In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. The term truncation comes from the fact that these simplifications often involve the truncation of an infinite series expansion so as to make the computation possible and practical.

Examples

Infinite series

Example A:

Given the following infinite series, find the truncation error for

Solution

For our series, and

Differentiation

to be finite is a truncation error in the mathematical process of differentiation.

Example A:

Solution:

Integration

is given as follows.

be a function defined on a closed interval be a partition of I.

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.

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

Solution

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

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

• Quantization error

References

• .