Polynomial roots by NAThompson · Pull Request #1452 · boostorg/math

NAThompson · 2026-09-02T15:51:21Z

polynomial_roots: the Imbach-Moroz piecewise root finder (arXiv:2302.06244), whose sector DFTs use tools::fft; polynomial_roots_companion_matrix: the Aurentz-Mach-Vandebril-Watkins companion QR, O(n^2) and normwise backward stable. Both take any random access container of built-in or Boost.Multiprecision reals, with Python bindings.

$timing$

Max root error / (κ ε), κ the coefficientwise condition number:

Polynomial	`polynomial_roots`	`polynomial_roots_companion_matrix`	numpy (LAPACK)
Kac, degree 1000	1.03	483	101
x^1000 - 1	0.71	192	393
Wilkinson 20	0.075	6.7e3	0.99
Chebyshev T_30	0.24	3.4e4	467
∏(x - 10^i), i = -8..8	0.31	4.3e20	166
x^50 - 1e300	0.74	15	1.3e17

🤖 Generated with Claude Code

NAThompson · 2026-09-02T15:53:11Z

https://arxiv.org/abs/2302.06244

amlel-el-mahrouss

I have some questions and remarks, so, per void check_residuals in test/polynomial_roots_test.cpp:

This could be const auto& for for(auto root : roots).

per void fft in polynomial_roots.hpp:

This could benefit of some comments, and more testing as well due to the nested loops.

jeremy-murphy · 2026-09-08T00:13:37Z

Just wondering naming, since there are so many algorithms for finding polynomial roots; you wouldn't call it imbach_moroz_polynomial_roots?

NAThompson · 2026-09-08T02:57:55Z

@jeremy-murphy : I believe this algorithm finishes polynomial rootfinding as it is nlogn and stable.

jeremy-murphy · 2026-09-09T02:05:58Z

@jeremy-murphy : I believe this algorithm finishes polynomial rootfinding as it is nlogn and stable.

Cool, that's pretty exciting! Do you have any results on how accurate it is compared to the existing algorithms we have or against some optimal answer?

NAThompson · 2026-09-09T03:40:50Z

Cool, that's pretty exciting! Do you have any results on how accurate it is compared to the existing algorithms we have or against some optimal answer?

Not yet-I think Aurentz's 2015 work (which is O(n^2)) is probably the reference for stability. I do not believe this algorithm is stable in quite the same sense.

JacobHass8 · 2026-09-10T03:59:56Z

+   assert(std::count(zero_roots.begin(), zero_roots.end(), std::complex<double>(0, 0)) == 2);
+
+   std::mt19937_64 generator(0x5eed);
+   std::uniform_real_distribution<double> distribution(-2, 2);


Have you tested how accurate this method is with multiprecision types like boost::multiprecision::cpp_bin_float?

JacobHass8 · 2026-09-10T04:02:32Z

+   using std::abs;
+   using std::log2;


Would adding BOOST_MATH_STD_USING here and below allow multiprecision types?

@JacobHass8 : BOOST_MATH_STD_USING just brings in all math functions. using std::abs allows ADL of abs.

JacobHass8 · 2026-09-10T04:07:25Z

+   try { polynomial_roots(std::vector<double>{0, 0}); }
+   catch(std::domain_error const&) { threw = true; }
+   assert(threw);


Should all tests here use boost test functions like BOOST_CHECK_THROW? Additionally, is there any benefit in adding policies to polynomial_roots?

Not sure why BOOST_CHECK_THROW would matter here, what's your rationale for it?

@JacobHass8 : I eliminated Boost.Test from the library because it takes forever to compile and its error messages are not actionable. We'll use math_unit_test.hpp throughout.

Looks like it needs a CHECK_THROW macro though.

math_unit_test.hpp does have CHECK_THROW now; the test uses it for the zero polynomial and non-finite coefficients.

NAThompson · 2026-10-12T15:30:02Z

+
+``
+#include <boost/math/tools/polynomial_roots.hpp>
+#include <boost/math/tools/amvw_roots.hpp>


@claude : I don't think amvw_roots will be scrutable to anyone. We can also do multiple implementations in a single file and dispatch to a detail, rather than have multiple files.

Done: both are in polynomial_roots.hpp now (amvw_roots.hpp is gone). The public functions are polynomial_roots(c) and polynomial_roots_companion_matrix(c), each a thin wrapper over polynomial_roots_detail::piecewise / polynomial_roots_detail::companion_qr.

NAThompson · 2026-10-12T15:30:45Z

+``
+
+   template <class Real>
+   std::vector<std::complex<Real> >


These should be templated on a RandomAccessContainer, and the return type deduced.

It should work with std::array, std::span, and std::vector.

Done: both take RandomAccessContainer const& (anything with value_type, begin and end) and return auto, a std::vector of std::complex<Real> for built-in types or the matching Boost.Multiprecision complex type. The test covers std::vector, std::array, and std::span under C++20.

NAThompson · 2026-10-12T15:31:13Z

+   template <class Real>
+   std::vector<std::complex<Real> >
+   polynomial_roots(std::vector<Real> coefficients,
+                    unsigned precision = std::numeric_limits<Real>::digits);


Precision is not a useful parameter here-how would a user set it?

Agreed, removed. The local expansion order is now fixed by std::numeric_limits<Real>::digits, which is what the default was.

polynomial_roots: the Imbach-Moroz piecewise root finder (arXiv:2302.06244), whose sector DFTs use tools::fft; polynomial_roots_companion_matrix: the Aurentz-Mach-Vandebril-Watkins companion QR, O(n^2) and normwise backward stable. Both take any random access container of built-in or Boost.Multiprecision reals, with Python bindings. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

NAThompson mentioned this pull request Sep 3, 2026

Gauss-Hermite Quadrature #1453

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amlel-el-mahrouss reviewed Sep 5, 2026

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JacobHass8 reviewed Sep 10, 2026

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NAThompson mentioned this pull request Oct 11, 2026

Add a fast Fourier transform #1558

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NAThompson changed the base branch from develop to fft October 11, 2026 18:20

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NAThompson commented Oct 12, 2026

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NAThompson merged commit 60b4ed1 into develop Oct 12, 2026
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