36 Hermite Interpolation The Hermite interpolation problem is given values of f

36 hermite interpolation the hermite interpolation

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3.6Hermite InterpolationThe Hermite interpolation problem is:given values offand some of itsderivatives at the nodesx0, x1, ..., xn, find the interpolating polynomial ofsmallest degree interpolating those values.This polynomial is called theHermite Interpolation Polynomialand can be obtained with a minor modifi-cation to the Newton’s form representation.For example: Suppose we look for a polynomialpof lowest degree whichsatisfies the interpolation conditions:p(x0) =f(x0),p0(x0) =f0(x0),p(x1) =f(x1),p0(x1) =f0(x1).We can view this problem as a limiting case of polynomial interpolation offat two pairs of coincident nodes,x0, x0, x1, x1and we can use Newton’sInterpolation form to obtainp. The table of divided dierences, in view of(3.57), isx0f(x0)x0f(x0)f0(x0)x1f(x1)f[x0, x1]f[x0, x0, x1]x1f(x1)f0(x1)f[x0, x1, x1]f[x0, x0, x1, x1](3.58)
52CHAPTER 3.INTERPOLATIONandp(x) =f(x0) +f0(x0)(x-x0) +f[x0, x0, x1](x-x0)2+f[x0, x0, x1, x1](x-x0)2(x-x1).(3.59)Example 8.Letf(0) = 1,f0(0) = 0andf(1) =p2.Find the HermiteInterpolation Polynomial.We construct the table of divided dierences as follows:010101p2p2-1p2-1(3.60)and thereforep(x) = 1 + 0(x-0) + (p2-1)(x-0)2= 1 + (p2-1)x2.(3.61)3.7Convergence and Accuracy of PolynomialInterpolationFrom the Cauchy Remainder formulaf(x)-pn(x) =1(n+ 1)!f(n+1)((x))(x-x0)(x-x1)· · ·(x-xn)(3.62)and the examples in the first homework, it is clear that the accuracy andconvergence of the interpolation polynomialpnoffdepends on both thesmoothness offand thedistribution of nodesx0, x1, . . . , xn.In the Runge examplef(x) =11 + 25x2x2[-1,1],is very sooth. It has an infinite number of continuous derivatives, i.e.f2C1[-1,1] (in factfis real analytic in the whole real line, i.e.it has aconvergent Taylor series tof(x) for everyx2R). Nevertheless, we observedthat for the equispaced nodes (3.15)pn(x) did not converge uniformly tof(x)
3.7. CONVERGENCE AND ACCURACY OF POLYNOMIAL INTERPOLATION53asn! 1. In fact it diverged quite dramatically toward the end points ofthe interval. On the other hand, we noticed fast and uniform convergence ofpntofwhen the Chebyshed nodes (3.16) were employed.It is then natural to ask: Given anyf2C[a, b], can we guarantee that ifwe choose the Chebyshev nodeskf-pnk1!0? The answer is no. Bernsteinand Faber proved in 1914 that given any distribution of points, organized ina triangular array asx(0)0x(1)0,x(1)1x(2)0,x(2)1,x(2)2...(3.63)it is possible to construct a continuous functionffor which its interpolatingpolynomialpn(corresponding to the nodes on then-th row of (3.63)) will notconverge uniformly tofasn! 1. However, iffis slightly smoother, forexamplef2C1[a, b], then for the Chebyshev array of nodeskf-pnk1!0.In one of the homework examplesf(x) =e-x2and we noticed convergenceofpneven with the equidistributed nodes.What is so special about thisfunction? The functionf(z) =e-z2,(3.64)z=x+iyis analytic in the entire complex plane. Using complex variablesanalysis it can be shown that iffis analytic in a sufficiently large region inthe complex plane containing [a, b] thenkf-p

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