The complete procedure may be summarized as follows:

1.  Calculate and , starting with , from the relations :

The quantities need not be saved, unless the same system is solved repeatedly for different non–homogeneous terms. The quantities and are not used after and have been computed.

2.  Calculate , stating with , form the relations:

The whole process requires approximately additions, multiplications and divisions. This compares very favorably to the multiplications alone that have to be performed in the solution of a system with a matrix of order that has no zero elements.

Solution of the difference scheme: nonlinear case.

If the function is not linear in , one can not hope to solve the system (10.27) by algebraic methods. Some iterative procedure must be resorted to. The method which is recommended for this purpose is generalization of the Newton-Raphson method to systems of transcendental equations.

In the case of a single equation, the Newton-Raphson method consists of linearizing the given function by replacing by its differential at a point believed to be close to the actual solution, and solving the linearized equation . The value is then accepted as a better approximation, and the process is continued if necessary. Quite analogously if is a vector believed to be close to the actual solution of the equation

(10.44)

so that the residual vector

(10.45)

is small, we replace the increments of the function by their differentials at the point and solve the resulting linear system of equations for the increment of the vector , which we shall call by . Since the expression is already linear in , the differential of the vector at is found to be , where denotes the diagonal matrix

(10.46)