alt Hacker News>The entire scientific and spectral differential equations community knows there are no numerical issues working with high degree polynomials over arbitrary intervals.
This is totally wrong. Of course there are enormous numerical problems with high degree polynomials. Computing large powers of large numbers is enormously unstable and needs to be avoided under basically all circumstances, that is what makes dealing with those polynomials difficult and cautioning people against this is obvious correct.
What you described are the ways to deal with those problems. But this isn't what the article does. My problem with the article is the following:
- Does not mention the most important issue with high degree polynomials, namely their numerical instability.
- Gives examples, but does not explain why they circumvent the numerical problems at all. The most important choice made (the interval of 0 to 1) is portrayed as essentially arbitrary, not an absolute necessity.
- Concludes that there are no problems with high degree polynomials, based on the fact that the experiments worked. Not based on the fact that the actual issues were circumvented, leaving the reader with a totally wrong impression of the issue.
This is terrible scholarship and makes for a terrible article. Not acknowledging the issue is a terrible thing to do and not explain why seemingly arbitrary choices are extremely important is another terrible thing to do. The whole article fails at actually portraying the issue at all.
To be clear, I am not saying that this approximation does not work or that with appropriate scaling and the right polynomials these issues can't be mostly circumvented. Or that high degree polynomials "in general" are incalculable. I am saying that this article completely fails to say what the issue is and why the examples work.
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> Computing large powers of large numbers is enormously unstable and needs to be avoided under basically all circumstances, that is what makes dealing with those polynomials difficult and cautioning people against this is obvious correct
But we don’t compute large powers of large numbers? Chebyshev is on [-1, 1]. Your large powers go to zero. And your coefficients almost always decrease as the degree goes up. Then to top it off, you usually compute the sum of your terms in descending order, so that the biggest ones are added last.
I believe the author assumes that it's clear to the reader that there is a distinction between how a mathematical object is defined, and how it's computationally used. A polynomial can be defined as a power series, but it's not how they are computationally used. In this sense, the author was mistaken.
But it's not that the problems are "circumvented", in the sense that it's a kind of a hack or a patch, but they are solved, in the sense that there is a systematic way to correctly compute with polynomials.