Numerical Analysis Spring 2001 Professor Diamond

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1 Numerical Analysis Spring 2001 Professor Diamond Polynomial interpolation: The Polynomial Interpolation Problem: Find a polynomial pÿx a 0 a 1 x... a n x n which satisfies the n 1 conditions pÿx 0 y 0, pÿx 1 y 1,...,pŸx n y n. It is easy to see that the unknown coefficients a 0,..,a n are to satisfy a system of n 1 equations in n 1 unknowns, which we write in shorthand as pÿx i y i, i 0,..., n I. The homogeneous system pÿx i 0, i 0,..., n has pÿx q 0,i.e. a 0 a 1... a n 0, as its only solution. This is obvious from algebra since a polynomial of degree n which is zero at the n 1 points x 0,..,x n must be the zero polynomial. This implies, from the theory of linear equations, that the general system pÿx i y i, i 0,..., n has a unique solution for any given choice of y 0,..,y n. Thus the solution of the polynomial interpolation problem exists, and is unique. II. Direct construction of an interpolating polynomial: The Lagrange Form, the Lagrange fundamental polynomials. Example: Consider this mysterious looking polynomial pÿx y 0 Ÿx " 1 Ÿx " 2 Ÿx " 3 Ÿ0 " 1 Ÿ0 " 2 Ÿ0 " 3 y 1 Ÿx " 0 Ÿx " 2 Ÿx " 3 Ÿ1 " 0 Ÿ1 " 2 Ÿ1 " 3 y 2 Ÿx " 0 Ÿx " 1 Ÿx " 3 Ÿ2 " 0 Ÿ2 " 1 Ÿ2 " 3 y 3 Ÿx " 0 Ÿx " 1 Ÿx " 2 Ÿ3 " 0 Ÿ3 " 1 Ÿ3 " 2 We can easily see that pÿ0 y 0,pŸ1 y 1,pŸ2 y 2,pŸ3 y 3. Generalizing this construction to a general set of interpolation points x 0,..,x n we obtain Lagrange form of the interpolating polynomial: pÿx y 0 L 0 Ÿx y 1 L 1 Ÿx... y n L n Ÿx where L i Ÿx is the Lagrange fundamental polynomial satisfying H. Diamond Numerical Analysis Spring

2 L i Ÿx 1ifx x i ;0 ifx x j for j p i and is given explicitly by L i Ÿx Ÿx " x 0 Ÿx " x i"1 Ÿx " x i1 Ÿx " x n Ÿx i " x 0 Ÿx i " x i"1 Ÿx i " x i1 Ÿx i " x n We have more efficient ways of computing than using the polynomial in this form, but there is a very important theoretical & practical importance to L i Ÿx ; it represents the sensitivity of the value of the interpolating polynomial p at x to changes in the datavalue y i,so that if y i changes by y i then pÿx changes by Ÿ y i L i Ÿx. Newton form of the interpolating polynomial: Here is another way of constructing the interpolating polynomial that leads to the so-called Newton form the and theory of divided differences. It s the best algorithm for constructing and evaluating the interpolating polynomial. For notation purposes, it helps to think of the y-values as coming from some function fÿx, so that we are interpolating the values fÿx i at the points x i, i.e. satisfying the equations pÿx i fÿx i,i 0,.., n. Note the following: p 0 Ÿx fÿx 0 interpolates the single value fÿx 0 with a polynomial of degree zero p 1 Ÿx fÿx 0 a 1 Ÿx " x 0 interpolates at x x 0 and with the right choice of a 1 fÿx 1 " fÿx 0 x 1 " x 0 also interpolates at x x 1 Now consider p 2 Ÿx p 1 Ÿx a 2 Ÿx " x 0 Ÿx " x 1 Because of the form of the last term we added, p 2 Ÿx still interpolates at x 0 and x 1.Ifwe now choose a 2 the right way we can make p 2 Ÿx interpolate at x 2 as well. We could write a 2 fÿx 2 " p 1 Ÿx 2 Ÿx 2 " x 0 Ÿx 2 " x 1 but there is a better way to calculate a 2 that we ll see later. For now, all we need to notice is that we can calculate the unique a 2 for which p 2 Ÿx H. Diamond Numerical Analysis Spring

3 interpolates at x 0,x 1,x 2. form In this way, we can construct, in principle, an interpolating polynomial of degree n of the p n Ÿx a 0 a 1 Ÿx " x 0... a n Ÿx " x 0 Ÿx " x n"1 p n"1 Ÿx a n Ÿx " x 0 Ÿx " x n"1 whose partial sums p k Ÿx a 0 a 1 Ÿx " x 0... a k Ÿx " x 0 Ÿx " x k"1 are interpolating polynomials of lower degree and which solve the interpolation problem at x i,i 0,.., k. Note that each a k can be described as the coefficient of x k in the interpolating polynomial p k Ÿx - this characterization does not depend on the form of the polynomial representation or the order in which the interpolation points are used. This is called the Newton form of the interpolating polynomial. The polynomial, once the coefficients have been determined, can be efficiently evaluated recursively using nested multiplication. The coefficients themselves are important in how they depend on the function values fÿx i and the x i themselves. They are called divided differences, and there is a nice theory that applies to them. Divided differences: Definition: Let t 0,.., t k be a vector of (for now, distinct) values. Given a function f, we define the divided difference of f at the values t 0,..,t k to be the coefficient of x k in the (interpolating) polynomial p of degree k that satisfies pÿt i fÿt i, i 0,.., k. We write f t 0,..,t k to denote this divided difference. A divided difference over k 1 points is referred to as a k th order divided difference. Using the divided difference notation, we can write the Newton form of the interpolating polynomial as H. Diamond Numerical Analysis Spring

4 p n Ÿx f x 0 f x 0,x 1 Ÿx " x 0... f x 0,..., x n Ÿx " x 0 Ÿx " x n"1 We will see that this is a discrete version of the Taylor polynomial P n, in the sense that we obtain P n as the polynomial that interpolates the derivatives of f at x a, and we add an n th degree term on to P n"1 to obtain P n. Also the k th order divided difference f t 0,..,t k will be seen to be closely related to the k th derivative of f. Next we have a nice theorem which provides a recursive formula for calculating divided differences. Theorem: f x 0,x 1,..., x n"1,x n f x 0,x 1,...,x n"1 " f x 1,...,x n"1,x n x n " x 0 This theorem espresses an n th order divided difference as a (first-order) divided difference of n " 1 st order divided differences. Proof: Suppose the polynomial PŸx interpolates f at x 0,x 1,...,x n"1 so that PŸx i fÿx i, i 0,.., n " 1; suppose the polynomial QŸx interpolates f at x 1,...,x n"1,x n so that QŸx i fÿx i, i 1,.., n Now we create as follows a polynomial pÿx that interpolates f at x 0,x 1,...,x n"1,x n : pÿx Ÿx n " x PŸx Ÿx " x 0 QŸx Clearly pÿx 0 PŸx 0 fÿx 0 and pÿx n QŸx n fÿx n and for i 1,.., n " 1 we have pÿx i Ÿx n " x i PŸx i Ÿx i " x 0 QŸx i Ÿx n " x i fÿx i Ÿx i " x 0 fÿx i fÿx i Now in pÿx the coefficient of x n is, by definition, the divided difference f x 0,x 1,..., x n"1,x n ; on the right, in the expression pÿx Ÿx n " x PŸx Ÿx " x 0 QŸx the coefficient of x n is the highest power coefficient of QŸx minus the highest power coefficient of H. Diamond Numerical Analysis Spring

5 PŸx, divided by,or f x 0, x 1,..., x n"1 " f x 1,...,x n"1,x n x n " x 0 coefficient of x n on both sides, we obtain f x 0,x 1,..., x n"1,x n f x 0,x 1,...,x n"1 " f x 1,..., x n"1,x n x n " x 0., and equating the Let s use this result for a practical purpose: Computing the coefficients a i f x 0,..,x i of the Newton polynomial. We begin with an array of x-values x 0,..., x n and an array of function values fÿx 0,...,fŸx n. We compute the first order differences between adjacent points, then the second order differences, and so on. In tabular form, this produces the divided difference table x i f x i f x i,x i1 f x i,x i1,x i2... x 0 fÿx 0 x 1 fÿx 1 f x 0,x 1 f x 0,x 1,x 2 f x 1,x 2 B B B B f x 0,..., x n f x n"2,x n"1 x n"1 fÿx n"1 f x n"2,x n"1,x n x n fÿx n f x n"1,x n The coefficients of the Newton interpolating polynomial appear on the top downward-sloping diagonal of the table. One might also note that any path through the table provides the coefficients of pÿx in (a different) Newton form. If one wishes to interpolate at an additional interpolation point, x n1, then this may be added to the bottom of the table and the bottom upward-sloping diagonal computed to find the coefficient a n1 f x 0,..,x n,x n1 of the additional term a n1 Ÿx " x 0 Ÿx " x n. H. Diamond Numerical Analysis Spring

6 Now we demonstrate the connection between divided differences and derivatives: Theorem: Assuming that f has an n th continuous derivative, f x 0,..., x n 1 n! f Ÿn Ÿ8 where 8 is between the smallest of the x i and the largest. Proof: Let pÿx interpolate fÿx at x 0,...,x n and consider the difference fÿx " pÿx. This function is zero at the n1 points x 0,...,x n. Now if a function is zero at two points, its derivative is zero somewhere in between (Rolle s theorem). If a function is zero at three points, then the first derivative is zero at two points (in between) and the second derivative is then zero at one point in between the three. Continuing, a function that is zero at n 1 points has its n th derivative zero at one point in between. Also note that the n th derivative of pÿx is n! ' coeff of x n n!f x 0,..., x n. We now have: Ÿf " p Ÿn Ÿ8 f Ÿn Ÿ8 " p Ÿn Ÿ8 0, f Ÿn Ÿ8 " n!f x 0,..., x n 0 and the theorem follows. Finally, we provide an important theorem analogous to Taylor s Theorem which says how well a polynomial interpolant p n Ÿx of degree n approximates the function fÿx Theorem: The error in polynomial interpolation satisfies fÿx " p n Ÿx f x 0,.., x n,x Ÿx " x 0 Ÿx " x n 1 Ÿn 1! f Ÿn1 Ÿ8 Ÿx " x 0 Ÿx " x n where 8 is between the smallest and largest values of the x i. Proof: We are given the points x 0,..,x n and values fÿx 1,..,fŸx n, and p n Ÿx interpolates the values of f at the given x-values. Now pick some other x-value, say x t and interpolate at x 0,.., x n,t with the polynomial p n1 Ÿx written in Newton form as p n1 Ÿx p n Ÿx f x 0,.., x n,t Ÿx " x 0 Ÿx " x n. Now since p n1 interpolates at x t we have fÿt p n1 Ÿt p n Ÿt f x 0,..,x n,t Ÿt " x 0 Ÿt " x n. Now simply replace t by x on H. Diamond Numerical Analysis Spring

7 both sides and obtain fÿx " p n Ÿx f x 0,..,x n,x Ÿx " x 0 Ÿx " x n and the latter expression can be replaced by 1 Ÿn 1! f Ÿn1 Ÿ8 Ÿx " x 0 Ÿx " x n by the previous theorem. We sometimes say that the remainder in polynomial interpolation is given by 1 Ÿn 1! f Ÿn1 Ÿ8 Ÿx " x 0 Ÿx " x n. This can be estimated in a manner similar to the remainder in Taylor s theorem. H. Diamond Numerical Analysis Spring