What Is Optimization Toolbox?

Large-Scale ExamplesNonlinear Equations with JacobianConsider the problem of finding a solution to a system of nonlinear equationswhose Jacobian is sparse. The dimension of the problem in this exampleis 1000. The goal is to find x such that F(x) =0. Assumingn = 1000, thenonlinear equations areTo solve a large nonlinear system of equations, F(x) = 0, use the large-scalemethod available in fsolve.Step 1: Write an M-file nlsf1.m that computes the objectivefunction values and the Jacobian.function [F,J] = nlsf1(x);% Evaluate the vector functionn = length(x);F = zeros(n,1);i = 2:(n-1);F(i) = (3-2*x(i)).*x(i)-x(i-1)-2*x(i+1)1+ 1;F(n) = (3-2*x(n)).*x(n)-x(n-1) + 1;F(1) = (3-2*x(1)).*x(1)-2*x(2) + 1;% Evaluate the Jacobian if nargout > 1if nargout > 1d = -4*x + 3*ones(n,1); D = sparse(1:n,1:n,d,n,n);c = -2*ones(n-1,1); C = sparse(1:n-1,2:n,c,n,n);e = -ones(n-1,1); E = sparse(2:n,1:n-1,e,n,n);J = C + D + E;endStep 2: Call the solve routine for the system of equations.xstart = -ones(1000,1);fun = @nlsf1;options =2-45