In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process.[1][2]

Examples

edit

Infinite series

edit

A summation series for   is given by an infinite series such as  

In reality, we can only use a finite number of these terms as it would take an infinite amount of computational time to make use of all of them. So let's suppose we use only three terms of the series, then  

In this case, the truncation error is  

Example A:

Given the following infinite series, find the truncation error for x = 0.75 if only the first three terms of the series are used.  

Solution

Using only first three terms of the series gives  

The sum of an infinite geometrical series   is given by  

For our series, a = 1 and r = 0.75, to give  

The truncation error hence is  

Differentiation

edit

The definition of the exact first derivative of the function is given by  

However, if we are calculating the derivative numerically,   has to be finite. The error caused by choosing   to be finite is a truncation error in the mathematical process of differentiation.

Example A:

Find the truncation in calculating the first derivative of   at   using a step size of  

Solution:

The first derivative of   is   and at  ,  

The approximate value is given by  

The truncation error hence is  

Integration

edit

The definition of the exact integral of a function   from   to   is given as follows.

Let   be a function defined on a closed interval   of the real numbers,  , and   be a partition of I, where     where   and  .

This implies that we are finding the area under the curve using infinite rectangles. However, if we are calculating the integral numerically, we can only use a finite number of rectangles. The error caused by choosing a finite number of rectangles as opposed to an infinite number of them is a truncation error in the mathematical process of integration.

Example A.

For the integral   find the truncation error if a two-segment left-hand Riemann sum is used with equal width of segments.

Solution

We have the exact value as  

Using two rectangles of equal width to approximate the area (see Figure 2) under the curve, the approximate value of the integral

 

 

Occasionally, by mistake, round-off error (the consequence of using finite precision floating point numbers on computers), is also called truncation error, especially if the number is rounded by chopping. That is not the correct use of "truncation error"; however calling it truncating a number may be acceptable.

Addition

edit

Truncation error can cause   within a computer when   because   (like it should), while  . Here,   has a truncation error equal to 1. This truncation error occurs because computers do not store the least significant digits of an extremely large integer.

See also

edit

References

edit
  1. ^ Atkinson, Kendall E. (1989). An Introduction to Numerical Analysis (2nd ed.). New York: Wiley. p. 20. ISBN 978-0-471-62489-9. OCLC 803318878.
  2. ^ Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Princeton, N.J.: Recording for the Blind & Dyslexic, OCLC 50556273, retrieved 2022-02-08