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mid2-review

# mid2-review - Midterm 2 Review Polynomial Interpolation...

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Midterm 2 Review

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Polynomial Interpolation • Polynomial interpolation can be used to derive a function p that passes through a given set of points • Typically, the goal is to achieve an approximation to an underlying function f (that was sampled to obtain the points being interpolated). In that case, it is desired not only that f ( x ) = p ( x ) at the nodes, but also that | f ( x )– p ( x )| is small between the nodes • This goal can be difFcult to achieve when p is a polynomial of high degree
Polynomial Interpolation • Given a table of points: the Lagrange form of the interpolating polynomial is deFned by: where (note that the cardinal functions depend only on the x i ) x 0 y 0 x n y n .. .. p ( x ) = l i i = 0 n " ( x ) # y i l i ( x ) = x " " # \$ % ( j = 0 j ) i n * , i = 0 K n x j x j x i

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Polynomial Interpolation • Given a table of points: one can compute a table of divided differences: x 0 f ( x 0 ) x n .. .. f x i [ ] = f ( x i ) f x i , x i + 1 [ ] = f x i + 1 [ ] " f x i [ ] x i + 1 " x i f ( x n ) f x i , x i + 1 , x i + 2 [ ] = f x i + 1 , x i + 2 [ ] " f x i , x i + 1 [ ] x i + 2 " x i f x i , K , x j [ ] = f x i + 1 , K , x j [ ] " f x i , K , x j " 1 [ ] x j " x i
Polynomial Interpolation • the Newton form of the interpolating polynomial is then deFned by: (note that the polynomial is built up incrementally , increasing in degree and adding one new term as it interpolates each additional point in the table) p ( x ) = f x 0 K x i [ ] ( x " x j ) j = 0 i # \$ % ( ) * i = 0 n +

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Polynomial Interpolation • One can attempt to determine the root of a sampled function via inverse interpolation , forming the interpolating polynomial x = p ( y ) and then calculating x = p (0), but the process can be prone to error.
Polynomial Interpolation Neville’s algorithm constructs the interpolating polynomial for a table of points as a linear combination of lower order polynomials that each interpolate subsets of the data: – begin with P i 0 ( x ) = f ( x i ) – deFne P i j ( x ) = x " x i " j x i " x i " j # \$ % % ( ( P i j " 1 ( x ) + x i " x x i " x i " j # \$ % % ( ( P i " 1 j " 1 ( x )

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Errors in Polynomial Interpolation • The Runge function: f ( x ) = 1 / (1 + x 2 ) is particularly poorly approximated by an interpolating polynomial when the nodes x i are evenly spaced
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mid2-review - Midterm 2 Review Polynomial Interpolation...

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