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wdlang
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i find that chebyshev polynomials are quite useful in numerical computations
is there any good references?
is there any good references?
wdlang said:i find that chebyshev polynomials are quite useful in numerical computations
is there any good references?
jasonRF said:I agree - they are very useful. I have used the discrete orthogonality of them to build nice routines for special functions or integrals I run across (often factor out leading order asymptotic and/or oscillating portions). I usually use something like Mathematica or Maxima to calculate the coefficients to high precision, which I then use in a c or MATLAB routine.
Options I am familiar with include:
Chebyshev and Fourier Spectral Methods, by Boyd (may be free online version). This is pretty high level (for grad course I think) but has tons of stuff in it.
Numerical Methods for Scientists and Engineers, by Hamming. I like this book, and it has a couple of nice chapters on this. Accessible to anyone who knows calculus.
Numerical Recipes, by Press et al., a nice general book that has good, practical sections on chebyshev polynomials. I am familiar with the 2nd edition, which is nice.
Prof. Trefethen has done some nice stuff recently, including leading the development of a nice package that can be used in recent versions of Matlab:
http://www2.maths.ox.ac.uk/chebfun/publications/
good luck!
jason
Chebyshev polynomials are a type of mathematical polynomial used in approximation theory. They are named after Russian mathematician Pafnuty Chebyshev and have various applications in fields such as physics, engineering, and computer graphics.
Chebyshev polynomials are typically represented in terms of their coefficients, which can be found through recursive formulas or through trigonometric functions. They can also be represented graphically as curves on a coordinate plane.
Chebyshev polynomials have a number of important properties, including being orthogonal on a given interval, having a recurrence relation, and being the minimax approximation for certain functions. They also have a specific set of roots on a given interval.
Chebyshev polynomials are often used in approximation theory to find the best approximation of a given function on a given interval. This is done by finding the coefficients of the polynomial that minimize the maximum error between the polynomial and the function.
Some popular books on chebyshev polynomials include "Chebyshev and Fourier Spectral Methods" by John Boyd, "Approximation Theory and Approximation Practice" by Lloyd Trefethen, and "Chebyshev Polynomials" by Riazuddin and Fayyazuddin. It is also helpful to consult mathematical textbooks on approximation theory or numerical analysis.