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Coefficients of Chevychev polynomial for sine function?
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
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?
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