Computational Methods in Physics ASU Physics PHY 494

07 Numbers

08 Feb 2018

When working with numerical code, one has to be aware of the limitations imposed by the representation of numbers in the computer and the numerical errors that algorithms invariably accumulate.

Representation of numbers

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 on Numbers

The class will be presented in a Jupyter notebook. The annotated notebook is 07-numbers.ipynb.

Problem: Sine Series (Optional)

For one problem you should obtain the notebook 07-problem-sine-series.ipynb, which already contains part of the code you will need. git pull the PHY494-resources repository and find it under ~/PHY494-resources/07_numbers/07-problem-sine-series.ipynb; as usual, copy it to your work directory (~/PHY494/07_numbers) and work on it there.

(The solution 07-solution-sine-series.ipynb will be made available later.)

Additional resources for Numbers

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, but the float128 is not really a true 128-bit number; on typical x86 machines, these are C long double which can provide up to 80 bit precision (but not 128 bit), es explained in the NumPy docs on extended precision. ↩

  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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