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README.md

K-compensated de Casteljau

Cite paper Cite code
Paper DOI DOI

This is the repository for my K-compensated de Casteljau paper. If you'd just like to read the paper, feel free.

This repository is laid out in a manner described in Good Enough Practices in Scientific Computing.

The content itself has been uploaded to the arXiv and was submitted to the journal AMC in May 2018. The paper has been accepted and was published on April 5, 2019.

Abstract

In computer aided geometric design a polynomial is usually represented in Bernstein form. This paper presents a family of compensated algorithms to accurately evaluate a polynomial in Bernstein form with floating point coefficients. The principle is to apply error-free transformations to improve the traditional de Casteljau algorithm. At each stage of computation, round-off error is passed on to first order errors, then to second order errors, and so on. After the computation has been "filtered" (K - 1) times via this process, the resulting output is as accurate as the de Casteljau algorithm performed in K times the working precision. Forward error analysis and numerical experiments illustrate the accuracy of this family of algorithms.

Citation

To cite this paper:

@article{Hermes2019,
  doi = {10.1016/j.amc.2019.03.047},
  url = {https://doi.org/10.1016/j.amc.2019.03.047},
  year = {2019},
  month = {Sep},
  publisher = {Elsevier {BV}},
  volume = {357},
  pages = {57--74},
  author = {Danny Hermes},
  title = {Compensated de {C}asteljau algorithm in {$K$} times the working precision},
  journal = {Applied Mathematics and Computation}
}

To cite the code in this repository:

@misc{KCompensatedGitHub,
  doi = {10.5281/zenodo.1405259},
  url = {https://zenodo.org/record/1405259},
  author = {Danny Hermes},
  title = {dhermes/k-compensated-de-casteljau: 2018.08.28},
  publisher = {Zenodo},
  year = {2018}
}

Implementation

The K-compensated de Casteljau algorithm is implemented in C, C++ and Python in this repository. The implementations are contained in the following source files:

C

src/
├── de_casteljau.c
├── de_casteljau.h
├── eft.c
└── eft.h

C++

src/
├── de_casteljau.cpp
├── de_casteljau.hpp
├── eft.cpp
└── eft.hpp

Python

src/
├── de_casteljau.py
└── eft.py

Installation

The code used to build the manuscript, generate images and verify computations is written in Python. To run the code, Python 3.6 should be installed, along with nox:

python -m pip install --upgrade 'nox >= 2018.10.17'

Once installed, the various build jobs can be listed. For example:

$ nox --list-sessions
Available sessions:
* build_tex
* flop_counts
* verify_table
* make_images
* update_requirements
* verify_cpp
* verify_c

To run nox -s build_tex (i.e. to build the PDF), pdflatex and bibtex are required.

Plots

The plots can be (re)generated via nox -s make_images.

Operation Counts

A "special" numeric type is used to track flops and the actual operation count for each algorithm is computed and verified via nox -s flop_counts.

Table of Computation

There is a table in the manuscript that details the exact floating point values during a sample evaluation (at the point s = 1/2 + 1001u). These table values are verified via nox -s verify_table.

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Evaluate de Casteljau's Algorithm in K-times the Working Precision

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