6.2 Newton-Raphson and Secant Methods

The Newton-Raphson method is a powerful tool for finding roots of equations. It uses function derivatives to iteratively refine root estimates, offering rapid convergence near the solution. This approach combines mathematical elegance with practical effectiveness in solving nonlinear equations.

The secant method provides an alternative root-finding technique that doesn't require explicit derivatives. By approximating the tangent line with a secant, it achieves good convergence while being more versatile. Comparing these methods reveals trade-offs between speed, robustness, and ease of implementation.

Newton-Raphson Method

Derivation of Newton-Raphson method

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• Taylor series expansion of $f(x)$ around $x_0$ approximates function behavior
• First-order Taylor approximation $f(x) \approx f(x_0) + f'(x_0)(x - x_0)$ linearizes function
• Setting approximation to zero $f(x_0) + f'(x_0)(x - x_0) = 0$ finds root estimate
• Solving for $x$ yields Newton-Raphson formula $x = x_0 - \frac{f(x_0)}{f'(x_0)}$
• Iterative form $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ refines root approximation

Application of Newton-Raphson method

• Method application involves:
  1. Choose initial guess $x_0$
  2. Calculate $f(x_0)$ and $f'(x_0)$
  3. Compute next approximation using formula
  4. Repeat until convergence or max iterations reached
• Convergence criteria assess solution accuracy (absolute error, function value)
• Method limitations include:
  • Requires differentiable function
  • May fail near horizontal tangents
  • Initial guess sensitivity (overshooting, divergence)

Secant Method and Comparison

Geometry of secant method

• Secant line passes through two points on function curve, approximates tangent
• Finite difference approximates derivative without explicit calculation
• Iterative formula $x_{n+1} = x_n - f(x_n) \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}$ updates root estimate
• Geometric interpretation shows secant-axis intersection as next approximation

Newton-Raphson vs secant convergence

• Newton-Raphson achieves quadratic convergence (order 2), faster near root
• Secant method exhibits superlinear convergence (order ≈ 1.618), slower but steady
• Newton-Raphson requires fewer iterations, needs derivative calculation (analytical or numerical)
• Secant method needs more iterations, avoids derivative computation
• Secant method works on non-differentiable functions, more robust in some cases

Implementation and sensitivity analysis

• Implementation steps:
  1. Define $f(x)$ (polynomial, transcendental)
  2. Set convergence criteria (tolerance) and iteration limit
  3. Implement iterative formulas (Newton-Raphson, secant)
  4. Add error handling (division by zero, non-convergence)
• Sensitivity analysis explores method behavior:
  • Test various initial guesses (near root, far from root)
  • Observe convergence patterns (rapid, slow, oscillatory)
  • Identify failure cases (divergence, stagnation)
• Visualization aids understanding:
  • Function plots with root approximations show convergence path
  • Error vs. iterations graphs illustrate convergence speed
• Performance metrics evaluate method efficiency:
  • Iteration count to reach convergence
  • Final root approximation accuracy
  • Computation time comparison (CPU time, floating-point operations)