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linear convergence (Definition)

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

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

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

$\displaystyle \vert\vert x_{i+1}-x^*\vert\vert \leq c^i \vert\vert x_1 -x^*\vert\vert. $
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.



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See Also: quadratic convergence

Also defines:  superlinear convergence
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Cross-references: index, inequality, natural number, converge, sequence
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This is version 10 of linear convergence, born on 2004-05-02, modified 2007-05-27.
Object id is 5824, canonical name is LinearConvergence.
Accessed 5155 times total.

Classification:
AMS MSC41A25 (Approximations and expansions :: Rate of convergence, degree of approximation)

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