Linear multistep method

Why does skipping steps save time but lose accuracy?

Imagine you're hiking a trail with a map that shows your previous steps. You want to reach the next campsite quickly but also make sure you're on the right path.

Think of walking the trail step by step versus skipping ahead. Skipping helps you move faster but might lead you off the trail. Multistep methods like Adams-Bashforth-Moulton keep track of your past steps to guide you more accurately.

Example

If you skip every second step, you might miss a turn. But if you remember the last three steps, you can predict your next move more reliably.

Remember this

Multistep methods remember past steps, improving accuracy and efficiency over methods that only look back once.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

Related concepts

Finite element method

Why does a straight-line guess fail in complex terrain?

second-order methods (Newton's) converge faster but are expensive: O(n³) per step

Second-order methods converge faster due to quadratic convergence but are expensive due to O(n³) per iteration

ill-conditioned matrices cause numerical instability: small input changes → large output changes

Small changes can lead to huge errors in math

iterative methods (CG, GMRES) do: solve Ax=b without explicitly inverting A

How can we find answers without solving problems head-on?

Euler method

Euler method approximates ODE solution with y_{n+1} = y_n + h·f(y_n)

approximation algorithms guarantee: solution within factor α of optimal

Can we always find the best solution quickly?

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