The Newton-Raphson Method

Introduction

• The Newton-Raphson method is a numerical technique for solving equations based on linear approximation.
• Known for its efficiency in finding roots.
• Key Sections:
  • 2.1: Derivation of the basic formula.
  • 2.2: Geometric interpretation.
  • 6: Problems associated with the method.

Using Linear Approximations to Solve Equations

• Start with an estimate (x_0) for the root (r) of (f(x) = 0).
• Iteratively improve the estimate: (x_1, x_2, \ldots)
• The method is most effective when (x_0) is close to (r).
• Alternative iterative method: Secant Method.

Newton-Raphson Iteration

• Formula: (x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)})_

Geometric Interpretation

• Visualize using tangent lines.
• Tangent line at current estimate provides the next estimate.
• Can be challenging if function geometry is complex.

Convergence

• Assumes (f'(x)) exists and is continuous.
• Issues arise with high acceleration (second derivatives) or if (f'(x)) is close to zero.
• Converges fast if conditions are favorable.

The Secant Method

• Variant of Newton’s method, does not require the derivative.
• Iterative formula differs slightly, uses previous two estimates.
• More stable but slightly slower.

Historical Context

• Newton’s method originated from Newton, but evolved over time.
• Initially designed for polynomial equations and improved by various mathematicians.
• Geometric interpretation and convergence analysis developed later.

Using the Newton-Raphson Method

Practical Tips

• Ensure the equation is properly set up (e.g., (f(x) = 0)).
• A good initial estimate (x_0) is crucial.
• Graphing can aid in choosing (x_0).

Challenges

• May not converge if (x_0) is far from the root.
• Successive estimates might converge slowly.

End Game

• Correct decimal places roughly double with each iteration.
• Rule of thumb: Continue until successive estimates agree to the desired precision.

Sample Calculation

• Example: Solve (x = 2\sin x) using the Newton Method.
• Importance of initial guess highlighted by varying outcomes with different starting points.

Problems

1. Find a fraction close to (\sqrt{10}).
2. Show recurrence for (f(x) = x^2 - a).
3. Solve Newton’s equation (y^3 - 2y - 5 = 0).
4. Solve (e^{2x} = x + 6).
5. Solve (5x + \ln x = 10000).
6. Approximate using Newton method with limited operations.
7. Compute convergences for certain equations.
8. Slow convergence for a complex function.
9. Find roots for (\tan x = 4x).
10. Geometry problem involving chord and arc length.
11. Minimize distance to origin for (y = \ln x).
12. Break-even analysis for production cost.
13. Calculate interest rate for a loan using Newton method.