Chebyshev

Introduction

A simple python module for approximating any sympy expression using the Taylor series and Chebyshev polynomials.

It was created as a project for the DisCont mathematics 2 course at the Faculty of Electrical Engineering and Computing, University of Zagreb.

Installation

Requires at least python 3.6.

pip install -r requirements.txt

Submodules

The functionality is divided into two submodules:

• chebyshev.polynomial which is used for computing and storing Chebyshev polynomials, as well as some other simple polynomial manipulation.
• chebyshev.approximation which is used for approximating any sympy expression using the Taylor series and Chebyshev polynomials.

Approximated Functions

• exp(x)
• log(x + 1)
• sin(x)/x
• cos(x)

exp(x)

Coefficients for exp(x) on the [0, 1] interval:

Coefficient	Term
+0.00228989065375017828	x6
+0.00686967196125053310	x5
+0.04293544975781583839	x4
+0.16601707239688789919	x3
+0.50019798967855455540	x2
+0.99996662485953080601	x
+1.00000240440099563700	1

Maximum error on that interval is 2.724750259197606e-06

log(x + 1)

Coefficients for log(x + 1) on the [0, 1] interval:

Coefficient	Term
-1.78206380208333333330	x6
+1.68432617187500000000	x5
+1.23596191406250000000	x4
-0.76288859049479166667	x3
-0.86048889160156250000	x2
+1.16706848144531250000	x
+0.01269240000891307044	1

Maximum error on that interval is 0.026814743150641585

sin(x)/x

Coefficients for sin(x)/x on the [-1, 1] interval:

Coefficient	Term
+0.00000269375975765659	x8
-0.00019835866408658445	x6
+0.00833331406945632250	x4
-0.16666666426123592319	x2
+0.99999999995192540491	1

Maximum error on that interval is 4.807454434541114e-11

cos(x)

Coefficients for cos(x) on the [-1, 1] interval:

Coefficient	Term
+0.00002412120108317053	x8
-0.00138829603431854838	x6
+0.04166645537534185744	x4
-0.49999997362171781040	x2
+0.99999999947287565593	1

Maximum error on that interval is 5.271243441740125e-10

Related Projects

• ChebyshevPolyfit by @LMesaric.