Briefly, the function f ( x ) to be integrated is evaluated at the N extrema or roots of a Chebyshev polynomial and these values are used to construct a polynomial approximation for the function.
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For " N " > " J ", orthogonal projection yields the standard overdetermined system of equations ( often called normal equations ) used to compute the coefficients in the polynomial approximation.
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His dissertation was entitled " Projections in Uniform Polynomial Approximation . " He held positions at the Institute of Mathematics and the Institute of Computer Sciences of the University of Wroclaw, and the Institute of Applied Mathematics Bonn in Germany.
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However, a quasi-polynomial approximation scheme is possible : for any constant & epsilon; > 0, a solution with approximation ratio 1 + & epsilon; can be found in quasi-polynomial time exp ( O ( ( log " n " ) 9 ).
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The Whewell equation and other types of so-called intrinsic equations have applications in the solution of certain physical problems, but for computer graphics it's apparently more practical to use compound polynomial approximations as in B�zier splines .-- talk ) 19 : 55, 31 August 2015 ( UTC)
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In mathematics, the "'Lebesgue constants "'( depending on a set of nodes and of its size ) give an idea of how good the function ( at the given nodes ) is in comparison with the best polynomial approximation of the function ( the degree of the polynomials are obviously fixed ).
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Although no computational method ( based on ABC or not ) seems to be able to break the curse-of-dimensionality, methods have recently been developed to handle high-dimensional parameter spaces under certain assumptions ( e . g ., based on polynomial approximation on sparse grids, which could potentially heavily reduce the simulation times for ABC ).
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It is important to note that, while in principle a QSS method of arbitrary order can be used to model a continuous-time system, it is seldom desirable to use methods of order higher than four, as the Abel Ruffini theorem implies that the time of the next quantization, t, cannot ( in general ) be algebraically when the polynomial approximation is of degree greater than four, and hence must be approximated iteratively using a root-finding algorithm.
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