Truncation error or local truncation error (LTE) is error made by numerical algorithms that arises from taking finite number of steps in computation. It is present even with infinite-precision arithmetic, because it is caused by truncation of the infinite Taylor series

Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 to form the algorithm.

Use of arbitrarily small steps in numerical computation is prevented by round-off error

Round-off error

A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finitely many...

, which are the consequence of using finite precision floating point numbers

Floating point

In computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...

on computers.

Local Truncation error for a single step method:
Τ_i+1(h)= ( y_i+1 - y_i ) / h - Φ(t_i,y_i)