# Julia analog to R's poly function, orthogonal polynomials by QR decomposition

**URL:** <https://discourse.julialang.org/t/julia-analog-to-rs-poly-function-orthogonal-polynomials-by-qr-decomposition/128640>\
**Category:** New to Julia\
**Tags:** question, linearalgebra, r, polynomials, qr\
**Created:** [May 2, 2025, 9:20pm UTC](https://discourse.julialang.org/t/julia-analog-to-rs-poly-function-orthogonal-polynomials-by-qr-decomposition/128640 "2025-05-02T21:20:12Z")\
**Posts on this page:** 5\
**Page:** 2

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [May 7, 2025, 12:07am UTC](https://discourse.julialang.org/t/julia-analog-to-rs-poly-function-orthogonal-polynomials-by-qr-decomposition/128640/21 "2025-05-07T00:07:02Z")

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> [@Daniel\_Johansson](#):
>
> @danielwe @stevengj do any of you have some reference (introductory if possible) for this kind of topics (i.e. obtaining orthogonal polynomials provided a collection of knots), or that talk about reparametrization of a function basis using design matrix in general?

[Trefethen and Bau lecture 37](https://www.stat.uchicago.edu/~lekheng/courses/309/books/Trefethen-Bau.pdf) talks about the connection of orthogonal polynomials to Lanczos. The only difference between e.g. Legendre polynomials and your case (discrete collection of points) is the inner product.

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**Author:** ![Daniel\_Johansson](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/daniel_johansson/32/216620_2.png) [@Daniel\_Johansson](https://discourse.julialang.org/u/Daniel_Johansson)\
**Post date:** [May 11, 2025, 9:05pm UTC](https://discourse.julialang.org/t/julia-analog-to-rs-poly-function-orthogonal-polynomials-by-qr-decomposition/128640/22 "2025-05-11T21:05:58Z")

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Great, I´ll have it in mind! Thanks for the help!!

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**Author:** ![Daniel\_Johansson](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/daniel_johansson/32/216620_2.png) [@Daniel\_Johansson](https://discourse.julialang.org/u/Daniel_Johansson)\
**Post date:** [May 11, 2025, 9:07pm UTC](https://discourse.julialang.org/t/julia-analog-to-rs-poly-function-orthogonal-polynomials-by-qr-decomposition/128640/23 "2025-05-11T21:07:27Z")

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Great, I’ll check out the reference!! Thanks for the help!!

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [April 22, 2026, 3:09pm UTC](https://discourse.julialang.org/t/julia-analog-to-rs-poly-function-orthogonal-polynomials-by-qr-decomposition/128640/24 "2026-04-22T15:09:49Z")

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**Update:** I chatted with Nick Trefethen about this last night, and it turns out that he published a review in 2021 about how Arnoldi-based orthogonal polynomials can be used for accurate high-degree polynomial fitting:

- Pablo D. Brubeck, Yuji Nakatsukasa, and Lloyd N. Trefethen, [“Vandermonde with Arnoldi”](https://people.maths.ox.ac.uk/trefethen/vandermonde.pdf), _SIAM Review_ 63, [doi.org/10.1137/19M130100X](https://doi.org/10.1137/19M130100X) (2021).

It would be nice to have a Julia package implementing `polyval` and `polyfit` using this technique (although Chebyshev regression is a good alternative too), as well as simply constructing and evaluating the orthogonal polynomials over a discrete measure as in the R function.

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**Author:** ![j\_verzani](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/j_verzani/32/8551_2.png) [@j\_verzani](https://discourse.julialang.org/u/j_verzani)\
**Post date:** [April 22, 2026, 3:57pm UTC](https://discourse.julialang.org/t/julia-analog-to-rs-poly-function-orthogonal-polynomials-by-qr-decomposition/128640/25 "2026-04-22T15:57:18Z")

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Some of that is in the ArnoldiFit method of Polynomials.jl. A preprint was consulted at the time. I’ll review the published version to see if there were improvements or if I dropped something.

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