linear convergence

A sequence {xi} is said to converge linearly to x* if there is a constant 1>c>0 such that ||xi+1-x*||c||xi-x*|| for all i>N for some natural numberMathworldPlanetmath N>0.

An alternative definition is that ||xi+1-xi||c||xi-xi-1|| for all i.

Notice that if N=1, then by iterating the first inequality we have


That is, the error decreases exponentially with the index i.

If the inequality holds for all c>0 then we say that the sequence {xi} has superlinear convergence.

Title linear convergence
Canonical name LinearConvergence
Date of creation 2013-03-22 14:20:55
Last modified on 2013-03-22 14:20:55
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 13
Author Mathprof (13753)
Entry type Definition
Classification msc 41A25
Related topic QuadraticConvergence
Defines superlinear convergence