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| Existence and Uniqueness of Interpolant |
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If too few parameters, does not exist.
If too many parameters, not unique
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An alternative method for computing coefficients xk of the Newton polynomial interpolant recursively.
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| Derivatives as well as values of interpolating function are specified at the data points. Specifying derivatives adds more equations. Usually piecewise cubic polynomials as interpolants. 4(n-1) parameters (n-1 cubics, each has 4 parameters). n free parameters (only use 3n-4) |
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| Convergence and Error Bound for Polynomial Interpolation of a Continuous Function |
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| The error diminishes as n increases (and hence h decreases) but only if |f(n)(t)| does not grow too rapidly with n. Polynomial interpolant of high degree -- expensive and poorly determined coefficients. Poly of degree n has n-1 inflection points. (Many wiggles). |
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| Oscillation at the edges of an interval occurs when using polynomial interpolation with polynomials of high degree over a set of equispaces interpolation points. Error is zero at the points, but between points is worse for higher degree. |
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The kth Chebyshev polynomial is definted on the interval [-1,1] by Tk(t) = cos(k arccos(t)).
T0(t) = 1
T1(t) = t
Tk+1(t)=2tTk(t)-Tk-1(t) |
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Given by truncated Taylor series
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| Hermite Cubic Interpolation |
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4(n-1) parameters:
n-1 different cubics, each with 4 parameters
Interpolating data gives 2(n-1) equations because each of the n-1 cubics must match on either end of subinterval. Derivative to be continuous gives n-2 additional because derivatives must match at each of n-2 interior points. |
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| A piecewise polynomial of degree k that is continuously differentiable k-1 times. A linear spline is piecewise linear poly, degree one, continuous, but not differentiable. |
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| Cubic Spline Interpolation |
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| Interpolating data and requiring continuity of first and second derivatives requires 4n-6, so 2 free parameters |
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| Use two free paramters to force the second derivative to be zero at the endpoints |
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| Discrete Fourier Transform |
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The sequence y = [y0, ..., yn-1]T given by
[image]
Can be written in matrix form y = Fnx where
{Fn}mk=ωnmk
Fn - complex symmetric Vandermonde |
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| The DFT of an n-point sequence can be computed by breaking it into two DFTs of half the length, provided n is even. |
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| Orthogonal Polynomial Inner Product |
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| Orthogonal Polynomials: Gram-Schmidt Orthogonalization |
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| Given a set of polynomials, Gram-Schmidt can be used to generate an orthonormal set spanning the same space. |
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| Apply Gram-Schmidt to set of monomials and scale results so that Pk(1) = 1 for all k. |
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| Weierstrass Approximation Theorem |
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| Every continuous function defined on a closed interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. |
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| Least Squares Polynomial Approximation |
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| Polynomial best fit in least-squares sense minimizes the sum of squares residuals (differences between observed value and fitted value). |
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