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Chapter 3Interpolation and Polynomial ApproximationPer-Olof Persson[email protected]Department of MathematicsUniversity of California, BerkeleyMath 128A Numerical Analysis
Polynomial InterpolationPolynomialsPolynomialsPn(x) =axxn+· · ·a1x+a0are commonly usedfor interpolation or approximation of functionsBenefits include efficient methods, simple differentiation, andsimple integrationAlso, Weierstrass Approximation Theorem says that for eachε >0, there is aP(x)such that|f(x)-p(x)|< εfor allxin[a, b]forfC[a, b]. In other words, polynomials are good atapproximating general functions.
The Lagrange PolynomialTheoremIfx0, . . . , xndistinct andfgiven at these numbers, a uniquepolynomialP(x)of degreenexists withf(xk) =P(xk),for eachk= 0,1, . . . , nThe polynomial isP(x) =f(x0)Ln,0(x) +. . .+f(xn)Ln,n(x) =nXk=0f(xk)Ln,k(x)whereLn,k(x) =(x-x0)(x-x1)· · ·(x-xk-1)(x-xk+1)· · ·(x-xn)(xk-x0)(xk-x1)· · ·(xk-xk-1)(xk-xk+1)· · ·(xk-xn)=Yi6=k(x-xi)(xk-xi)
Lagrange Polynomial Error TermTheoremx0, . . . , xndistinct in[a, b],fCn+1[a, b], then forx[a, b]thereexistsξ(x)in(a, b)withf(x) =P(x) +f(n+1)(ξ(x))(n+ 1)!(x-x0)(x-x1)· · ·(x-xn)whereP(x)is the interpolating polynomial.
Recursive Generation of Lagrange PolynomialsDefinitionLetfbe defined atx0, . . . , xn, supposem1, . . . , mkdistinctintegers with0min. The Lagrange polynomial that agreeswithf(x)atxm1, xm2, . . . , xmkis denotedPm1,m2,...,mk(x).
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