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MathidiaMathematics Academy
Numerical methods
Lesson 57 of 75
Easy practice

Root-finding methods

Objective. Compare bisection, fixed-point and Newton methods.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Root-finding methods locate x with f(x)=0 numerically when no algebraic formula exists. Bisection repeatedly halves an interval where f changes sign, guaranteeing convergence but slowly; fixed-point iteration writes x = g(x) and iterates x_{n+1}=g(x_n), converging only if |g'(x)|<1 near the root; Newton's method uses x_{n+1} = x_n − f(x_n)/f'(x_n), converging very fast near a simple root but needing a good start and f'(x)≠0. For f(x)=x²−2 starting at x₀=1.5, Newton gives x₁ = 1.5 − (0.25)/3 = 1.41667, already close to √2.

Worked example

Starting from x₀ = 2, perform one Newton's method iteration for root-finding on f(x) = x² − 5.

Starting from x₀ = 2
Illustrated card showing Starting from x₀ = 2
  1. 1Newton formula: x₁ = x₀ − f(x₀)/f'(x₀)
  2. 2Compute f(2)=4−5=−1, f'(2)=4
  3. 3Substitute: x₁ = 2 − (−1)/4
  4. 4Simplify: x₁ = 2.25

Answer: x₁ = 2.25

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Common mistake. Applying Newton's method where f'(x) is zero or very small near the current estimate; the update divides by nearly zero and the iteration can diverge wildly instead of converging.

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