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Homelinear convergence

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# linear convergence

A sequence $\{x_{i}\}$ is said to converge linearly to $x^{*}$ if there is a constant $1>c>0$ such that $||x_{{i+1}}-x^{*}||\leq c||x_{i}-x^{*}||$ for all $i>N$ for some natural number $N>0$.

An alternative definition is that $||x_{{i+1}}-x_{i}||\leq c||x_{i}-x_{{i-1}}||$ for all $i$.

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

$||x_{{i+1}}-x^{*}||\leq c^{i}||x_{1}-x^{*}||.$ |

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

If the inequality holds for all $c>0$ then we say that the sequence $\{x_{i}\}$
has *superlinear convergence*.

Defines:

superlinear convergence

Related:

QuadraticConvergence

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

41A25*no label found*

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