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MathidiaMathematics Academy
Numerical analysis
Lesson 45 of 75
Review

Convergence rates

Objective. Measure and justify algorithmic convergence.

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

Convergence rate measures how quickly an iterative sequence xₙ approaches its limit x*, typically classified as linear, superlinear or quadratic depending on how the error eₙ = |xₙ − x*| shrinks. A method has order p if eₙ₊₁ ≈ C eₙ^p for some constant C; Newton's method typically achieves quadratic convergence (p = 2) near a simple root, while Jacobi iteration is only linear (p = 1). For example, if eₙ halves each step the convergence is linear, but if eₙ squares relative to a constant (like eₙ₊₁ = eₙ²) the error shrinks dramatically faster once eₙ < 1. Justifying convergence rate requires estimating C and p from successive error ratios, often via log-log plots.

Worked example

A sequence has errors e₁ = 0.1, e₂ = 0.01, e₃ = 0.0001. Determine the apparent order p of convergence using the ratio test eₙ₊₁ ≈ C eₙ^p.

A sequence has errors e₁ …
Illustrated card showing A sequence has errors e₁ = 0
  1. 1From e₁ to e₂: 0.01 = C(0.1)^p.
  2. 2From e₂ to e₃: 0.0001 = C(0.01)^p.
  3. 3Divide the two relations: 0.0001/0.01 = (0.01/0.1)^p, giving 0.01 = (0.1)^p.
  4. 4Solve: p = 2, since (0.1)² = 0.01, so convergence is quadratic.

Answer: p = 2 (quadratic convergence)

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Common mistake. Students confuse 'converges faster' with 'converges in fewer iterations' without checking the order p; a linearly convergent method with very small C can beat a quadratic one for a few iterations, so always compare actual error ratios, not just classification.

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