21-Polynomial-interpolation - Interpolation Polynomial interpolation Monomial basis Interpolation Polynomial interpolation Monomial basis Polynomial

# 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 f that satisfies 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 f is an interpolating function or interpolant x k are interpolation points or nodes or abscissa Desirable to have f smooth, differentiable, easy to compute Two particular sources of interpolation problems: Data from experiments (interpolation perhaps too stringent) 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
Interpolation Polynomial interpolation Monomial basis Classes of interpolating functions Polynomial interpolant f ( x ) = n k = 0 a k x k = a 0 + a 1 x + · · · + a n x n Trigonometric interpolant f ( x ) = M k = - M c k e ikx ( M = n /2 ) = c - M e - iMx + · · · + c 0 + · · · + c n e iMx Rational interpolant 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 Monomial basis Interpolation As stated, problem does not specify f uniquely Need to restrict class of functions Choose