Article

Free access

On the method of minimum (or "best") approximation and the method of least nth powers

Authors:

Allen A. Goldstein,

James B. Herreshoff,

Norman LevineAuthors Info & Claims

ACM '56: Proceedings of the 1956 11th ACM national meeting

Pages 17 - 20

https://doi.org/10.1145/800258.808938

Published: 01 January 1956 Publication History

PDF eReader

Abstract

There has been a renewed interest of late in the problem of approximation in the Tchebycheff sense, that is of minimizing the maximum residues. Several methods 1, 2, 3, have already been discussed and many examples of approximating functions of one variable by polynomials and rational functions, etc. have been published. On the other hand the important problem of solving an overdetermined system of linear equations in the Tchebycheff sense has not been given the attention it deserves. The possession of a practical method for the solution of this problem would make possible “best” approximations of functions of several variables by linear combinations of arbitrary functions.

It is not widely known that an algorithm for solving this problem exists and is due to de la Vallee Poussin.⁴ Unfortunately the algorithm is unwieldy but it is still useful in the computation of a minimum approximation. We shall therefore outline the results of de la Vallee Poussin before discussing some new methods with which we have been experimenting.

References

[1]

Hastings Cecil, Jr. "Approximations for Digital Computers" Princeton University Press 1955.

Digital Library

Google Scholar

[2]

Selfridge, R. G. "Approximations With Least Maximum Error", Pacific Journal of Mathematics, Vol. 3, No. 1, March 1953, pp 247-255.

Crossref

Google Scholar

[3]

Carlson, Bengt and Goldstein, Max. "Rational Approximations of Functions" U.S. Atomic Energy Commission, Los Alamos Scientific Lab. August 1955

Google Scholar

[4]

Poussin, Ch. J. de la Vallee "Sur la Methode de I'approximation Minimum", Societe Scientifiaue de Bruxelles, Annales, Seconde Partie, Memires, Vol. 35 1911, pp 1-16. The above paper has been translated with a commentary by H. E. Salzer. Copies may be obtained by writing him.

Google Scholar

Index Terms

On the method of minimum (or "best") approximation and the method of least nth powers
1. Mathematics of computing
  1. Mathematical analysis
    1. Functional analysis
      1. Approximation
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Recommendations

The Carathéodory–Fejér Method for Real Rational Approximation

A “Carathéodory–Fejér method” is presented for near-best real rational approximation on intervals, based on the eigenvalue (or singular value) analysis of a Hankel matrix of Chebyshev coefficients. In approximation of a smooth function F, the CF ...
Best uniform polynomial approximation of some rational functions

In this research paper using the Chebyshev expansion, we explicitly determine the best uniform polynomial approximation out of P"q"n (the space of polynomials of degree at most qn) to a class of rational functions of the form 1/(T"q(a)+/-T"q(x)) on [-1,...
Completing nth powers of polynomials
SYMSAC '76: Proceedings of the third ACM symposium on Symbolic and algebraic computation

A frequent exercise in high school algebra courses is completing the square of some given polynomial. The goal is to find terms involving only constants independent of the main variable, which when added to the given polynomial will result in a perfect ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

ACM '56: Proceedings of the 1956 11th ACM national meeting

January 1956

178 pages

ISBN:9781450379229

DOI:10.1145/800258

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 1956

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

0
Total Citations
77
Total Downloads

Downloads (Last 12 months)52
Downloads (Last 6 weeks)11

Reflects downloads up to 04 Jan 2025

Other Metrics

View Author Metrics

Citations

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Abstract

References

Index Terms

Recommendations

The Carathéodory–Fejér Method for Real Rational Approximation

Best uniform polynomial approximation of some rational functions

Completing nth powers of polynomials

Comments

Information

Published In

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

View options

PDF

eReader

Login options

Full Access

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations