How is the Jacobian used to solve the non-linear systems of equations?

Nonlinear Equations with JacobianConsider the problem of finding a solution to a system of nonlinear equations whose Jacobian is sparse. The dimension of the problem in this example is 1000. The goal is to find x such that F(x) = 0. Assuming n = 1000, the nonlinear equations areF(1)F(i)F(n)=3x1−2x21−2x2+1,=3xi−2x2i−xi−1−2xi+1+1,=3xn−2x2n−xn−1+1.To solve a large nonlinear system of equations, F(x) = 0, you can use the trust-region reflective algorithm available in fsolve, a large-scale algorithm (Large-Scale vs. Medium-Scale Algorithms).Step 1: Write a file nlsf1.m that computes the objective function values and the Jacobian.function [F,J] = nlsf1(x)

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How is the Jacobian used to solve the non-linear systems of equations?
Nonlinear Equations with JacobianConsider the problem of finding a solution to a system of nonlinear equations whose Jacobian is sparse. The dimension of the problem in this example is 1000. The goal is to find x such that F(x) = 0. Assuming n = 1000, the nonlinear equations areF(1)F(i)F(n)=3x1−2x21−2x2+1,=3xi−2x2i−xi−1−2xi+1+1,=3xn−2x2n−xn−1+1.To solve a large nonlinear system of equations, F(x) = 0, you can use the trust-region reflective algorithm available in fsolve, a large-scale algorithm (Large-Scale vs. Medium-Scale Algorithms).Step 1: Write a file nlsf1.m that computes the objective function values and the Jacobian.function [F,J] = nlsf1(x)

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