In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process.[1][2]
Examples
editInfinite series
editA 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
editThe 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
editThe 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
editTruncation 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
editReferences
edit- ^ Atkinson, Kendall E. (1989). An Introduction to Numerical Analysis (2nd ed.). New York: Wiley. p. 20. ISBN 978-0-471-62489-9. OCLC 803318878.
- ^ 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
- Atkinson, Kendall E. (1989), An Introduction to Numerical Analysis (2nd ed.), New York: John Wiley & Sons, p. 20, ISBN 978-0-471-50023-0
- Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, New York: Springer-Verlag, p. 1, ISBN 978-0-387-95452-3.