# Chapter4 - Universityof Houston Department of Mathematics...

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Unformatted text preview: Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 4 Interpolation 4.1 Polynomial interpolation Problem: Let P n ( I ) , n ∈ lN , I :=[ a , b ] ⊂ lR , be the linear space of polynomials of degree ≤ n on I , P n ( I ) := { p n : I → lR | p n ( x ) = n summationdisplay i = a i x i , a i ∈ lR , ≤ i ≤ n } . Wehave dimP n ( I ) = n + 1 , since any p n ∈ P n ( I ) is uniquelydeterminedbythecoefficients a i , ≤ i ≤ n . Interpolation problem: x x x 5 6 2 x y p(x) 4 x x x x 1 3 Given n + 1 points ( x i , f i ) , ≤ i ≤ n , find p n ∈ P n ( I ) such that (+) p n ( x i ) = f i , ≤ i ≤ n . Definition 4.1 Interpolating polynomial The points x i (resp. f i , ≤ i ≤ n , ) are called nodal points (resp. nodal values ) of the interpolation problem. The n + 1 equations (+) are called the interpolating conditions . Apolynomial p n ∈ P n ( I ) satisfying (+) is called an interpolating polynomial . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 4.1.1 Existence and Uniqueness In explicit form, the interpolating conditions are given as follows: n summationdisplay j = a j x j i = f i , ≤ i ≤ n . They represent a linear systemin the coefficients a j , ≤ j ≤ n , : ( ⋆ ) 1 x x 2 · · · x n 1 x 1 x 2 1 · · · x n 1 · · · · · · · · · · · · 1 x n x 2 n · · · x n n bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright =: V a a 1 · · · a n = f f 1 · · · f n . Definition 4.2 Vandermonde’s determinant The coeffizient matrix V in ( ⋆ ) is called Vandermonde’s matrix . Its determinant is referred to as Vandermonde’s determinant . Universityof Houston, Department of Mathematics Numerical Analysis, Fall 2005 Lemma 4.3 Vandermonde’s determinant Vandermonde’s determinant is given by: detV = productdisplay ≤ j < k ≤ n ( x k − x j ) . Proof: We consider V n ( x ) := 1 x x 2 · · · x n 1 x 1 x 2 1 · · · x n 1 · · · · · · · · 1 x n − 1 x 2 n − 1 · · · x n n − 1 1 x x 2 · · · x n . Expansion of the determinant w.r.t. the ( n + 1 , n + 1 )-st element x n results in: detV n ( x ) = detV n − 1 ( x n − 1 ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright leading coeff....
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Chapter4 - Universityof Houston Department of Mathematics...

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