---
tag: BX4D85A79P
kind: concept
name: Linear multistep method
msc: [65L06, 65L05]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_multistep_method
same_as:
  wikidata: Q1462003
  wikipedia: Linear multistep method
---

## Summary

A linear multistep method solves an [[initial value problem|0FXDW992DS]] $y' = f(t, y)$ numerically.
It computes the next approximation from a fixed number of earlier approximations and the values of $f$ there.
It needs only one new evaluation of $f$ per step, which makes it cheap compared with [[Runge–Kutta methods|7DBPJKYC6C]] of the same order.
The Adams methods and the [[backward differentiation formulas|TDA94C2JYT]] are the main families.
Its convergence is governed by consistency and zero-stability, and its possible order is limited by the [[Dahlquist barriers|BQBZ7QMJE2]].

## Explanations

### L1 human

When you walk along a curve and have already recorded several points, you can use all of them to guess where the curve goes next.
A linear multistep method does this: it combines the last few positions and slopes with fixed weights.
For example, the two-step Adams–Bashforth method steps forward by $h \left( \frac32 f_{k+1} - \frac12 f_k \right)$, an extrapolation of the last two slopes.
The first few values must be supplied by another method, such as a Runge–Kutta method.