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单词 IntervalHalvingConvergesLinearly
释义

interval halving converges linearly


Theorem 1.

The interval halving algorithm converges linearly.

Proof.

To see that interval halving (or bisection) converges linearly we use the alternative definition of linear convergence that says that |xi+1-xi|<c|xi-xi-1| for some constant 1>c>0.

In the case of interval halving, |xi+1-xi| is the length of the intervalMathworldPlanetmath we should search for the solution in and has xi+2 as its midpointMathworldPlanetmathPlanetmathPlanetmath. We have then that this interval has half the length of the previous interval which means, 𝑙𝑒𝑛𝑔𝑡ℎi+1=12𝑙𝑒𝑛𝑔𝑡ℎi. Thus c=1/2 and we have exact linear convergence.∎

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更新时间:2025/5/4 12:09:33