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Kronrod

Kronrod

Gauss-Kronrod Quadrature Rules for Python

Tests Documentation Ruff

codecov License: LGPL-2.1

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kronrod is a Python library for computing the abscissas and weights of Gauss-Kronrod quadrature rules. Given an N-point Gauss quadrature rule, the library optimally adds N+1 points to produce a 2N+1 point Gauss-Kronrod rule for numerical integration.

The advantage of using a Gauss and Gauss-Kronrod pair is that the second rule, which uses 2*N+1 points, actually includes the N points in the previous Gauss rule. This means that the function values from that computation can be reused. This efficiency comes at the cost of a mild reduction in the degree of polynomial precision of the Gauss-Kronrod rule.

Quick example

import kronrod

# Compute a 3-point Gauss / 7-point Gauss-Kronrod rule
x, w1, w2 = kronrod.kronrod(3)

print("Abscissas:", x)
print("Gauss-Kronrod weights:", w1)
print("Gauss weights:", w2)

# Adjust the rule from [-1, +1] to [0, 1]
x_adj, w1_adj, w2_adj = kronrod.kronrod_adjust(0.0, 1.0, 3, x, w1, w2)

Installation

pip install kronrod

Requires Python 3.10+ and NumPy. See the full installation guide for uv, poetry, and source builds.

Documentation

  • Theory — Gauss-Kronrod quadrature, Kronrod extension, error estimation
  • Quickstart — runnable examples
  • API Reference — function signatures and parameters
  • References — literature citations

Routines

Function Description
abwe1 Calculate a Kronrod abscissa and weight
abwe2 Calculate a Gaussian abscissa and two weights
kronrod Add N+1 points to an N-point Gaussian rule
r8_abs Return the absolute value of a double
timestamp Print the current YMDHMS date as a time stamp

References

  • Robert Piessens, Maria Branders, "A Note on the Optimal Addition of Abscissas to Quadrature Formulas of Gauss and Lobatto," Mathematics of Computation, Volume 28, Number 125, January 1974, pages 135-139.

Attribution

The original FORTRAN77 code was written by Robert Piessens and Maria Branders. The C translation was made by John Burkardt and is distributed under the LGPL-2.1 license.

License

LGPL-2.1 — see LICENSE.

Release files for kronrod 0.1.2

For a detailed explanation of source distributions (sdists) and built distributions (wheels), please see the package formats documentation.

Built distributions (wheels)

Table of built distributions (wheels) for kronrod 0.1.2
File
kronrod-0.1.2-cp315-cp315t-win_amd64.whl CPython 3.15 CPython 3.15 free-threading Windows x86-64 Details
kronrod-0.1.2-cp315-cp315t-musllinux_1_2_x86_64.whl CPython 3.15 CPython 3.15 free-threading Linux musl 1.2+ x86-64 Details
kronrod-0.1.2-cp315-cp315t-manylinux2014_x86_64.manylinux_2_17_x86_64.manylinux_2_28_x86_64.whl CPython 3.15 CPython 3.15 free-threading Linux glibc 2.17+ x86-64, Linux glibc 2.28+ x86-64 Details
kronrod-0.1.2-cp315-cp315t-macosx_15_0_arm64.whl CPython 3.15 CPython 3.15 free-threading macOS 15.0+ ARM64 Details
kronrod-0.1.2-cp315-cp315-win_amd64.whl CPython 3.15 CPython 3.15 Windows x86-64 Details
kronrod-0.1.2-cp315-cp315-musllinux_1_2_x86_64.whl CPython 3.15 CPython 3.15 Linux musl 1.2+ x86-64 Details
kronrod-0.1.2-cp315-cp315-manylinux2014_x86_64.manylinux_2_17_x86_64.manylinux_2_28_x86_64.whl CPython 3.15 CPython 3.15 Linux glibc 2.28+ x86-64, Linux glibc 2.17+ x86-64 Details
kronrod-0.1.2-cp315-cp315-macosx_15_0_arm64.whl CPython 3.15 CPython 3.15 macOS 15.0+ ARM64 Details
kronrod-0.1.2-cp314-cp314t-win_amd64.whl CPython 3.14 CPython 3.14 free-threading Windows x86-64 Details
kronrod-0.1.2-cp314-cp314t-musllinux_1_2_x86_64.whl CPython 3.14 CPython 3.14 free-threading Linux musl 1.2+ x86-64 Details
kronrod-0.1.2-cp314-cp314t-manylinux2014_x86_64.manylinux_2_17_x86_64.manylinux_2_28_x86_64.whl CPython 3.14 CPython 3.14 free-threading Linux glibc 2.17+ x86-64, Linux glibc 2.28+ x86-64 Details
kronrod-0.1.2-cp314-cp314t-macosx_15_0_arm64.whl CPython 3.14 CPython 3.14 free-threading macOS 15.0+ ARM64 Details
kronrod-0.1.2-cp314-cp314-win_amd64.whl CPython 3.14 CPython 3.14 Windows x86-64 Details
kronrod-0.1.2-cp314-cp314-musllinux_1_2_x86_64.whl CPython 3.14 CPython 3.14 Linux musl 1.2+ x86-64 Details
kronrod-0.1.2-cp314-cp314-manylinux2014_x86_64.manylinux_2_17_x86_64.manylinux_2_28_x86_64.whl CPython 3.14 CPython 3.14 Linux glibc 2.17+ x86-64, Linux glibc 2.28+ x86-64 Details
kronrod-0.1.2-cp314-cp314-macosx_15_0_arm64.whl CPython 3.14 CPython 3.14 macOS 15.0+ ARM64 Details
kronrod-0.1.2-cp313-cp313-win_amd64.whl CPython 3.13 CPython 3.13 Windows x86-64 Details
kronrod-0.1.2-cp313-cp313-musllinux_1_2_x86_64.whl CPython 3.13 CPython 3.13 Linux musl 1.2+ x86-64 Details
kronrod-0.1.2-cp313-cp313-manylinux2014_x86_64.manylinux_2_17_x86_64.manylinux_2_28_x86_64.whl CPython 3.13 CPython 3.13 Linux glibc 2.28+ x86-64, Linux glibc 2.17+ x86-64 Details
kronrod-0.1.2-cp313-cp313-macosx_15_0_arm64.whl CPython 3.13 CPython 3.13 macOS 15.0+ ARM64 Details
kronrod-0.1.2-cp312-cp312-win_amd64.whl CPython 3.12 CPython 3.12 Windows x86-64 Details
kronrod-0.1.2-cp312-cp312-musllinux_1_2_x86_64.whl CPython 3.12 CPython 3.12 Linux musl 1.2+ x86-64 Details
kronrod-0.1.2-cp312-cp312-manylinux2014_x86_64.manylinux_2_17_x86_64.manylinux_2_28_x86_64.whl CPython 3.12 CPython 3.12 Linux glibc 2.28+ x86-64, Linux glibc 2.17+ x86-64 Details
kronrod-0.1.2-cp312-cp312-macosx_15_0_arm64.whl CPython 3.12 CPython 3.12 macOS 15.0+ ARM64 Details
kronrod-0.1.2-cp311-cp311-win_amd64.whl CPython 3.11 CPython 3.11 Windows x86-64 Details
kronrod-0.1.2-cp311-cp311-musllinux_1_2_x86_64.whl CPython 3.11 CPython 3.11 Linux musl 1.2+ x86-64 Details
kronrod-0.1.2-cp311-cp311-manylinux2014_x86_64.manylinux_2_17_x86_64.manylinux_2_28_x86_64.whl CPython 3.11 CPython 3.11 Linux glibc 2.17+ x86-64, Linux glibc 2.28+ x86-64 Details
kronrod-0.1.2-cp311-cp311-macosx_15_0_arm64.whl CPython 3.11 CPython 3.11 macOS 15.0+ ARM64 Details
kronrod-0.1.2-cp310-cp310-win_amd64.whl CPython 3.10 CPython 3.10 Windows x86-64 Details
kronrod-0.1.2-cp310-cp310-musllinux_1_2_x86_64.whl CPython 3.10 CPython 3.10 Linux musl 1.2+ x86-64 Details
kronrod-0.1.2-cp310-cp310-manylinux2014_x86_64.manylinux_2_17_x86_64.manylinux_2_28_x86_64.whl CPython 3.10 CPython 3.10 Linux glibc 2.28+ x86-64, Linux glibc 2.17+ x86-64 Details
kronrod-0.1.2-cp310-cp310-macosx_15_0_arm64.whl CPython 3.10 CPython 3.10 macOS 15.0+ ARM64 Details

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