Overview of Newton-Raphson Method

I. Overview of Newton-Raphson Method

First, let us briefly introduce the general Newton-Raphson method. Given a function of a single variable X (Equation 4-6), the solution is found by starting from an appropriate initial approximation X(0) and iteratively applying Equation 4-7. The iteration continues until X(n) satisfies specified convergence criteria, at which point X(n) becomes the root of Equation 4-6. This approach is known as the Newton-Raphson method.

Equation 4-7 geometrically represents finding the next approximation X(n+1) as the intersection point between the tangent line at the curve point corresponding to the nth approximation X(n) and the X-axis. Refer to Figure 4-2(a). For effective implementation, proper initial value X(0) and function f(X) selection are crucial to ensure convergence to the true root. Inappropriate choices may lead to divergence or convergence to alternative roots, as illustrated in Figure 4-2(b).

Alternatively, the method can be interpreted through error analysis. Defining the error e(n) as the difference between the nth iterative solution and the true value, we obtain:
e(n+1) = f(X(n+1)) = f(X(n)) + f'(X(n))(X(n+1)-X(n)) = 0
Expanding f(X(n)) using Taylor series around X(n) yields:
f(X(n)) + f'(X(n))(X(n+1)-X(n)) + O((X(n+1)-X(n))^2) = 0
Neglecting higher-order terms results in the approximation:
X(n+1) ≈ X(n) - f(X(n))/f'(X(n))
Comparing Equations 4-7 and 4-11 reveals that the Newton-Raphson correction term equals the first-order error term e(n). This methodology can be extended to multivariable systems through analogous mathematical formulation.