Numerical methods and problem solving
Numerical methods and problem solving tests the approximation of roots and integrals when exact methods are unavailable or awkward. It covers locating a root by sign change, fixed-point iteration, the Newton–Raphson method, and the trapezium rule, together with judging whether an approximation is reliable.
Locating roots
If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then a root lies in (a, b). Repeatedly halving the interval, keeping the half with the sign change, gives a bisection method whose accuracy doubles with each step. A sign change alone is not proof of a unique root, since several roots may lie in the interval.
Iteration and Newton–Raphson
Fixed-point iteration rearranges f(x) = 0 into x = g(x) and computes successive values. It converges when the gradient of g near the root has magnitude less than 1; otherwise it diverges or oscillates. Newton–Raphson uses x₁ = x₀ − f(x₀)/f′(x₀) and usually converges very quickly when the initial estimate is close to the root, but it fails where the derivative is zero or the starting point is poorly chosen.
Numerical integration
The trapezium rule with strip width h approximates an integral as h/2[y₀ + 2(y₁ + ⋯ + yₙ₋₁) + yₙ]. Increasing the number of strips reduces the error. For a concave-up curve the trapezium rule overestimates; for a concave-down curve it underestimates.
Worked example
Use one step of Newton–Raphson, starting from x₀ = 2, to estimate a root of f(x) = x³ − 2x − 5.
Here f′(x) = 3x² − 2. Then f(2) = 8 − 4 − 5 = −1 and f′(2) = 10, so x₁ = 2 − (−1)/10 = 2.1.
Traps that cost marks
- Using Newton–Raphson without differentiating correctly, or with a derivative that vanishes at the estimate.
- Treating a sign change as proof of an exact or unique root.
- Forgetting to state an accuracy or tolerance when asked for a root to a given number of decimal places.
- Miscounting the number of strips when applying the trapezium rule.