Types of Errors in Numerical Analysis

In the world of math, numerical analysis is well known for focusing on the algorithms used to solve issues in continuous math. The practice is familiar territory for engineers and those who work with physical science, but it's beginning to expand further into liberal arts areas as well. You can see this in astrology, stock portfolio analysis, data analysis and medicine. Part of the application of numerical analysis involves the use of errors. Specific errors are sought out and applied to arrive at mathematical conclusions.

Mother and daughter using an abacus together
A mother and daughter are using an abacus.
credit: Ariel Skelley/Blend Images/Getty Images

Rounding Off

The round-off error is used because representing every number as a real number isn't possible. So rounding is introduced to adjust for this situation. A round-off error represents the numerical amount between what a figure actually is versus its closest real number value, depending on how the round is applied. For instance, rounding to the nearest whole number means you round up or down to what is the closest whole figure. So if your result is 3.31 then you would round to 3. Rounding the highest amount would be a bit different. In this approach, if your figure is 3.31, your rounding would be to 4. In terms of numerical analysis the round-off error is an attempt to identify what the rounding distance is when it comes up in algorithms. It’s also known as a quantization error.

Truncation Error

A truncation error occurs when approximation is involved in numerical analysis. The error factor is related to how much the approximate value is at variance from the actual value in a formula or math result. For example, take the formula of 3 x 3 + 4. The calculation equals 28. Now, break it down and the root is close to 1.99. The truncation error value is therefore equal to 0.01.

Discretization Error

Discretization involves converting or partitioning variables or continuous attributes to nominal attributes, intervals and variables. As a type of truncation error, the discretization error focuses on how much a discrete math problem is not consistent with a continuous math problem.

Numerical Stability

If an error stays at one point in an algorithm and doesn’t aggregate further as the calculation continues, then it's considered a numerically stable error. This happens when the error causes only a very small variation in the formula result. If the opposite occurs and the error propagates bigger as the calculation continues, then it is considered numerically unstable.

Error Benefits

Errors are usually regarded as negative, but math errors come in useful in statistics, computer programming, advanced mathematics and much more. Evaluating errors provides significantly useful information, especially when chance and probability is required.