write a quasi-newton solver using schubert's sparse inverse modification. in matlab

main.m
function [x, fval] = quasiNewtonSolver(func, x0)
    % Initialize variables
    x = x0;
    Hinv = eye(length(x0)); % Initial approximation of the Hessian inverse
    
    maxIterations = 1000;
    tolerance = 1e-6;
    step = 1e-4;
    iter = 0;
    
    while iter < maxIterations
        % Compute the gradient at the current point
        g = gradient(func, x);
        
        % Check the norm of the gradient for convergence
        if norm(g) < tolerance
            break;
        end

        % Compute search direction d
        d = -Hinv*g;
        
        % Perform line search
        alpha = linesearch(func, x, d, step);
        
        % Update x
        x_new = x + alpha*d;
        
        % Compute s and y for Schubert's sparse inverse modification
        s = x_new - x;
        y = gradient(func, x_new) - g;
        
        % Update the Hessian inverse using Schubert's formula
        rho = 1 / (y' * s);
        V = eye(length(x0)) - rho*s*y';
        
        Hinv = V' * Hinv * V + rho*s*s';
        
        % Update x
        x = x_new;
        
        iter = iter + 1;
    end
    
    fval = func(x);
end

function g = gradient(func, x)
    g = gradient(func, x, 'CentralDifference');
end

function alpha = linesearch(func, x, d, step)
    alpha = step;
    while func(x + alpha*d) > func(x)
        alpha = alpha / 2;
    end
end
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