What is Truncation Error and How is it Defined?

Truncation error is a type of error that occurs when an infinite mathematical process is approximated by a finite one. It is the difference between the true (analytical) derivative of a function and its derivative obtained by numerical approximation. In numerical analysis and scientific computation, truncation error is the error caused by the approximation of a mathematical process. 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 is present even with infinite precision arithmetic, because it is caused by the truncation of Taylor's infinite series to form the algorithm. Often, the truncation error 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. Sometimes the rounding error is also called a truncation error, especially if the number is rounded by truncation. Truncation errors are defined as errors that result from the use of an approximation rather than an exact mathematical procedure.

The global truncation error (GTE) is one order smaller than the local truncation error (LTE). The truncation error generally increases as the step size increases, while the rounding error decreases as the step size increases. For example, 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. Find the truncation error if you use a Reimann sum of two segments on the left with the same width of segments.

Therefore, we can conclude that the global truncation error is a smaller order than the local truncation error. Occasionally, by mistake, the rounding error (the consequence of using finite-precision floating-point numbers in computers) is also called a truncation error, especially if the number is rounded by cutting. 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 infinite series, find the truncation error of x% 3d0,75 if only the first three terms of the series are used.