A simple polynomial construction about chebfun

[hejun Huang]

Hello Chebfun friends!

I have a very simple question but since I could not figure out after searching a lot, I have to bother you.

The core idea of the Chebyshev approximation is: $\sum^{N}_{i=0}{C_i \times T_i} = P_N \approx f(x)$, where f(x) is the target function. Now suppose the f is an non-polynomial function.

And now we want to represent the polynomial function directly to do sum-of-squares optimization via chebfun-toolbox, like f = chebfun('sqrt(sqrt((exp(x)*cos(x))^2))',[-1,1],'splitting','on'). Clearly, f is chebfun-structure data. 

(1) We can obtain the coefficients C by chebcoeffs(f),;

(2) Suppose the truncation degree is 20, we can obtain the Chebshev polynomials T by chebpoly(20,[0,4]).

Till now, I get confused for the next process. What I want is an univariate polynomials functions with clear coefficients as this form: $\sum^{N}_{i=0}{C_i \times T_i} = P_N \approx f(x)$. And I believe the clear guys designed chebfun-toolbox must consider it. But I just don't know how to figure it. Can anyone help?

I look forward to hearing from you soon! Thanks in advance!

Hejun Huang     Chinese University of Hong Kong

[Nick Hale]

You can construct a degree 20 interpolant to the given function with

>> f = chebfun('sqrt(sqrt((exp(x)*cos(x))^2))',[-1,1], 20)

f =

   chebfun column (1 smooth piece)

       interval       length     endpoint values  

[      -1,       1]       20      0.45      1.2 

vertical scale = 1.2 

Alternatively you can construct the degree 20 truncated Chebyshev series with

>> g = chebfun('sqrt(sqrt((exp(x)*cos(x))^2))',[-1,1], 'trunc', 20)

g =

   chebfun column (1 smooth piece)

       interval       length     endpoint values  

[      -1,       1]       20      0.45      1.2 

vertical scale = 1.2 

Note that these are not the same thing:

>> norm(f-g)

ans =

   2.5011e-11

For more details see >> help chebfun or the Chebfun Guide.

[hejun Huang]

Hi, Professor Nick. Thanks for your reply. I have fixed my questions before. But again I met another really weird question in approximation via Chebfun, the code has shown in figure 1.

>> For  convenience, I have attached the original code at the post end. 

And the result of the polynomial approximation is not accessible as figure 2.

>> The green and black line has perfect approximation performance, while the red line seems a disaster.

I have followed the Chebfun-guide but still not figure it out. Really appreciate for your suggestions if available!

Thanks advanced!

Jun 

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

clear;clf;clc;

%% Preliminaries

syms x;

deg = 5; % Series degree

% Non-polynomial to polynomial via chebyshev approximation 

f1 = chebfun('sqrt(sqrt((exp(x)*cos(x))^2))-1',[0,4],'splitting','on');

% Construct the compare approximation degree

[f2,e2] = minimax(f1,deg);

%% Construct the polynomials

C = transpose(flip(chebcoeffs(f2,deg+1))); % Obtain the coefficients of $T_n$

A = poly(chebpoly(deg,[0,4])); % Obtain the coefficients of polynomials

X = transpose(flip(monomials(x,0:deg))); % Create the polynomials

ft = A.*X; % Construct the $T_n$

k = sum(C.*ft); % Construct the $P_n = \sum^{6}_{i=1}C(i)*T(i)  

%% Plot

plot(f1,'g',f2,'k');

hold on; grid on;

fplot(k,'r',[0,4]);

title('$$\sqrt{\vert \exp{(x)} \times \cos{(x)} \vert}-1$$','interpreter','latex');

legend('Chebfun approximation','Chebfun approximation of degree 5','Polynomial approximation of degree 5')