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How can you ensure your algorithm implementation is numerically stable?

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Numerical stability is a crucial property of any algorithm that involves computations with finite precision, such as floating-point arithmetic. It means that the algorithm produces accurate and consistent results despite the inevitable errors and rounding that occur during the calculations. In this article, you will learn some tips and techniques to ensure your algorithm implementation is numerically stable, whether you are using Python, MATLAB, R, or any other programming language.

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Ashkan Mohammadi, Ph.D.

Data Scientist | Predictive Modeler | Applied Mathematician | Operation Researcher | Math Olympiad Medalist

1 Avoid subtraction of nearly equal numbers

One of the most common sources of numerical instability is the subtraction of two numbers that are very close in magnitude. This can lead to a significant loss of precision and accuracy, as the difference may be much smaller than the smallest representable unit (or machine epsilon) of the floating-point system. For example, consider the following code in Python:

a = 1.23456789
b = 1.23456788
c = a - b
print(c)

The output is 1.000000082740371e-08, which is much larger than the true difference of 1e-08. To avoid this problem, you should try to reformulate your algorithm or equation in a way that avoids such subtraction, or use a higher-precision data type if available.

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Dr. Gbenga Obokhai Folami(Reg)

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Numerical stability could be achieved by the following: Including numerical diffusion which refers to mathematical terms that ensures round offs and other errors in the calculations get spread out and do not cause the calculations to blow up. By keeping ?t small enough to satisfy the inequality.

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2 Choose appropriate scaling and units

Another way to improve numerical stability is to choose appropriate scaling and units for your variables and parameters. Scaling means adjusting the magnitude or order of magnitude of your values, so that they are within a reasonable range for the floating-point system. For example, if you are working with distances in meters, you may want to scale them by 1000 to convert them to kilometers, or by 0.001 to convert them to millimeters, depending on the context and the expected results. Units mean selecting a consistent and meaningful unit of measurement for your values, such as meters, seconds, kilograms, etc. This can help you avoid confusion and errors when performing calculations and conversions.

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3 Use stable algorithms and libraries

Not all algorithms are equally stable, even if they are mathematically equivalent. Some algorithms may amplify the errors and rounding that occur during the computations, while others may reduce or control them. For example, when solving a system of linear equations, the Gaussian elimination method may be unstable if the matrix is ill-conditioned, while the LU decomposition method may be more stable. Similarly, when computing the roots of a polynomial, the classical formula may be unstable if the coefficients are large or the roots are close, while the Jenkins-Traub algorithm may be more robust. Therefore, you should always choose a stable algorithm for your problem, and use reliable and tested libraries or packages that implement them.

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4 Test and compare your results

Finally, you should always test and compare your results with different inputs, data types, algorithms, and libraries, to check for numerical stability and accuracy. You can use tools such as unit testing, debugging, profiling, and benchmarking to verify and validate your implementation. You can also compare your results with analytical solutions, if available, or with other sources of reference, such as textbooks, papers, or online calculators. By doing so, you can identify and correct any potential sources of numerical instability, and ensure your algorithm implementation is reliable and consistent.

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5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Ashkan Mohammadi, Ph.D.

Data Scientist | Predictive Modeler | Applied Mathematician | Operation Researcher | Math Olympiad Medalist
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Stability depends on the pair of the problem itself and the associated algorithm to solve the problem. Solely looking at one of the components of this pair we cannot determine the stability. Some algorithms require computing the inverse of a matrix on each iteration may create instability but the same algorithm can be very efficient on problems having sparsity essence.

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How can you ensure your algorithm implementation is numerically stable?

1

2

3

4

5

1 Avoid subtraction of nearly equal numbers

2 Choose appropriate scaling and units

3 Use stable algorithms and libraries

4 Test and compare your results

5 Here’s what else to consider

Operations Research

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How can you ensure your algorithm implementation is numerically stable?

1

2

3

4

5

1 Avoid subtraction of nearly equal numbers

2 Choose appropriate scaling and units

3 Use stable algorithms and libraries

4 Test and compare your results

5 Here’s what else to consider

Operations Research

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