Chapter 3. Interpolation. 3.1 Introduction

Transcription

1 Chapter 3 Interpolation 3 Introduction One of the fundamental problems in Numerical Methods is the problem of interpolation, that is given a set of data points ( k, k ) for k =,, n, how do we find a function P () that connects the dots such that the function is guaranteed to go through the data (P ( k ) = k ) while being faithful to our understanding between the data points The first task is relativel straightforward, as Figure 3 shows, there are lots of was to thread 8 points ranging from polnomial interpolation to piecewise linear interpolation to cubic splines The more important issue is to understand what controls the behavior of the interpolants between the nodes so as to make wise choices that ma depend on the underling data 5 Interpolation Figure 3: Several interpolation schemes discussed in this chapter Figure produced b Moler s NCM routine interpguim

2 2 CHAPTER 3: Interpolation 32 The interpolating polnomial The place to begin however is with the Interpolating Polnomial Definition Through an n + distinct points (, ), (, ),, ( n, n ) with i j there is a unique interpolating polnomial P n () of degree n such that P n ( k ) = k There are man different was to epress and evaluate this polnomial depending on the choice of polnomial basis, however, as we ll show, as long as the coordinates are distinct, this polnomial is unique 32 Polnomial bases The standard wa we usuall think of Polnomials is as a linear combination of the monomials,, 2 ie P n () = p + p + p 2 2 p n n (32) with undetermined coefficients p,, p n To determine these coefficients we could solve the (n + ) (n + ) linear sstem P n ( ) = P n ( ) = P n ( 2 ) = 2 (322) P n ( ) = or in Matri-Vector notation 2 n 2 n n 2 n 2 3 n 3 p p p 2 p n = 2 n (323) as A = b In Matlab, this matri could be rapidl constructed given a column vector of coordinates n = length(); A = ones(n) for j = 2:n A(:,j) = A(:,j-)*; end If is a column vector of values, then p=a\; c 27, Marc Spiegelman, Columbia Universit

3 32 THE INTERPOLATING POLYNOMIAL 3 will solve the linear sstem for the coefficients p The actual polnomial can then be evaluated using Horner s method Clearl if there is an pair of points such that i = j, the Vandermonde matri A will be singular as there will be repeated rows While less obvious here, it can be shown that if all the coordinates are distinct, then A is invertible and therefore the coefficients are unique 322 Other Bases: Lagrange Polnomials While there is nothing particularl wrong with this approach, there is nothing terribl right either For large values of n, (which are often a bad idea for other reasons we will discuss), the matri A can be poorl conditioned and it still requires order n 3 /3 operations to solve the linear sstem However, there are man other bases for the space of polnomials of degree n and these can often be more convenient or efficient for working with An important basis is the Lagrange Polnomials which are defined b L k () = n i=,i k ( i ) ( k i ) (324) which is the product of the fractional distances of the point from the point k For eample, the three Lagrange Polnomials defined b the nodes, and 2 are L () = ( )( 2 ) L () = ( )( 2 ) L 2 () = ( )( ) ( )( 2 ) ( )( 2 ) ( 2 )( 2 ) (325) The critical propert of the Lagrange Polnomials is that the are either eactl or zero at the nodes i More specificall L k ( i ) = { i = k i k (326) Figure 32 shows the Lagrange Polnomials that pass through 4 and 8 evenl spaced points In the Lagrange basis, the interpolating polnomial can be written as P n () = n k= p kl k and the linear sstem defined b Eq (322) becomes = (327) or simpl Ip = Thus in the lagrange basis, we can write the interpolating polnomial as P n () = k L k () (328) k= which is just a linear combination of the Lagrange polnomials weighted b the values at the nodes This formulation can be a bit unwield to evaluate but provides a ver important framework for other applications of interpolation such as numerical quadrature or differentiation This is actuall what Matlab s functions polfit and polval do although the order of the columns are reversed, see help vander in Matlab p p p 2 p n 2 n c 27, Marc Spiegelman, Columbia Universit

4 4 CHAPTER 3: Interpolation Lagrange Interpolating Polnomials: n=4 k= k=2 k=3 k= Lagrange Interpolating Polnomials: n=8 k= k=2 k=3 k=4 k=5 k=6 k=7 k= a b Figure 32: Eample of the Lagrange Polnomials through (a) 4 Points, (b) 8 points Each polnomial L k is guaranteed to be eactl at node k and zero at all the rest of the nodes (here the numbering of the polnomials starts at rather than zero sorr for the confusion) Note that, for evenl spaced points, the Lagrange polnomials can oscillate wildl between the nodes 323 Newton Polnomials A third basis for polnomials that is quite useful for numerical work is the Newton Polnomials defined b { k = N k () = k i= ( (329) i) k =,, n Thus, the interpolating polnomial can be written as P n () = p k N k () = p +p ( )+p 2 ( )( )+p 3 ( )( )( 2 ) k= and the linear sstem defined b Eq (322) becomes the Lower Triangular sstem ( ) ( 2 ) ( 2 )( 2 ) N ( n ) N 2 ( n ) N n ( n ) p p p 2 p n = 2 n (32) (32) The triangular nature of this sstem allows for solution of the coefficients p i in onl n 2 operations (and leads to several other nice recursive properties of the Newton Polnomials) In addition, this basis shows clearl that for the polnomial to be unique, the nodes k must be distinct For the matri in Eq (32) to be invertible, it s determinant must not be zero However, the determinant of a triangular matri is simpl the product of the diagonal terms Clearl, this product will onl vanish if one or more points are equal, ie i = j for some i, j If all the points are distinct the Newton Polnomials are linearl independent and the solution for p is unique c 27, Marc Spiegelman, Columbia Universit

5 32 THE INTERPOLATING POLYNOMIAL Uniqueness of the interpolating polnomial Independent of the choice of basis however, it is important to understand that the actual polnomial P n () is unique if the nodes are distinct While I think the linear algebra representation makes this reasonabl clear, this can be shown more generall b the properties of polnomials Theorem The Interpolating polnomial P n () that satisfies P n ( k ) = k for k =,, n distinct points is unique Proof Assume that there are two distinct polnomials of degree n that both interpolate the same n + points P n () = p k k (322) and Q n () = k= q k k (323) such that P n ( k ) = Q n ( k ) = k If we form a new polnomial b subtraction ie k= R n () = P n () Q n () = (p k q k ) k (324) k= we will produce another n degree polnomial, which should have at most n roots However, R n () is guaranteed to vanish at the n + points k The onl polnomial of degree n that is zero at more than n points is the zero polnomial R n () = This implies that p k = q k and the interpolating polnomial is unique 325 Errors While the interpolating polnomial is unique, and guaranteed to pass eactl through the n + data points, it is not necessaril a good approimation for a function between the nodes For eample, Figure 322 compares the interpolating polnomial for regularl spaced points for against two functions e and /( ) Similar to Talor s theorem, the Lagrange Error theorem sas that, for an function f() can be approimated b the interpolating polnomial plus a remainder term, ie where P n () is the interpolating polnomial and f() = P n () + R n () (325) R n () = f (n+) (ζ) (n + )! ( )( )( 2 ) ( n ) (326) This Lagrange remainder term is similar to the Talor Remainder term ecept that it is guaranteed to vanish at each of the nodes (whereas the Talor remainder is onl zero at the point of epansion) Again, f (n+) (ζ) is the n + derivative of f evaluated at some (unknown) point ζ [, n ] As usual, this constant cannot be eactl evaluated but often can be bounded b c 27, Marc Spiegelman, Columbia Universit

6 6 CHAPTER 3: Interpolation 3 25 Interpolant Nodes true function F(), P(), N= 2 5 F(), P(), N= Interpolant Nodes true function a Error: f() P(), N= b Error: f() P(), N= 2 5 error error 2 5 c d Figure 322: Comparison of polnomial interpolation of two functions a comparison of the function f() = e and its interpolating polnomial for evenl spaced points in [ ] (c) the error R n () = f() P n () in interpolation For functions that are well approimated b high-order polnomials, this interpolant can work well However, (b,d) comparison of interpolant for same points and f() = /( ) For this function the Runge effects are etremel evident and the interpolating polnomial is a poor approimation Increasing the number of points actuall makes the error worse c 27, Marc Spiegelman, Columbia Universit

7 32 THE INTERPOLATING POLYNOMIAL 7 8 F(), P(), N= Interpolant Nodes true function 9 8 F(), P(), N=2 Interpolant Nodes true function a Error: f() P(), N= b Error: f() P(), N=2 5 5 error 5 error 5 5 c d Figure 323: Polnomial interpolation and error of f() = /( ) with (a) points located at the zeros of the Chebshev Polnomial T and (b) N = 2 Chebshev points For high-order polnomial approimation of functions on a fied interval, the Chebshev points are optimal for minimizing the error c 27, Marc Spiegelman, Columbia Universit

8 8 CHAPTER 3: Interpolation some constant M For evenl spaced points, there are eplicit error formulas (eg see [?]) However, in general, high-order polnomial interpolation approimation with evenl spaced points is a Bad! idea unless the underling function is well described b high-order polnomials (eg e, see Figure 322) The reason for this can be seen b eploring the behavior of the Lagrange interpolating polnomials for evenl spaced points as a function of the number of points N (see problem set 2), which have ver large ecursions from in the interval [, + h] 326 High-order polnomial interpolation using Chebshev points Unless ou reall understand our function, ou should rarel use polnomial approimation with evenl spaced points for more than about 5 points (m rule of thumb) If ou do, need to approimate a function b high-order polnomials over a fied interval, there is an optimal choice of points to interpolate A full description is beond the scope of these notes but Mathews and Fink provide a good beginning discussion The basic idea however is to tr and minimize the size of the Lagrange remainder term Inspection of Eq (326), however, shows that the remainder has two parts, the first depends on the properties of the function itself (which ou have no control over), but the second is the size of the monic polnomial 2 Q n+ () = ( )( )( 2 ) ( n ) (327) 33 Practical Algorithms: Piecewise Polnomial Interpolation 33 C : linear interpolation, overlapping cubic 332 C : piecewise cubic hermite polnomials 333 C 2 : Cubic Splines 34 Multi-dimensional interpolation 2 Monic polnomials are all polnomials whose highest order coefficient is c 27, Marc Spiegelman, Columbia Universit