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# 16 - Interpolation Polynomial interpolation Limitations of...

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Interpolation Polynomial interpolation Limitations of interpolation Piecewise interpolation Polynomial Interpolation Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 1 / 32

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Interpolation Polynomial interpolation Limitations of interpolation Piecewise interpolation Polynomial Interpolation 1 Interpolation of data 2 Polynomial interpolation Polynomial interpolation in a monomial basis 3 Limitations of polynomial interpolation 4 Piecewise polynomial interpolation c D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 2 / 32
Interpolation Polynomial interpolation Limitations of interpolation Piecewise interpolation Interpolation Interpolation problem Given n data points { ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) } with x k distinct ( k = 1 : n ), determine a function e f that satisfies e f ( x k ) = y k ( k = 1: n ) . -5 -4 -3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 c D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 4 / 32

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Interpolation Polynomial interpolation Limitations of interpolation Piecewise interpolation Remarks e f is an interpolating function or interpolant x k are interpolation points or nodes or abscissa Desirable to have e f smooth, differentiable, easy to compute Two particular sources of interpolation problems: I Data from experiments (interpolation perhaps too stringent) I Tabulating values of prescribed function x 0 π /6 π /4 π /3 π /2 sin ( x ) 0 1/2 1/ 2 3/2 1 cos ( x ) 1 3/2 1/ 2 1/2 0 c D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 5 / 32
Interpolation Polynomial interpolation Limitations of interpolation Piecewise interpolation Interpolation As stated, problem does not specify e f uniquely Need to restrict class of functions Choose e f to lie from vector space of dimension n e f ( x ) = n k = 1 a k φ k ( x ) = a 1 φ 1 ( x ) + a 2 φ 2 ( x ) + · · · + a n φ n ( x ) φ k are basis functions , a k are coefficients For e f = n k = 1 a k φ k , a k depend linearly on data y ( = 1: n ) c D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 6 / 32

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Interpolation Polynomial interpolation Limitations of interpolation Piecewise interpolation Polynomial interpolation Polynomial interpolation problem Given n data points { ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) } with x k distinct ( k = 1 : n ), determine a polynomial function e f of degree at most n that satisfies e f ( x k ) = y k ( k = 1: n ) . n - 1 is (maximum) degree of interpolant Basis functions are φ k ( x ) = x n - k ( k = 1: n ) n is number of data points c D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 8 / 32
Interpolation Polynomial interpolation Limitations of interpolation Piecewise interpolation Polynomial interpolation Polynomials easy to evaluate, differentiate, etc. e f lies in vector space of polynomials of degree at most n n coefficients to determine as deg ( e f ) n - 1 Reduces to linear algebra: solution of linear system of equations Theorem (Existence/Uniqueness of polynomial interpolation) There exists a unique polynomial e f of degree at most n - 1 that satisfies the polynomial interpolation problem.

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16 - Interpolation Polynomial interpolation Limitations of...

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