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set5 - $ Set 5 Piecewise Polynomial Interpolation Kyle A...

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a39 a38 a36 a37 Set 5: Piecewise Polynomial Interpolation Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 2 Spring 2011 1
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a39 a38 a36 a37 Summary Approximation of f ( x ) ∈ C (0) with polynomials. Various metrics possible: i | f ( x i ) p ( x i ) | i | f ( x i ) p ( x i ) | + · · · + | f ( k ) ( x i ) p ( k ) ( x i ) | bardbl f p bardbl bardbl f p bardbl L 2 2
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a39 a38 a36 a37 Summary Interpolation f ( x i ) = p ( x i ) f ( x i ) = p ( x i ) , ..., f ( k ) ( x i ) = p ( k ) ( x i ) more general combinations of function values and derivatives Various interpolation forms of unique polynomials Lagrange – standard or barycentric Newton Hermite-Birkoff • bardbl f p bardbl 0 : convergent sequence of polynomial family representations Bernstein polynomials for f ∈ C (0) interpolatory strategies for more constrained class of f 3
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a39 a38 a36 a37 Polynomial Interpolation Problems: Pointwise error too large at important points • bardbl f p bardbl too large on interval of interest erratic variation, i.e., not smooth enough excessive computational complexity ill-conditioning and instability 4
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a39 a38 a36 a37 Polynomial Interpolation Solutions – Complications: choose better points – may not be possible increase n – may or may not improve error, may not converge interpolate derivatives – values may not be available 5
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a39 a38 a36 a37 Piecewise Lagrange Interpolation Use local interpolants of lower order rather than one global polynomial. a = x 0 <x 1 < · · · <x n = b [ a,b ] = s I s : union of disjoint subintervals (intersect only at subset of grid points) g k ( x ) , on I s = [ x i s ,x i s + k ] is in P k g k ( x ) is a piecewise polynomial local interpolant p k,i s ( x j ) = f ( x j ) , i s j i s + k global interpolant g k ( x i ) = f ( x i ) , 0 i n 6
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a39 a38 a36 a37 Choices Form of p k,i ( x ) In practice, each interval is independent in construction and evaluation. For analysis the form matters, e.g., basis choice When used to define a set of relationships between unknown f ( x i ) ,...,f ( k ) ( x i ) the form determines the structure of equations to be solved. 7
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a39 a38 a36 a37 Forms and Bases monomial p k,i s ( x ) = α ( i s ) 0 + α ( i s ) 1 x + · · · + α ( i s ) k - 1 x k - 1 + α ( i s ) k x k Newton p k,i s ( x ) = f i s + f [ x i s ,x i s +1 ]( x - x i s ) + · · · + f [ x i s ,...,x i s + k ] ω ( i s ) k Lagrange p k,i s ( x ) = k X j =0 ( i s ) j ( x ) f i s + j basis form for analysis and implicit equations g k ( x ) = n X i =0 f i φ i ( x ) = n X i =0 γ i ψ i ( x ) where φ i ( x ) and ψ i ( x ) are piecewise polynomials. 8
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a39 a38 a36 a37 Error If f ∈ C ( k +1) [ a,b ] a x b, f ( x ) g k ( x ) = f ( x ) p k,i s ( x ) = f ( k +1) ( ξ ) ( k + 1)! ω ( i s ) k +1 ( x ) x [ x i s ,x i s + k ] The local error expressions can be combined to get a global error bardbl f g k bardbl Ch k +1 bardbl f ( k +1) bardbl where h is maximum size of intervals I i 9
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a39 a38 a36 a37 Error This is easily shown: | f ( k +1) ( ξ ) ( k + 1)!
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