Accuracy and stability of numerical algorithms
by Nicholas J. Higham · first published 1996
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Why HN recommends it
A thorough treatment of floating-point error and numerical stability for anyone writing accurate numerical algorithms. It serves as a follow-up to Goldberg's floating-point paper.
Summarised by Claude from recommending comments
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If you are interested in writing accurate numerical algorithms, as further reading I highly recommend reading "Accuracy and Stability of Numerical Algorithms" by Nick Higham, which is a very good comprehensive book. Like this article says, floating-point arithmetic is often thought of as mysterious, but it really isn't: it just obeys its own precisely specified…
on “Practically Accurate Floating-Point Math” ↗…that provide good background for this stuff, read Goldberg's paper "What Every Computer Scientist Should Know About Floating-Point Arithmetic" [1]. The next good thing to dig through is Higham's book "Accuracy and Stability of Numerical Algorithms" [2] Those will give you lots of places to look for issues, and the ability to analyze algorithms more concretely for floating point issues. [1] https://docs.oracle.com/cd/E19957-01/800-7895/800-7895.pdf [2]…
on “C library for multiple-precision floating-point arithmetic with correct rounding” ↗…with a dynamic scale is presented along the way motivating floating point in probably every computer architecture class. It’s a floating point. This guy needs to open a book. I recommend Nick Higham’s _Accuracy and Stability of Numerical Algorithms_.
on “Detached Point Arithmetic” ↗…there could be mistakes but that’s the gist. I’ve had to do this for my work on a much more complicated example, and it took a few tries. There is a good linear algebra book about this, I think called Accuracy and Stability of Numerical Algorithms (Nicholas Higham). The famous PBR graphics book by Pharr et. al. also talks about error estimation techniques.
on “Unit Testing Numerical Routines” ↗He also wrote "Accuracy and Stability of Numerical Algorithms" which is part of the canon of the field.
on “Seven sins of numerical linear algebra” ↗…http://lipforge.ens-lyon.fr/www/crlibm/documents/cern.pdf. Floating point numbers are _not_ intervals. If you read carefully any formal definition of floating point numbers (The IEEE standards, TAoCP Chapter 4, or Higham's _Accuracy and Stability of Numerical Algorithms_, to name just three possible references), you will see that floating point numbers by definition form an exact rational subset of the extended real line. > All numbers in that range alias to the same…
on “0.30000000000000004” ↗Floating point is Nick Higham's thing isn't? I have his book "Accuracy and Stability of Numerical Algorithms, Second Edition" http://www.ma.man.ac.uk/~higham/asna/ but I haven't got very far with it yet. Maybe the article in the Princeton Companion is a summarized version of his book?
on “How many floating-point numbers are in the interval [0,1]?” ↗