Section V:   Solution of a Set of Nonlinear Equations by Newton-Raphson Method

In this section we shall discuss the solution of a set of nonlinear equations through Newton-Raphson method. Let us consider that we have a set of n nonlinear equations of a total number of n variables x₁ , x₂ , ... , x_n. Let these equations be given by

(4.22)

where f₁, ... , f_n are functions of the variables x₁ , x₂ , ... , x_n. We can then define another set of functions g₁ , ... , g_n as given below

(4.23)

Let us assume that the initial estimates of the n variables are x₁⁽⁰⁾ , x₂⁽⁰⁾ , ... , x_n⁽⁰⁾ . Let us add corrections Δx₁⁽⁰⁾ , Δx₂⁽⁰⁾ , ... , Δx_n⁽⁰⁾ to these variables such that we get the correct solution of these variables defined by

(4.24)

The functions in (4.23) then can be written in terms of the variables given in (4.24) as

(4.25)

We can then expand the above equation in Taylor 's series around the nominal values of x₁⁽⁰⁾ , x₂⁽⁰⁾ , ... , x_n⁽⁰⁾ . Neglecting the second and higher order terms of the series, the expansion of g_k , k = 1, ... , n is given as

(4.26)

where is the partial derivative of g_k evaluated at x₂⁽¹⁾ , ... , x_n⁽¹⁾ .

Equation (4.26) can be written in vector-matrix form as

(4.27)

The square matrix of partial derivatives is called the Jacobian matrix J with J ⁽¹⁾ indicating that the matrix is evaluated for the initial values of x₂⁽⁰⁾ , ... , x_n⁽⁰⁾ . We can then write the solution of (4.27) as

(4.28)

,   k= 1, ....n

Since the Taylor 's series is truncated by neglecting the 2^nd and higher order terms, we cannot expect to find the correct solution at the end of first iteration. We shall then have

(4.29)

These are then used to find J ⁽¹⁾ and Δg_k ⁽¹⁾ , k = 1, ... , n . We can then find Δx₂⁽¹⁾ , ... , Δx_n⁽¹⁾ from an equation like (4.28) and subsequently calculate x₂⁽¹⁾ , ... , x_n⁽¹⁾. The process continues till Δg_k , k = 1, ... , n becomes less than a small quantity.