An n × n linear system of equations Ax = b is modified in the following manner: for each i, i = 1,...,n, the value bi on the right-hand side of the ith equation is replaced by bi - x 3 i . Obviously, the modified system of equations (for the unknowns xi) is now nonlinear.
(a) Find the corresponding Jacobian matrix.
(b) Given that A is strictly diagonally dominant with positive elements on its diagonal, state whether or not it is guaranteed that the Jacobian matrix at each iterate is nonsingular.
(c) Suppose that A is symmetric positive definite (not necessarily diagonally dominant) and that Newton's method is applied to solve the nonlinear system. Is it guaranteed to converge?