21-Polynomial-interpolation

21-Polynomial-interpolation - Interpolation Polynomial...

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Interpolation Polynomial interpolation Monomial basis Polynomial Interpolation Dhavide Aruliah UOIT MATH 2070U c ± D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 1 / 19 Interpolation Polynomial interpolation Monomial basis Polynomial Interpolation 1 Interpolation of data 2 Polynomial interpolation 3 Polynomial interpolation in a monomial basis c ± D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 2 / 19 Interpolation Polynomial interpolation Monomial basis Interpolation Interpolation problem Given n + 1 data points { ( x 0 , y 0 ) , ( x 1 , y 1 ) , . . . , ( x n , y n ) } with x k distinct ( k = 0 : n ), determine a function e f that satisfies e f ( x k ) = y k ( k = 0: 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 / 19 Interpolation Polynomial interpolation Monomial basis 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 / 19
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Interpolation Polynomial interpolation Monomial basis Classes of interpolating functions Polynomial interpolant e f ( x ) = n k = 0 a k x k = a 0 + a 1 x + ··· + a n x n Trigonometric interpolant e f ( x ) = M k = - M c k e ikx ( M = b n /2 c ) = c - M e - iMx + ··· + c 0 + ··· + c n e iMx Rational interpolant e f ( x ) = k j = 0 a j x j n - k - 1 ` = 0 a k + ` + 1 x ` ( k < n ) = a 0 + a 1 x 1 + ··· + a k x k a k + 1 + a k + 2 x 1 + ··· + a n x n - k - 1 c ± D. Aruliah (UOIT) Polynomial Interpolation MATH 2070U 6 / 19 Interpolation Polynomial interpolation
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This note was uploaded on 02/23/2010 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Spring '10 term at UOIT.

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21-Polynomial-interpolation - Interpolation Polynomial...

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