Numerical Methods (A2 Unit 3: Pure Mathematics B) — WJEC A-Level Mathematics
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Numerical Methods (A2 Unit 3: Pure Mathematics B) explained
To solve f(x) = 0 iteratively, rearrange into the fixed-point form x = g(x), generating the recurrence relation xₙ₊₁ = g(xₙ) from an initial approximation x₁.
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Graphically, the intersection of y = x and y = g(x) gives the root. Starting at (x₁, 0), draw vertically to (x₁, g(x₁)), then horizontally to y = x at (x₂, x₂); repeating this generates an iterative path. If the gradient g'(x) near the root α is positive (0 < g'(α) < 1), successive iterations approach the root monotonically from one side, producing a staircase diagram. If the gradient is negative (−1 < g'(α) < 0), iterations alternate above and below the root, producing an inward spiralling cobweb diagram. If |g'(α)| > 1, the iteration diverges away from the root.
Your focus
- Rearrange equations into fixed-point form x = g(x) and generate iterative sequences.
- Construct staircase and cobweb diagrams to illustrate convergence or divergence of iterations.
- Relate the convergence of iterative methods to the gradient of g(x) near the root.
Numerical Methods (A2 Unit 3: Pure Mathematics B) exam tips
Marking Points
- rearranging f(x) = 0 into fixed-point form x = g(x)
- generating successive iterative approximations x₂, x₃ using the recurrence relation
- drawing an accurate staircase diagram showing monotonic convergence where 0 < g'(x) < 1
- drawing an accurate cobweb diagram showing alternating spiral convergence where −1 < g'(x) < 0
Examiner Tips
- 💡Draw lines strictly vertically to the curve y = g(x) and horizontally to the line y = x.
- 💡Use the ANS key on your calculator to calculate iterative values quickly and avoid rounding errors.
Common Mistakes
- drawing lines between curves at arbitrary angles instead of alternating strictly vertical and horizontal lines
- confusing staircase diagrams (monotonic approach) with cobweb diagrams (oscillating spiral)