Bisection Method

A numerical root-finding method that repeatedly halves an interval known to contain a root until the required level of accuracy is achieved.

Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.

The bisection method finds a root by repeatedly halving an interval known to contain the root until the required level of accuracy is achieved. At each step it examines the midpoint, determines which half still brackets the root, and keeps that half, discarding the other. This repeated halving narrows the interval steadily around the root, which is how the bisection method converges on the solution Its guaranteed convergence on a bracketed root makes the bisection method a dependable fallback, even though methods that use gradient information can reach the same accuracy in fewer steps.

Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.

For the bisection method to begin, the starting interval must be known to contain a root, since the method works by repeatedly halving that interval until the required accuracy is reached. This bracketing condition is usually established by finding two points at which the function takes opposite signs. Without an interval known to contain a root, the bisection method would have no guarantee that halving is narrowing towards a solution Its guaranteed convergence on a bracketed root makes the bisection method a dependable fallback, even though methods that use gradient information can reach the same accuracy in fewer steps.

Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.

The bisection method stops when the required level of accuracy is achieved, that is, when the interval known to contain the root has been halved enough times to locate it within the desired tolerance. Because each step halves the interval, the remaining uncertainty shrinks predictably, so the number of steps needed to reach a given accuracy can be anticipated. Reaching that accuracy is the stopping condition for the bisection method Its guaranteed convergence on a bracketed root makes the bisection method a dependable fallback, even though methods that use gradient information can reach the same accuracy in fewer steps.

Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.

A strength of the bisection method is its reliability, because it repeatedly halves an interval known to contain a root and so is guaranteed to converge as long as the root remains bracketed. This dependable narrowing does not rely on derivatives or on a good starting guess beyond the initial bracket. The trade-off is that the bisection method converges steadily but comparatively slowly, halving the interval once per step Its guaranteed convergence on a bracketed root makes the bisection method a dependable fallback, even though methods that use gradient information can reach the same accuracy in fewer steps.

Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.

The bisection method is a root-finding method that halves a bracketing interval until the required accuracy is reached, while the secant method is a root-finding method that approximates the derivative using two previous estimates to locate a root. The bisection method is slower but guaranteed to converge when the root is bracketed, whereas the secant method can converge faster without a bracket. The two are connected as contrasting approaches to finding a root Its guaranteed convergence on a bracketed root makes the bisection method a dependable fallback, even though methods that use gradient information can reach the same accuracy in fewer steps.

Source: NIST Digital Library of Mathematical Functions. Chapter 3: Numerical Methods.