Truncation error is a concept used in numerical analysis and scientific computation to describe the difference between the true (analytical) derivative of a function and its derivative obtained by numerical approximation. It is the error that is made when truncating an infinite sum and approximating it to a finite sum. 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. 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.
Sometimes the rounding error is also called a truncation error, especially if the number is rounded by truncation. The error that occurs when shortening an infinite process after many finite terms or iterations, or when not reaching the limit. It 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 the numerical solution of differential equations, it is closely related to the concept of discretization error. It seems that the truncation error is relatively simple to calculate by hand or by symbolic software without specifying the differential equation and the discrete model in a special case. Truncation errors are defined as errors that result from the use of an approximation rather than an exact mathematical procedure.
For one-step methods, the local truncation error gives us a measure to determine how the solution to the differential equation does not solve the difference equation. 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. 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. Knowing truncation error or other error measures is important for program verification by empirically establishing convergence rates.
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. The truncation error generally increases as the step size increases, while the rounding error decreases as the step size increases.
We will be concerned with calculating truncation errors that arise in finite difference formulas and in finite difference discretizations of differential equations. Find the truncation error if you use a Reimann sum of two segments on the left with the same width of segments.