Numerical methodsLesson 3 of 3 in the course Numerical integration and error

Round-Off Error in Floating-Point Arithmetic and Numerical Stability

5 flashcards · 5 quiz questions · 5 match pairs · 3 fill-in sentences · English

Save course
Lesson progress
0 / 5

Your progress is kept in this browser only.Sign up to save it

Summary

A round-off error is the difference between the result an algorithm would produce using exact arithmetic and the result it produces using finite-precision, rounded arithmetic. Computers store numbers using a limited number of digits or bits, so some values must be approximated.

If xx is the exact value and fl(x)fl(x) is its computer representation, the absolute representation error is ∣x−fl(x)∣|x-fl(x)|. The relative error compares that difference with the size of the exact value: ∣x−fl(x)∣/∣x∣|x-fl(x)|/|x|, for x≠0x\ne 0.

All flashcards

Round-off error
The difference between an algorithm’s exact-arithmetic result and its finite-precision, rounded result.
Truncation error
Error from replacing an exact mathematical process with a finite approximation.
Machine epsilon
A precision-related scale for relative representation error or the smallest increment distinguishable from 1.
Subtractive cancellation
Loss of reliable digits when nearly equal values are subtracted.
Conditioning and stability
Conditioning describes a problem’s sensitivity to input changes; stability describes an algorithm’s response to them.

Quiz questions

  1. When an exact value lies halfway between two representable values, how does IEEE tie-to-even rounding choose?Answer: It selects the value whose last stored digit is even.
  2. Which quantities characterize a floating-point system’s format?Answer: Base, precision, and exponent range.
  3. What is absorption in floating-point addition?Answer: A small addend’s contribution is lost when its low-order digits exceed available precision.
  4. If two significands each have pp digits, how many digits can their exact product have at most?Answer: Up to 2p2p digits.
  5. What does the condition number describe?Answer: How sensitive a problem’s solution is to changes in its input.

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

More in Numerical methods

All Numerical methods lessons
Numerical methodsLagrange Polynomial Interpolation: Construction, Evaluation, and Error5 flashcards · 5 quiz questions · 4 gamesPlay free →Numerical methodsBisection: Root Bracketing, Stopping Criteria, and Higher-Dimensional Variants5 flashcards · 5 quiz questions · 4 gamesPlay free →Numerical methodsNumerical Integration: Quadrature, Error Control, and Multidimensional Methods5 flashcards · 5 quiz questions · 4 gamesPlay free →Numerical methodsPolynomial Interpolation: Forms, Error, Node Choice, and Applications5 flashcards · 5 quiz questions · 4 gamesPlay free →Numerical methodsInterpolation: Methods, Error, and Applications5 flashcards · 5 quiz questions · 4 gamesPlay free →Numerical methodsSimpson's 1/3 and 3/8 Rules: Composite and Adaptive Integration5 flashcards · 5 quiz questions · 4 gamesPlay free →