# remez

Published articles for remez.

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## Carathéodory-Fejér Approximation as an Alternative to Remez

DevFeed: [Carathéodory-Fejér Approximation as an Alternative to Remez](<https://devfeed.tech/articles/anyone-know-of-an-approachable-proof-of-the-caratheodory-fejer-theorem-40515.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/shortform/2025-02-09-1947/>)

Published: 2025-02-10T03:47:59Z

Content type: opinion

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Algorithms](<https://devfeed.tech/topics/algorithms.md>), [Compiler](<https://devfeed.tech/topics/compiler.md>), [Python](<https://devfeed.tech/topics/python.md>), [C++](<https://devfeed.tech/topics/c-plus-plus.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [c-plus-plus](<https://devfeed.tech/tags/c-plus-plus.md>), [compiler](<https://devfeed.tech/tags/compiler.md>), [python](<https://devfeed.tech/tags/python.md>), [remez](<https://devfeed.tech/tags/remez.md>), [shortform](<https://devfeed.tech/tags/shortform.md>)

### AI overview

The author describes replacing a brittle Remez approximation prototype with the Carathéodory-Fejér method, which a colleague prototyped and the author is porting to C++ for the HEIR compiler. The author seeks an approachable proof of the Carathéodory-Fejér theorem and notes known limitations of the method.

### Source excerpt

A colleague of mine recently lent a hand implementing a polynomial approximation routine I could port to our compiler, though it wasn't the method I was expecting. As I had written about previously, I was studying the Remez algorithm and implementing a prototype in Python. Remez approximation involves an iterated loop that alternates between root-finding and linear-system solving, and as such it can be rather brittle and difficult. Numerical errors and accuracy limits in these subsolvers contribute to weird edge cases that make the algorithm fail to converge.

## Remez and function approximations

DevFeed: [Remez and function approximations](<https://devfeed.tech/articles/remez-and-function-approximations-40500.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/shortform/2024-05-06-1018/>)

Published: 2024-05-06T17:18:29Z

Content type: article

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [FHE](<https://devfeed.tech/topics/fhe.md>), [homomorphic encryption](<https://devfeed.tech/topics/homomorphic-encryption.md>), [polynomials](<https://devfeed.tech/topics/polynomials.md>), [Algorithm](<https://devfeed.tech/topics/algorithm.md>), [Python](<https://devfeed.tech/topics/python.md>)

Tags: [algorithm](<https://devfeed.tech/tags/algorithm.md>), [approximation](<https://devfeed.tech/tags/approximation.md>), [article](<https://devfeed.tech/tags/article.md>), [fhe](<https://devfeed.tech/tags/fhe.md>), [homomorphic-encryption](<https://devfeed.tech/tags/homomorphic-encryption.md>), [how-to](<https://devfeed.tech/tags/how-to.md>), [library](<https://devfeed.tech/tags/library.md>), [optimization](<https://devfeed.tech/tags/optimization.md>), [polynomials](<https://devfeed.tech/tags/polynomials.md>), [python](<https://devfeed.tech/tags/python.md>), [remez](<https://devfeed.tech/tags/remez.md>), [shortform](<https://devfeed.tech/tags/shortform.md>)

### AI overview

The article discusses approximating functions with low-degree polynomials for arithmetic fully homomorphic encryption. It examines lolremez, the Remez algorithm, Paterson-Stockmeyer evaluation, and a multi-interval Remez method for approximating discontinuous functions such as sign.

### Source excerpt

I've been learning recently about how to approximate functions by low-degree polynomials. This is useful in fully homomorphic encryption (FHE) in the context of "arithmetic FHE" (see my FHE overview article), where the computational model makes low-degree polynomials cheap to evaluate and non-polynomial functions expensive or impossible. In browsing the state of the art I came across two interesting things. The first is the software package lolremez that implements polynomial (and rational polynomial $f(x) / g(x)$) function approximation using the so-called Remez algorithm.