What is Truncation Error and How to Calculate it?

Truncation error is the difference between a true (analytical) derivative of a function and its derivative obtained by numerical approximation. It is the discrepancy that arises when performing a finite number of steps to approximate a process with infinite steps.

Truncation

error is present even with infinite precision arithmetic, because it is caused by the truncation of Taylor's infinite series to form the algorithm. It is also known as the error of truncation, and it is one of the main sources of error in numerical methods for the algorithmic solution of continuous problems.

In numerical analysis and scientific computation, the truncation error is the error caused by the approximation of a mathematical process. It can be seen when shortening an infinite process after many finite terms or iterations, or when not reaching the limit. For example, if we approximate the sine function by the first two non-zero terms of its Taylor series, as in small, the resulting error is a truncation error. It also includes the discretization error, which is the error that arises when taking a finite number of steps in a calculation to approximate an infinite process. In general, the global truncation error (GTE) is one order smaller than the local truncation error (LTE).

The error that occurs when shortening an infinite process after many finite terms or iterations, or when not reaching the limit, represents one of the main sources of error in numerical methods for the algorithmic solution of continuous problems. Its analysis and methods for its estimation and control are central problems in numerical analysis. In numerical methods for ordinary differential equations, the continuously varying function that is the solution of the differential equation is approximated by a process that progresses step by step, and the error that this implies is a discretization or truncation error. The error caused by choosing a finite number of rectangles instead of an infinite number of them is a truncation error in the mathematical process of integration. Given an example such as an infinite series truncation, finite precision arithmetic, finite differences, and differential equations, it is possible to calculate its truncation errors. For instance, if we take into account an example such as x% 3d0,75 and only use its first three terms of its series, then we can calculate its truncation errors.

We can also list down all common finite difference approximation formulas and their corresponding truncation errors. Therefore, we can conclude that the global truncation error is a smaller order than the local truncation error. In general, the term truncation error refers to the discrepancy that arises when performing a finite number of steps to approximate a process with infinite steps.