Evaluating numerical methods
Objective. Compare accuracy, convergence and computational cost.
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Learn: the key idea
Evaluating numerical methods means comparing techniques like fixed-point iteration, Newton–Raphson and the trapezium rule on accuracy, speed of convergence, and computational effort. Newton–Raphson typically converges fastest near a root but needs a derivative and a good starting value, while fixed-point iteration is simpler but can converge slowly or diverge. For integration, increasing the number of strips in the trapezium rule improves accuracy but increases the number of calculations required. A good evaluation states a specific trade-off, such as speed versus reliability, rather than a vague preference.
Worked example
Evaluate why Newton–Raphson may fail for the equation x³ − 3x + 2 = 0 at x₀ = 1.
- 1Differentiate: f′(x) = 3x² − 3
- 2Evaluate f′(1) = 3(1) − 3 = 0
- 3Note the iteration formula divides by f′(x₀), which is zero
- 4Conclude the method fails because the tangent at x₀ = 1 is horizontal
Answer: Fails because f′(1) = 0, making the tangent horizontal and the iteration undefined
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Practice
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Start practisingCommon mistake. Students often state a method is simply 'better' without naming the trade-off, such as accuracy against computational cost. Always justify an evaluation by comparing a concrete factor, like convergence speed or sensitivity to the starting value.
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