysis.
Hope this helps.
Regards,
Nipun
