Newton's method is a simple and widely used root-finding technique. It converges rapidly in many cases, though it may fail for poor initial guesses. It also has a geometric interpretation that helps visualize the iterative process.

Derivation from Taylor Expansion

Suppose \( f(x) \) has a root \( r \) and an initial guess \( x_0 \). We seek a correction \( h \) such that:
\[ f(x_0 + h) = 0 \] Expanding \( f \) in Taylor series around \( x_0 \):
\[ f(x_0 + h) = f(x_0) + f'(x_0) h + \frac{1}{2} f''(x_0) h^2 + \dots \] Neglecting higher-order terms (assuming \( h \) is small):
\[ 0 \approx f(x_0) + f'(x_0) h \quad \Rightarrow \quad h = -\frac{f(x_0)}{f'(x_0)} \] The next iterate is:
\[ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \] In general:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

Example

Solve \( f(x) = e^x - x = 0 \) with initial guess \( x_0 = 0 \). The derivative is:
\[ f'(x) = e^x - 1 \] The iteration becomes:
\[ x_{n+1} = x_n - \frac{e^{x_n} - x_n}{e^{x_n} - 1} \]

n	xn	|xn - xn-1|
0	0.00000000	–
1	0.50000000	0.50000000
2	0.56631100	0.06631100
3	0.56714329	0.00083229
4	0.56714329	≈10-6

Geometric Interpretation

Let \( x_0 \) be an initial guess. The tangent line to \( f \) at \( x_0 \) is:
\[ y = f(x_0) + f'(x_0)(x - x_0) \] Its intersection with the x-axis gives \( x_1 \):
\[ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \] This process is repeated to improve the approximation successively: \( x_2, x_3, \dots \)