# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Relating the Arnoldi approximation to the best Chebyshev approximation of the characteristic polynomial - A study using a complex valued variant of the Remez algorithm

(2022) In Bachelor's Theses in Mathematical Sciences NUMK11 20212
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
The Remez algorithm is a beautiful algorithm that finds the best approximation to a function by finding points satisfying the alternation condition. In 1987 Ping Tang published his doctoral thesis detailing a version of the Remez algorithm modified for functions of complex values with complex coefficients. He later followed the thesis with an article presenting an algorithm for finding the coefficients of a complex valued function. The Arnoldi approximation is a well-known iterative method for approximating the m eigenvalues of a matrix by a smaller set of n eigenvalues. The accuracy of the Arnoldi approximation is however dependent on an initial choice of a vector. The best choice of this initial vector leads to the so called ideal... (More)
The Remez algorithm is a beautiful algorithm that finds the best approximation to a function by finding points satisfying the alternation condition. In 1987 Ping Tang published his doctoral thesis detailing a version of the Remez algorithm modified for functions of complex values with complex coefficients. He later followed the thesis with an article presenting an algorithm for finding the coefficients of a complex valued function. The Arnoldi approximation is a well-known iterative method for approximating the m eigenvalues of a matrix by a smaller set of n eigenvalues. The accuracy of the Arnoldi approximation is however dependent on an initial choice of a vector. The best choice of this initial vector leads to the so called ideal Arnoldi approximation. In 1994 Anne Greenbum and Lloyd Trefethen published an article discussing the possibility of finding the ideal Arnoldi approximation using Chebyshev approximation. The purpose of this thesis is to experiment with Ping Tangs algorithm to approximate the characteristic equation of a matrix on a circle set and compare the results to that of the ideal Arnoldi approximation to see if the ideal Arnoldi is a best approximation in a Chebyshev sense. (Less)
Popular Abstract (Swedish)
Beräkningar med stora matriser är svåra och långsamma att utföra, även för datorer. Med hjälp av en matrisens egenvärden kan man istället arbeta med dess karaktäristiska polynom, där egenvärdena är nollställen. För mycket stora matriser får detta polynom en hög grad och vi skulle vilja approximera det med ett av lägre grad för att förenkla beräkningarna ytterligare. En populär metod för att göra detta är den så kallade Arnoldimetoden som ger oss ett set av färre egenvärden som approximerar de ursprungliga. Den metoden kan dock ge olika resultat beroende på dess startparametrar. I detta arbete kommer vi jämföra det bästa möjliga resultatet ifrån Arnoldimetoden, med optimala startparametrar, med en annan metod presenterad av Ping Tang som... (More)
Beräkningar med stora matriser är svåra och långsamma att utföra, även för datorer. Med hjälp av en matrisens egenvärden kan man istället arbeta med dess karaktäristiska polynom, där egenvärdena är nollställen. För mycket stora matriser får detta polynom en hög grad och vi skulle vilja approximera det med ett av lägre grad för att förenkla beräkningarna ytterligare. En populär metod för att göra detta är den så kallade Arnoldimetoden som ger oss ett set av färre egenvärden som approximerar de ursprungliga. Den metoden kan dock ge olika resultat beroende på dess startparametrar. I detta arbete kommer vi jämföra det bästa möjliga resultatet ifrån Arnoldimetoden, med optimala startparametrar, med en annan metod presenterad av Ping Tang som approximerar komplexvärda funktioner med polynom. Vi kommer att applicera Tangs metod på den ursprungliga matrisens karaktäristiska polynom och se hur de olika metodernas resultat skiljer sig åt. (Less)
author
supervisor
organization
alternative title
Om Arnoldiapproximationens relation till det karakteristiska polynomets bästa Chebyshevapproximation - en experimentell studie med en komplexvärd variant av Remez algoritm
course
NUMK11 20212
year
type
M2 - Bachelor Degree
subject
keywords
Chebyshev, Arnoldi, Remez, Tang, approximation, matrix, complex, characteristic polynomial
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFNA-4042-2022
ISSN
1654-6229
other publication id
2022:K16
language
English
id
9098223
2022-09-19 15:32:16
date last changed
2022-09-19 15:32:16
```@misc{9098223,
abstract     = {{The Remez algorithm is a beautiful algorithm that finds the best approximation to a function by finding points satisfying the alternation condition. In 1987 Ping Tang published his doctoral thesis detailing a version of the Remez algorithm modified for functions of complex values with complex coefficients. He later followed the thesis with an article presenting an algorithm for finding the coefficients of a complex valued function. The Arnoldi approximation is a well-known iterative method for approximating the m eigenvalues of a matrix by a smaller set of n eigenvalues. The accuracy of the Arnoldi approximation is however dependent on an initial choice of a vector. The best choice of this initial vector leads to the so called ideal Arnoldi approximation. In 1994 Anne Greenbum and Lloyd Trefethen published an article discussing the possibility of finding the ideal Arnoldi approximation using Chebyshev approximation. The purpose of this thesis is to experiment with Ping Tangs algorithm to approximate the characteristic equation of a matrix on a circle set and compare the results to that of the ideal Arnoldi approximation to see if the ideal Arnoldi is a best approximation in a Chebyshev sense.}},
author       = {{Renström, Thomas}},
issn         = {{1654-6229}},
language     = {{eng}},
note         = {{Student Paper}},
series       = {{Bachelor's Theses in Mathematical Sciences}},
title        = {{Relating the Arnoldi approximation to the best Chebyshev approximation of the characteristic polynomial - A study using a complex valued variant of the Remez algorithm}},
year         = {{2022}},
}

```