The linearized system (10.44) thus reads

(10.47)

and its solution is given by

provided that the inverse of the matrix

exists for . If all goes well, the vector will be a better approximation to the exact solution, the residual vector will be smaller, and the process

can be repeated with taking the place of , etc. until the convergence is achieved. Since for problems of class M, the system (10.44) has a unique solution for sufficiently small values of , and it is known that Newton's method produces a sequence of vectors which converges rapidly to provided that the initial approximation is not too bad. Here we are mainly concerned with the question of computational technique. In this respect it should be noted that for the solution of (10.47) it is not necessary to calculate the inverse of the matrix . All that is required is the solution of the system of linear equations

For the components of . This solution is greatly facilitated by the fact that the matrix is again tri-diagonal. In fact, if , we have

and all other elements are zero.

The method described for the linear case thus is immediately applicable. The only work that is required for one step of Newton's method in addition to the work involved in the solution of a linear system is the evaluation of the residual vector and of the partial derivative (n = 1, 2, … , N -1).