How to Solve: Numerical Methods (Iteration, Newton-Raphson, Trapezium Rule)

How to Solve: Numerical Methods (Iteration, Newton-Raphson, Trapezium Rule)

For GCSE & A-Level Maths (Edexcel/AQA/OCR)

Introduction

"Mastering numerical methods lets you solve equations that algebra can’t—like finding the root of x³ + 2x – 5 = 0 or estimating the area under a curve when integration is impossible. On your exam, these questions are worth 8–12 marks (10–15% of your paper), and they’re often the easiest marks to lose if you don’t follow the steps exactly. Today, you’ll learn the foolproof method to ace them every time."

What You Need To Know First

1. Rearranging equations – You must be able to rewrite f(x) = 0 into the form x = g(x) for iteration.
2. Differentiation – For Newton-Raphson, you need to find f’(x).
3. Area under a curve – The Trapezium Rule approximates this when exact integration is too hard.

Key Vocabulary

Formulas To Know

1. Iteration (Fixed-Point Iteration)

Formula: xₙ₊₁ = g(xₙ) - xₙ = current guess - g(x) = rearranged function (e.g., x = √(5 – 2x) for x² + 2x – 5 = 0)

MEMORISE THIS: You must rearrange f(x) = 0 into x = g(x) first.

2. Newton-Raphson Method

Formula: xₙ₊₁ = xₙ – f(xₙ)/f’(xₙ) - f(x) = original function - f’(x) = derivative of f(x) - xₙ = current guess

MEMORISE THIS: You must differentiate f(x) correctly.

3. Trapezium Rule

Formula: Area ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ] - h = width of each strip (h = (b – a)/n) - n = number of strips (trapezoids) - y₀, y₁, …, yₙ = heights at x = a, a+h, a+2h, …, b

GIVEN ON EXAM SHEET: You don’t need to memorise this, but you must know how to apply it.

Step-by-Step Method

A. Iteration (Fixed-Point Iteration)

When to use: When you’re given f(x) = 0 and asked to find a root using iteration.

Steps: 1. Rearrange f(x) = 0 into x = g(x) (e.g., x² – 3x + 1 = 0 → x = (x² + 1)/3). 2. Choose a starting value x₀ (usually given in the question). 3. Plug x₀ into g(x) to get x₁. 4. Repeat using x₁ to get x₂, then x₃, etc. 5. Stop when the values round to the same number (e.g., 1.532, 1.532, 1.532). 6. Check convergence: If the values get closer, it’s working. If they explode, try a different g(x) or x₀.

B. Newton-Raphson Method

When to use: When you need a fast approximation of a root (fewer iterations than fixed-point).

Steps: 1. Write down f(x) and differentiate to get f’(x). 2. Choose a starting value x₀ (close to the root if possible). 3. Plug into the formula:
x₁ = x₀ – f(x₀)/f’(x₀) 4. Repeat using x₁ to get x₂, etc. 5. Stop when the values round to the same number (e.g., 2.345, 2.345). 6. Watch for failure: If f’(xₙ) = 0, the method breaks (tangent is horizontal).

C. Trapezium Rule

When to use: When asked to estimate the area under a curve y = f(x) between x = a and x = b.

Steps: 1. Find h: h = (b – a)/n (where n = number of strips). 2. List x-values: x₀ = a, x₁ = a + h, x₂ = a + 2h, …, xₙ = b. 3. Find y-values: Calculate y = f(x) for each x. 4. Apply the formula:
Area ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ] 5. Simplify and give the answer to the required decimal places.

Worked Examples

Example 1 – Iteration (Basic)

Question: Use iteration to solve x² – 4x + 1 = 0 with x₀ = 3. Give the root to 3 decimal places.

Steps: 1. Rearrange: x² – 4x + 1 = 0 → x = (x² + 1)/4 → g(x) = (x² + 1)/4 2. x₀ = 3 3. x₁ = (3² + 1)/4 = (9 + 1)/4 = 2.5 4. x₂ = (2.5² + 1)/4 = (6.25 + 1)/4 = 1.8125 5. x₃ = (1.8125² + 1)/4 ≈ (3.285 + 1)/4 ≈ 1.071 6. x₄ = (1.071² + 1)/4 ≈ (1.147 + 1)/4 ≈ 0.537 7. x₅ = (0.537² + 1)/4 ≈ (0.288 + 1)/4 ≈ 0.322 8. x₆ = (0.322² + 1)/4 ≈ (0.104 + 1)/4 ≈ 0.276 9. Stop when stable: The values are diverging! Try a different g(x) or x₀.

Alternative rearrangement: x = √(4x – 1) - x₀ = 3 - x₁ = √(43 – 1) = √11 ≈ 3.317 - x₂ = √(43.317 – 1) ≈ √12.268 ≈ 3.503 - x₃ ≈ √(43.503 – 1) ≈ √13.012 ≈ 3.607 - x₄ ≈ √(43.607 – 1) ≈ √13.428 ≈ 3.664 - x₅ ≈ √(43.664 – 1) ≈ √13.656 ≈ 3.695 - x₆ ≈ √(43.695 – 1) ≈ √13.78 ≈ 3.712 - Converging to ≈ 3.732 (correct root).

What we did and why: - The first rearrangement (x = (x² + 1)/4) failed because it diverged. - The second (x = √(4x – 1)) worked because it converged. - Key lesson: If one g(x) fails, try another!

Example 2 – Newton-Raphson (Medium)

Question: Use Newton-Raphson to find the root of f(x) = x³ – 2x – 5 near x = 2. Give the answer to 4 decimal places.

Steps: 1. f(x) = x³ – 2x – 5 2. f’(x) = 3x² – 2 (differentiate) 3. x₀ = 2 4. f(2) = 8 – 4 – 5 = –1 5. f’(2) = 12 – 2 = 10 6. x₁ = 2 – (–1)/10 = 2 + 0.1 = 2.1 7. f(2.1) = 9.261 – 4.2 – 5 = 0.061 8. f’(2.1) = 13.23 – 2 = 11.23 9. x₂ = 2.1 – 0.061/11.23 ≈ 2.1 – 0.0054 ≈ 2.0946 10. f(2.0946) ≈ 9.173 – 4.189 – 5 ≈ –0.016 11. f’(2.0946) ≈ 13.16 – 2 ≈ 11.16 12. x₃ ≈ 2.0946 – (–0.016)/11.16 ≈ 2.0946 + 0.0014 ≈ 2.0960 13. Stop when stable: x₃ ≈ 2.0960 (to 4 d.p.).

What we did and why: - We used the Newton-Raphson formula xₙ₊₁ = xₙ – f(xₙ)/f’(xₙ). - The method converged quickly because f’(x) was never zero. - Key lesson: Always check f’(xₙ) ≠ 0 to avoid division by zero.

Example 3 – Trapezium Rule (Exam-Style)

Question: Estimate the area under y = eˣ between x = 0 and x = 2 using 4 strips. Give your answer to 3 decimal places.

Steps: 1. a = 0, b = 2, n = 4 2. h = (2 – 0)/4 = 0.5 3. x-values: 0, 0.5, 1, 1.5, 2 4. y-values (y = eˣ):
- y₀ = e⁰ = 1
- y₁ = e⁰·⁵ ≈ 1.6487
- y₂ = e¹ ≈ 2.7183
- y₃ = e¹·⁵ ≈ 4.4817
- y₄ = e² ≈ 7.3891 5. Apply formula:
Area ≈ 0.5/2 [1 + 2(1.6487 + 2.7183 + 4.4817) + 7.3891]
≈ 0.25 [1 + 2(8.8487) + 7.3891]
≈ 0.25 [1 + 17.6974 + 7.3891]
≈ 0.25 [26.0865] ≈ 6.5216 6. Final answer: 6.522 (to 3 d.p.).

What we did and why: - We split the area into 4 trapezoids and applied the Trapezium Rule. - Key lesson: More strips (n) = more accurate result (but more work!).

Common Mistakes

Exam Traps

1-Minute Recap (Night Before the Exam)

"Listen up—this is your 60-second survival guide for numerical methods: 1. Iteration: Rearrange f(x) = 0 into x = g(x), pick x₀, and keep plugging in until the numbers stop changing. If it diverges, try a different g(x). 2. Newton-Raphson: Use xₙ₊₁ = xₙ – f(xₙ)/f’(xₙ). Differentiate f(x) first, and never let f’(x) = 0. 3. Trapezium Rule: Split the area into strips, calculate h, list x and y values, then plug into h/2 [y₀ + 2(y₁ + …) + yₙ]. More strips = more accuracy. 4. Exam traps: Watch for f’(x) = 0 in Newton-Raphson, divergence in iteration, and whether the Trapezium Rule over/underestimates. 5. Always show working—examiners give marks for method, not just the answer!

Now go smash those 12-mark questions!