What does it mean for an interval to bracket a root?

For a continuous function, its endpoint values have opposite signs, guaranteeing at least one root in the interval.

How does bracket width bound the midpoint's root error?

If the bracket contains a root, the midpoint is at most half the bracket width from that root.

Why does a small residual not always imply an accurate root approximation?

A function may change very slowly near a root, so a small function value can occur far from it.

What is a sign vector in two-dimensional characteristic bisection?

The pair of signs of the two component-function values at a point.

What does relative change measure in an iterative computation?

The absolute change between successive approximations divided by the magnitude of the current approximation.

Numerical methodsLesson 1 of 1 in the course Root finding

Bisection: Root Bracketing, Stopping Criteria, and Higher-Dimensional Variants

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

Save course

Lesson progress

0 / 5

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

Summary

The bisection method is a reliable way to approximate a root of a continuous function when you know an interval whose endpoints give function values with opposite signs. This lesson introduces the bracketing principle, explains each iteration, and works through a polynomial example.

A root of $f$ is a value $x$ for which $f(x)=0$ . Suppose $f$ is continuous on $[a,b]$ and $f(a)$ and $f(b)$ have opposite signs. The intermediate value theorem guarantees that $f$ takes the value zero somewhere between the endpoints. We say that $[a,b]$ brackets a root.

All flashcards

What does it mean for an interval to bracket a root?: For a continuous function, its endpoint values have opposite signs, guaranteeing at least one root in the interval.
How does bracket width bound the midpoint's root error?: If the bracket contains a root, the midpoint is at most half the bracket width from that root.
Why does a small residual not always imply an accurate root approximation?: A function may change very slowly near a root, so a small function value can occur far from it.
What is a sign vector in two-dimensional characteristic bisection?: The pair of signs of the two component-function values at a point.
What does relative change measure in an iterative computation?: The absolute change between successive approximations divided by the magnitude of the current approximation.

Quiz questions

For successive approximations $p_N$ and $p_{N-1}$ , which condition expresses an absolute-change stopping criterion?Answer: $|p_N-p_{N-1}|<\epsilon$
A root is known to lie in a current bisection interval of width $w$ . What is the greatest possible distance from its midpoint to that root?Answer: At most $w/2$ .
In characteristic bisection, what makes an edge proper?Answer: Its endpoint sign vectors differ in exactly one component.
What concept can help establish that a region contains a common zero in a higher-dimensional generalized bisection method?Answer: Topological degree.
In two dimensions, which sign vectors must appear among the vertices of a characteristic quadrilateral?Answer: The four combinations $(-,-)$ , $(-,+)$ , $(+,-)$ , and $(+,+)$ .

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

All Numerical methods lessons

Numerical methodsLagrange Polynomial Interpolation: Construction, Evaluation, and Error5 flashcards · 5 quiz questions · 4 gamesPlay free →Numerical methodsNumerical Integration: Quadrature, Error Control, and Multidimensional Methods5 flashcards · 5 quiz questions · 4 gamesPlay free →

Summary

All flashcards

Quiz questions

More in Numerical methods