1 The Lagrange interpolation formula


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1 Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x j x k, j k. (1.1) We wnt to determine polynomil p of degree t most N hving the interpoltion property p(x j ) y j, j 0, 1,..., N. (1.2) We hve tht there is unique solution to this problem. It cn be obtined in mny different wys. Here we give the Lgrnge 1 form of the interpolting polynomil. We define the Lgrnge polynomils s follows. l k (x) N (x x j ) j k, k 0, 1,..., N. (1.3) N (x k x j ) j k It follows from the definition tht ech polynomil l k (x) hs degree N nd stisfies { 1 for k j, l k (x j ) 0 for k j. (1.4) Thus the interpolting polynomil cn be given s p(x) y k l k (x). (1.5) k0 2 Smpling, interpoltion, differentition, nd integrtion There is generl scheme used in performing the bsic opertions in clculus numericlly. The scheme cn be described s follows. Given relvlued continuous function f on n intervl [, b], choose smpling points x 0 < x 1 < x 2 < < x N b nd let y j f(x j ), j 0, 1,..., N. Bsed on these points (x j, y j ), compute the interpolting polynomil p. Then we cn pproximte the derivtive t x j of f by p (x j ), i.e. f (x j ) p (x j ). Furthermore, one cn pproximte the integrl over the intervl by integrting the interpolting polynomil, i.e. f(x) p(x)dx. These two schemes include mny clssicl nd well known methods of numericl differentition nd integrtion. In the following section we will look in detil t one clss of numericl integrtion schemes, usully clled Guss qudrture. 1 Lgrnge ws not the first mthemticin stting this formul. It ws Wring in Pge 1 of 8
2 Notes on Qudrture 3 Qudrture rules Using the smpled vlues of the function f the interpolting polynomil cn be written s p(x) f(x j )l j (x). Integrtion leds to the formul p(x)dx f(x j ) l j (x)dx c j f(x j ). (3.1) If one fixes the smpling points x j, then one needs to compute the constnts only once. A formul of the form c j l j (x)dx c j f(x j ) (3.2) for determining n pproximtion to the integrl of function over specified intervl is clled qudrture rule. Now we discuss for which polynomils this formul (or rule) gives n exct nswer, not n pproximte nswer. Since the interpolting polynomil is unique, the formul is exct for ll polynomils of degree less thn or equl to N. We introduce the following definition. Definition 3.1. A qudrture rule is sid to be precise of degree m, if it is exct for polynomils of degree m, but not for x m+1. The three most bsic qudrture rules, often encountered in clculus course, re the following: Midpoint rule. Here one chooses N 0 for the interpolting polynomil nd the midpoint of the integrtion intervl s the smpling point. This leds to the formul f(x)dx (b )f( 1 ( + b)). (3.3) 2 Trpezoidl rule. Here one chooses N 1 for the interpolting polynomil nd the endpoints of the integrtion intervl s the smpling points. This leds to the formul f(x)dx 1 (b )(f() + f(b)). (3.4) 2 Simpson s rule. Here one chooses N 2 for the interpolting polynomil nd the midpoint of the integrtion intervl together with the end points s the smpling points. This leds to the formul f(x)dx 1 6 (b )( f() + 4f( 1 2 ( + b) + f(b)). (3.5) Pge 2 of 8
3 Notes on Qudrture The two first rules re exct of degree 0 nd 1, respectively, wheres Simpson s rule is exct of degree 3. In qudrture rule one uses the terminology nodes for the smpling points x j nd weights for the constnts c j in (3.2). If one tries to optimize the choice of both the nodes nd the weights, then one cn get degree of precision substntilly higher thn N. Mny of these rules re known s Guss qudrture rules. To derive them we need some tools. 4 Orthogonl polynomils A convenient wy to estblish generl scheme for deriving Guss qudrture rules uses orthogonl polynomils. We give some of the detils of this vst theory. We use slightly modified version of the presenttion in [3]. We fix n intervl (, b). In our discussion this cn be finite intervl or n infinite intervl. We introduce weight function in our integrls (sometimes clled density). It is function w : (, b) R, w(x) > 0 for ll x (, b). We denote by P the polynomils with rel coefficients, nd by P N the subspce of polynomils of degree t most N. It is well known tht both P nd P N re rel vector spces, nd tht dim P N N + 1. The Lgrnge polynomils l j (x) constitute bsis for P N, irrespective of the choice of the nodes x j, s long s they re distinct. We introduce n inner product on the spce P, given by p, q p(x)q(x)w(x)dx, p, q P. (4.1) We impose the condition tht the integrl is finite for ll polynomils. This is n implicit condition on the weight function w. We will give exmples lter. Definition 4.1. A sequence of polynomils {p n } n0 is clled sequence of orthogonl polynomils, if the following two conditions re stisfied. (1) degree(p n ) n, n 0, 1,..., (2) p n, p m 0 for ll n m. Note tht this concept depends on both the intervl (, b) nd the weight function w under considertion. Some sort of normliztion of orthogonl polynomils is convenient. We choose convenient one, where we require our orthogonl polynomils to be monic. This is not the stndrd normliztion. This mens tht the coefficient to the leding power x n in p n is equl to 1, i.e. the polynomil hs the form p n (x) x n + n 1 x n x + 0. One wy of getting sequence of orthogonl polynomils would be to choose s strting point the sequence 1, x, x 2,..., x n,... nd then pply the GrmSchmidt orthogonliztion procedure. However, this is not convenient wy to obtin the sequence of orthogonl polynomils ssocited with given choice of intervl (, b) nd weight function w. We will describe procedure for obtining the sequence of orthogonl polynomils ssocited with given choice of intervl (, b) nd weight function w. They will be given in terms of recursion reltion. Pge 3 of 8
4 Notes on Qudrture In the sequel we fix (, b) nd w, nd will not mention the dependence on these prmeters in our results below. Lemm 4.2. Let {p n } n0 be sequence of orthogonl polynomils. Then ny q P n cn be written uniquely s q b n p n + b n 1 p n b 0 p 0. (4.2) Proof. This follows from the fct tht {p n,..., p 0 } is n orthogonl bsis of P n. Corollry 4.3. We hve p n+1, q 0 for ll q P n. We now strt the construction process, nd t the end we summrize the construction. The first polynomil p 0 is uniquely determined by the requirement tht ll polynomils in our sequence be monic. Thus p 0 (x) 1. (4.3) The sme requirement tells us tht we must hve p 1 (x) x α 1 for some constnt α 1. This constnt is determined by the orthogonlity requirement p 1, p 0 0, leding to p 1 (x) x α 1, α 1 x, 1 1, 1 xw(x)dx. (4.4) w(x)dx The construction now proceeds inductively. Assume tht we hve constructed the polynomils p 0, p 1,..., p n. Then p n+1 must hve the form p n+1 (x) xp n (x) α n+1 p n (x) β n+1 p n 1 (x) γ n+1 p n 2 (x) +. (4.5) We determine the unknown coefficients using the orthogonlity requirements. We strt by tking inner product with p n. Using Corollry 4.3 we get which leds to 0 p n+1, p n xp n, p n α n+1 p n, p n, α n+1 xp n, p n p n, p n. Next we tke inner product with p n 1. Using tht p n 1 is orthogonl to both p n nd p n 2, we get the result β n+1 xp n, p n 1 p n 1, p n 1. Repeting the rguments, tking inner product with p n 2 this time, we get Now we rewrite s follows xp n, p n 2 xp n (x)p n 2 (x)w(x)dx γ n+1 xp n, p n 2 p n 2, p n 2. p n (x)xp n 2 (x)w(x)dx p n, xp n 2 0, where the zero follows from the fct tht xp n 2 (x) is of degree n 1 nd Corollry 4.3. The sme rgument shows tht the remining coefficients re lso zero. We summrize: Pge 4 of 8
5 Notes on Qudrture Theorem 4.4. Given (, b) nd w(x) subject to the conditions bove, there exists sequence of orthogonl polynomils, which cn be obtined recursively s follows where p 0 (x) 1, (4.6) p 1 (x) x α 1, (4.7) p n+1 (x) xp n (x) α n+1 p n (x) β n+1 p n 1 (x), n 1, 2,..., (4.8) α n+1 xp n, p n p n, p n β n+1 xp n, p n 1 p n 1, p n 1 xp n(x) 2 w(x)dx p n(x) 2 w(x)dx, n 0, 1, 2,..., (4.9) xp n(x)p n 1 (x)w(x)dx p, n 1, 2,.... (4.10) n 1(x) 2 w(x)dx The recursion reltion (4.8) is clled three term recursion reltion. We cn lso sy something bout the zeroes of orthogonl polynomils. We ssume tht the reder is fmilir with the fundmentl theorem of lgebr nd the fctoriztion of polynomil over its zeroes (in generl complex). Theorem 4.5. Given (, b) nd w(x) subject to the conditions bove, let {p n } n0 be the orthogonl polynomils. The zeroes of p n, n 1, 2,..., re ll rel nd simple, nd lie in the intervl [, b]. Proof. Choose fixed n 1. Let t 1, t 2,..., t k denote the zeroes of p n in [, b] with odd multiplicity. Here we ssume t j t k, j k. If we cn show k n, we re done. Assume k < n. We define q(x) (x t 1 )(x t 2 ) (x t k ). By ssumption degree(q) < n, such tht by Corollry 4.3 we must hve q, p n 0. But q(x)p n (x) cnnot chnge signs in [, b], since sign chnge must occur t zero of odd multiplicity nd this product hs no zeroes of odd multiplicity. Thus we conclude contrdiction. Therefore we must hve k n. q(x)p n (x)w(x)dx 0, The Guss qudrture scheme. We ssume tht we re given (, b) nd w subject to the bove conditions, nd construct the corresponding sequence of orthogonl polynomils, {p n } n0. For ech n we let x 0, x 1,..., x n denote the zeroes of the orthogonl polynomil p n+1 (x). Note tht we now strt our indexing t zero, s we did bove in Lgrnge interpoltion. Let l j denote the Lgrnge polynomils given in (1.3). Let Define qudrture rule by A j l j (x)w(x)dx, j 0, 1,..., n. (4.11) G n f A 0 f(x 0 ) + A 1 f(x 1 ) + + A n f(x n ). (4.12) Theorem 4.6. The Guss qudrture rule defined bove hs degree of precision (t lest) 2n + 1. Furthermore, we hve tht A j > 0, j 0, 1,..., n nd A 0 + A A n c, where c w(x)dx. Pge 5 of 8
6 Notes on Qudrture Proof. As mentioned bove in connection with the definition of the concept of precision, ny qudrture rule bsed on polynomil interpoltion hs precision t lest n. Thus we know tht G n f gives the exct result for ll f P n. Now ssume f P 2n+1. Using division of polynomils with reminder ( version of Euclid s lgorithm), we cn write Then we compute s follows n G n f A j f(x j ) f p n+1 q + r, with degree(q) n nd degree(r) n. n ( A j pn+1 (x j )q(x j ) + r(x j ) ) n A j r(x j ) since p n+1 (x j ) 0 r(x)w(x)dx since degree(r) n ( pn+1 (x)q(x) + r(x) ) w(x)dx since p n+1, q 0 f(x)w(x)dx. Next we prove tht A j > 0. We strt by noting tht { l k (x j ) l k (x j ) 2 1 for k j, 0 for k j. Furthermore, l k (x) 2 0 nd l k (x) 2 is polynomil f degree 2n. Thus our qudrture rule is exct for this polynomil. It follows tht n 0 < l k (x) 2 w(x)dx G n l 2 k A j l k (x j ) 2 A k. Since the qudrture rule is exct for the polynomil 1, we hve G n 1 A 0 + A A n 1w(x)dx. We hve the following error estimte for the qudrture rule. Theorem 4.7. Assume tht f is 2n + 2 times continuously differentible on the intervl (, b). Then there exists ξ (, b) such tht f(x)w(x)dx G n f f (2n+2) (ξ) (2n + 2)! We lso hve convergence result. p n+1 (x) 2 w(x)dx. (4.13) Theorem 4.8. Assume tht f is continuous on (, b) nd f(x) w(x)dx <. Then lim G nf n f(x)w(x)dx. (4.14) Pge 6 of 8
7 Notes on Qudrture 5 Exmples of orthogonl polynomils We will give few exmples of clssicl orthogonl polynomils. This involves specifying the intervl nd the weight function. Computtion of the recursion coefficients cn be quite cumbersome. Fortuntely for the clssicl orthogonl polynomils they hve been computed nd re found in mny hndbooks on mthemtics. We will just stte the results. The ssumption tht the polynomils re monic is convenient in the derivtion of the recursion reltion. However, the hndbooks do not stte the formuls in tht form. Below we give the form found in the reference works [1] nd [2]. If you relly need to use the reference works, it is importnt to spend some time getting cquinted with the nottion nd the conventions used. 5.1 Legendre polynomils In this cse the intervl is [ 1, 1] nd w(x) 1, x [ 1, 1]. The usul nottion is P n (x) nd the recursion is P 0 (x) 1, P 1 (x) x, (5.1) (n + 1)P n+1 (x) (2n + 1)xP n (x) np n 1 (x). (5.2) Using the scheme bove we cn derive clssicl qudrture formul bsed on the Legendre polynomil of degree 3. The recursion formul gives The qudrture rule is P 3 (x) 5 2 x3 3 2 x. 1 1 f(x) 1 9 ( 5f( 3/5) + 8f(0) + 5f( ) 3/5) (5.3) It is exct for ll polynomils of degree t most 5. Let us lso stte the recursion formul for the monic version of the Legendre polynomils, which we denote by Q n here. These polynomils stisfy Q n+1 (x) xq n (x) 2n 1 2n + 1 Q n 1(x). (5.4) 1 1 Q n (x) 2 dx 1 2n + 1. Thus if we define q n (x) 2n + 1Q n (x), we get n orthonorml bsis {q n } n0 for the Hilbert spce L 2 ([ 1, 1], dx). 5.2 Lguerre polynomils In this cse the intervl is [0, ), nd w(x) e x. The usul nottion is L n (x) nd the recursion is L 0 (x) 1, L 1 (x) 1 x, (5.5) (n + 1)L n+1 (x) (2n + 1 x)l n (x) nl n 1 (x). (5.6) Pge 7 of 8
8 Notes on Qudrture 5.3 Hermite polynomils In this cse the intervl is (, ), nd w(x) e x2. The usul nottion is H n (x) nd the recursion is H 0 (x) 1, H 1 (x) 2x, (5.7) H n+1 (x) 2xH n (x) 2nH n 1 (x). (5.8) 5.4 Chebyshev polynomils In this cse the intervl is ( 1, 1) nd w(x) (1 x 2 ) 1/2, x ( 1, 1). The usul nottion is T n (x) nd the recursion is T 0 (x) 1, T 1 (x) x, (5.9) T n+1 (x) 2xT n (x) T n 1 (x). (5.10) For this prticulr clss of orthogonl polynomils there re simple closed formuls for ll k. Let x cos(θ), θ rccos(x). Then T k (x) cos(kθ). Associted with the Chebyshev polynomils re points clled the Chebyshev points. They re given s x j cos(jπ/k), j 0, 1,..., k. At these points the polynomil ssumes its extreme vlues ±1 on the intervl [ 1, 1]. A detiled ccount of the usefulness of these points in numericl computtions cn be found in [4]. References [1] M. Abrmowitz nd I. A. Stegun (Eds.), Hndbook of Mthemticl Functions, Dover [2] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, nd C. W. Clrk (Eds.), NIST Hndbook of Mthemticl Functions, Cmbridge University Press Avilble online t dlmf. nist.gov [3] G. W. Stewrt, Afternotes on Numericl Anlysis, SIAM [4] L. N. Trefethen, Approximtion Theory nd Approximtion Prctice, SIAM Pge 8 of 8
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