Numerical methodsLesson 4 of 4 in the course Numerical integration and error

Numerical Stability: Error, Iterations, and Differential Equations

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Numerical algorithms use finite computations to approximate mathematical answers. Small rounding or approximation errors are often unavoidable, but a reliable algorithm should not let those errors grow without control. This lesson introduces numerical stability and the main ways to describe error in numerical linear algebra.

Suppose a problem maps input data xx to an exact solution y=f(x)y=f(x). A numerical algorithm returns an approximation y∗y^*. The difference between the computed and exact answers can arise from round-off error, caused by finite precision, and truncation error, caused by replacing an exact mathematical process with an approximation.

All flashcards

What is forward error?
Forward error is the computed output minus the exact output, y∗−yy^* - y.
What is backward error?
It is the smallest input perturbation that makes the computed result exact for a nearby problem.
What does a condition number describe?
It describes how sensitive a problem’s solution is to changes in its input.
What is backward stability?
An algorithm is backward stable when its result solves a nearby problem with only a small input perturbation.
How do conditioning and algorithm stability differ?
Conditioning is inherent problem sensitivity; stability concerns how an algorithm controls computational errors.

Quiz questions

  1. The iteration xk+1=xk+2/xk2x_{k+1} = \frac{x_k + 2/x_k}{2} is used to approximate which value?Answer: 2\sqrt{2}
  2. Why can evaluating x(x+1−x)x(\sqrt{x+1}-\sqrt{x}) be inaccurate for large xx in finite precision?Answer: It subtracts nearby approximations, which can discard significant digits.
  3. What is von Neumann stability analysis commonly used to study?Answer: Finite-difference schemes for linear partial differential equations.
  4. Under its applicable setting for linear evolutionary problems, what does the Lax equivalence theorem connect?Answer: Consistency and stability with convergence.
  5. Why is numerical stability especially important when solving a stiff ordinary differential equation?Answer: An unsuitable method may amplify errors or require impractically small steps.

Based on the Wikipedia article Numerical stability, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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