Coefficients of Chevychev polynomial for sine function?

Question

How do I find & calculation the coeffiecients if a Chevychev Polynomial for a sine wave? It seems as though the coeffiecients vary greatly depending on the precision you want the computer to calculate them (4-byte, 8-byte, etc.) Why? Is there such a polynomial for exp, ln, ATN, etc. Any easy-to-understand reference materials?

I know next to nothing about Chevychev Polynomials other then their range is from zero to one.

I would like to know how to calculate the co-efficients for the Chevychev polynomial equivalent of y=sin(x) I've read that computers calculate the value of sin(x) using a Chevychev polynomial as opposed to the series:

x - (x^3)/3! + (x^5)/5! - (x^7)/7! + etc.

I saw an interesting article about the cos function, but have yet to get the coefficients for the sin function:

http://www.avocetsystems.com/company/articles/magazine/atrig.htm

Answer 1

What kind of Chebyshev polynomial are you interested in? There are two types. The first is given by the formula

T[n,Cos[t]] = Cos[nt],

where n is a nonnegative integer. So for example,

T[2,Cos[t]] = Cos[2t] = 2 Cos[t]^2 - 1,

and it follows that

T[2,x] = 2x^2 - 1.

Similarly, we can use the angle addition formulae to find

T[3,x] = 4x^3 - 3x,
T[4,x] = 8x^4 - 8x^2 + 1,

and so forth.

The Chebyshev polynomials of the second kind are defined by the relation

U[n,Cos[t]] = Sin[nt]/Sin[t],

again for nonnegative integers n. We then have for example

U[2,Cos[t]] = Sin[2t]/Sin[t] = 2Sin[t]Cos[t]/Sin[t] = 2Cos[t]

so it follows that U[2,x] = 2x. Again, we can calculate higher degree polynomials using the angle addition formulae:

U[3,x] = 8x^3 - 4x
U[4,x] = 16x^4 - 12x^2 + 1,

and so on. Other than defining the polynomials for you, I can't figure out what your question is asking, since it is so poorly worded. Could you please clarify exactly what you want to calculate?

Answer 2

you get coeffecients by computing the functions at certain points (roots), then fitting a mutlivariate regression with function being the Y, and values of polynomials at those roots - the X's.

Also, you normally use chebyshev polynomials when you do not know how to write out the function. So no need to develop coefficients for sin, ln or any known function.

Answer 3

Chebyshev's is a probability theorem that only shows the minimum probability that a particular occurence will happen. It is not accurate, by nature, because it is used to find lower bound only.

That is about all I can say about the subject in relation to your problem. I have only done straight-up probability problems with Chebyshev's.