Computational Methods in Physics ASU Physics PHY 494

04 Numbers and Errors

28 Jan 2016

Only a finite number of numbers (integers and floating point) can be exactly represented in binary in the computer. This leads to problems of overflow1 and underflow and errors in floating point arithmetic that one needs to be aware of for numerical calculations. In particular, there is a machine precision , within which two mathematically different floating point numbers are represented by the same number in the computer. A common standard to represent floating point numbers is the IEEE 754 standard2, which defines 32 bit floats and 64 bit doubles 3.

Certain floating point arithmetic operations such as subtracting numbers of similar or very different magnitude, repeated summation, or attempts at establishing exact equivalence, can have unexpected consequences.4

Class Material

The class will be live-coded in a Jupyter notebook. The annotated notebook will be available after the class as 04-numbers-and-errors.ipynb.

For one problem you should obtain the notebook 04-problem-sine-series.ipynb, which already contains part of the code you will need. Pull your PHY494-resources-2016 repository:

cd ~/PHY494-resources-2016
git pull

and find it under ~/PHY494-resources-2016/04_numbers/04-problem-sine-series.ipynb.

Additional resources

Footnotes

  1. Python integers can be used for arbitrary precision integer arithmetic; they will not overflow. NumPy integer data types such as int32, however, will wrap around. ↩

  2. For everything you ever wanted to know about floating point arithmetic see the paper

    D. Goldberg. What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv., 23(1):5–48, 1991. doi: 10.1145/103162.103163. ↩

  3. The Python float is a IEEE 754 double. NumPy has a wider range of numeric data types, including float32 (like float), float64 (like double) and also float128. ↩

  4. See Bruce M Bush's The Perils of Floating Point and a notebook Perils_of_Floating_Point.ipynb based on that article. ↩

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