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MATH 3795 Lecture 14. Polynomial Interpolation.

MATH 3795 Lecture 14. Polynomial LeykekhmanFall 2008 GoalsILearn about Polynomial of the Interpolating of the Interpolating Polynomial Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 1 Polynomial datax1x2 xnf1f2 fn(think offi=f(xi)) we want to compute a polynomialpn 1ofdegree at mostn 1such thatpn 1(xi) =fi, i= 1, .. , Polynomial that satisfies these conditions is calledinterpolatingpolynomial. The pointsxiare calledinterpolation points orinterpolation will show that there exists a unique interpolation on how we represent the interpolation Polynomial it canbe computed more or less : We denote the interpolating Polynomial byP(f|x1.)

MATH 3795 Lecture 14. Polynomial Interpolation. Dmitriy Leykekhman Fall 2008 Goals I Learn about Polynomial Interpolation. I Uniqueness of the Interpolating Polynomial. I Computation of the Interpolating Polynomials. I Di erent Polynomial Basis. D. Leykekhman - MATH 3795 Introduction to Computational MathematicsLinear Least Squares { 1

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Transcription of MATH 3795 Lecture 14. Polynomial Interpolation.

1 MATH 3795 Lecture 14. Polynomial LeykekhmanFall 2008 GoalsILearn about Polynomial of the Interpolating of the Interpolating Polynomial Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 1 Polynomial datax1x2 xnf1f2 fn(think offi=f(xi)) we want to compute a polynomialpn 1ofdegree at mostn 1such thatpn 1(xi) =fi, i= 1, .. , Polynomial that satisfies these conditions is calledinterpolatingpolynomial. The pointsxiare calledinterpolation points orinterpolation will show that there exists a unique interpolation on how we represent the interpolation Polynomial it canbe computed more or less : We denote the interpolating Polynomial byP(f|x1.)

2 , xn)(x)D. Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 2 Polynomial datax1x2 xnf1f2 fn(think offi=f(xi)) we want to compute a polynomialpn 1ofdegree at mostn 1such thatpn 1(xi) =fi, i= 1, .. , Polynomial that satisfies these conditions is calledinterpolatingpolynomial. The pointsxiare calledinterpolation points orinterpolation will show that there exists a unique interpolation on how we represent the interpolation Polynomial it canbe computed more or less : We denote the interpolating Polynomial byP(f|x1.

3 , xn)(x)D. Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 2 Uniqueness of the Interpolating (Fundamental Theorem of Algebra)Every Polynomial of degree n that is not identically zero, has exactly nroots (including multiplicities). These roots may be real of (Uniqueness of the Interpolating Polynomial )Givennunequal pointsx1, x2, .. , xnand arbitrary valuesf1, f2, .. , fnthere is at most one polynomialpof degree less or equal ton 1suchthatp(xi) =fi, i= 1.

4 , there exist two polynomialsp1, p2of degree less or equal ton 1withp1(xi) =p2(xi) =fifori= 1, .. , n. Then the differencepolynomialq=p1 p2is a Polynomial of degree less or equal ton 1that satisfiesq(xi) = 0fori= 1, .. , n. Since the number of roots of anonzero Polynomial is equal to its degree, it follows thatq=p1 p2= Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 3 Uniqueness of the Interpolating (Fundamental Theorem of Algebra)Every Polynomial of degree n that is not identically zero, has exactly nroots (including multiplicities).

5 These roots may be real of (Uniqueness of the Interpolating Polynomial )Givennunequal pointsx1, x2, .. , xnand arbitrary valuesf1, f2, .. , fnthere is at most one polynomialpof degree less or equal ton 1suchthatp(xi) =fi, i= 1, .. , there exist two polynomialsp1, p2of degree less or equal ton 1withp1(xi) =p2(xi) =fifori= 1, .. , n. Then the differencepolynomialq=p1 p2is a Polynomial of degree less or equal ton 1that satisfiesq(xi) = 0fori= 1, .. , n. Since the number of roots of anonzero Polynomial is equal to its degree, it follows thatq=p1 p2= Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 3 Construction of the Interpolating a basisp1, p2.

6 , pnof the space of polynomials of degreeless or equal ton 1, we writep(x) =a1p1(x) +a2p2(x) + +anpn(x).IWe want to find coefficientsa1, a2, .. , ansuch thatp(x1) =a1p1(x1) +a2p2(x1) + +anpn(x1) =f1p(x2) =a1p1(x2) +a2p2(x2) + +anpn(x2) = (xn) =a1p1(xn) +a2p2(xn) + +anpn(xn) = leads to the linear system p1(x1)p2(x1).. pn(x1)p1(x2)p2(x2).. pn(x2)..p1(xn)p2(xn).. pn(xn) = .D. Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 4 Construction of the Interpolating a basisp1, p2.

7 , pnof the space of polynomials of degreeless or equal ton 1, we writep(x) =a1p1(x) +a2p2(x) + +anpn(x).IWe want to find coefficientsa1, a2, .. , ansuch thatp(x1) =a1p1(x1) +a2p2(x1) + +anpn(x1) =f1p(x2) =a1p1(x2) +a2p2(x2) + +anpn(x2) = (xn) =a1p1(xn) +a2p2(xn) + +anpn(xn) = leads to the linear system p1(x1)p2(x1).. pn(x1)p1(x2)p2(x2).. pn(x2)..p1(xn)p2(xn).. pn(xn) = .D. Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 4 Construction of the Interpolating a basisp1, p2.

8 , pnof the space of polynomials of degreeless or equal ton 1, we writep(x) =a1p1(x) +a2p2(x) + +anpn(x).IWe want to find coefficientsa1, a2, .. , ansuch thatp(x1) =a1p1(x1) +a2p2(x1) + +anpn(x1) =f1p(x2) =a1p1(x2) +a2p2(x2) + +anpn(x2) = (xn) =a1p1(xn) +a2p2(xn) + +anpn(xn) = leads to the linear system p1(x1)p2(x1).. pn(x1)p1(x2)p2(x2).. pn(x2)..p1(xn)p2(xn).. pn(xn) = .D. Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 4 Construction of the Interpolating the linear system p1(x1)p2(x1).

9 Pn(x1)p1(x2)p2(x2).. pn(x2)..p1(xn)p2(xn).. pn(xn) = .ifxi=xjfori6=j, then theith and thejth row of the systemsmatrix above are identical. Iffi6=fj, there is no solution. Iffi=fj, there are infinitely many assume thatxi6=xjfori6= Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 5 Construction of the Interpolating choice of the basis polynomialsp1, .. , pndetermines how easily p1(x1)p2(x1).. pn(x1)p1(x2)p2(x2).. pn(x2)..p1(xn)p2(xn).

10 Pn(xn) = .can be considerMonomial Basis:pi(x) =Mi(x) =xi 1, i= 1, .. , nLagrange Basis:pi(x) =Li(x) =n j=1j6=ix xjxi xj, i= 1, .. , nNewton Basis:pi(x) =Ni(x) =i 1 j=1(x xj), i= 1, .. , nD. Leykekhman - MATH 3795 introduction to Computational MathematicsLinear Least Squares 6 Construction of the Interpolating choice of the basis polynomialsp1, .. , pndetermines how easily p1(x1)p2(x1).. pn(x1)p1(x2)p2(x2).. pn(x2)..p1(xn)p2(xn).. pn(xn) = .can be considerMonomial Basis:pi(x) =Mi(x) =xi 1, i= 1.

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