Newton-Raphson Method

The Newton-Raphson Method is an iterative algorithm designed to find the roots of a real-valued function $f(x)$. A root is defined as any value $x$ such that the function evaluates to zero, representing the point where the graph crosses the x-axis.

Unlike simple sign-change methods, this approach uses calculus to speed up convergence. It requires the function to be differentiable, as the method relies on the gradient of the function at specific points.

The process begins with an initial approximation $x_0$, which is then refined through a recurrence relation to produce a sequence of values $x_1, x_2, \dots, x_n$ that ideally converge toward the true root $\alpha$.

The method is based on the geometric idea that a tangent line to a curve at a point near the root will intersect the x-axis at a point even closer to that root. By repeating this process, the x-intercepts of successive tangents 'walk' toward the solution.

Mathematically, the equation of the tangent at $x_n$ is given by $y - f(x_n) = f'(x_n)(x - x_n)$. To find where this tangent hits the x-axis, we set $y = 0$ and solve for $x$, which becomes our next estimate $x_{n+1}$.

This derivation leads to the standard Newton-Raphson formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ where $f'(x_n)$ is the derivative of the function evaluated at the current estimate.

Step 1: Rearrangement: Ensure the equation is in the form $f(x) = 0$. If you are solving $g(x) = h(x)$, you must first rewrite it as $f(x) = g(x) - h(x) = 0$ before proceeding.

Step 2: Differentiation: Calculate the first derivative $f'(x)$ using standard differentiation rules. This derivative is essential as it determines the slope of the tangent line used in the iteration.

Step 3: Iteration: Choose a starting value $x_0$ and substitute it into the formula to find $x_1$. Use $x_1$ to find $x_2$, and continue this process until the values stabilize to the required degree of accuracy.

Step 4: Verification: Once the sequence converges to a specific value (e.g., the same digits repeat), verify the root using a sign-change test on the boundaries of the rounded value to ensure the solution is correct.

Calculator Efficiency: Always use the 'ANS' button on your calculator to perform iterations. Store your $x_0$ value, then type the formula using the 'ANS' key to quickly generate $x_1, x_2, x_3$ by repeatedly pressing the equals sign.

Show Your Working: Even when using a calculator, examiners require you to write down the specific formula with your $f(x)$ and $f'(x)$ substituted in, as well as the values of the first few iterations (usually $x_1$ and $x_2$).

Accuracy Checks: If a question asks for a root to 3 decimal places, continue iterating until the 4th or 5th decimal place remains constant. Always perform a final check by substituting the bounds (e.g., $x \pm 0.0005$) into $f(x)$ to confirm a sign change.