Jim Lambers MAT 772 Fall Semester 2010-11 Lecture 13 Notes These notes correspond to Sections 9.4 in the text. Orthogonal Polynomials Previously, we learned that the problem of finding the polynomial fn(x), of degree n, that best approximates a function f(x) on an interval [a; b] in the least squares sense, i.e., that minimizes Z b 1=2 2 kfn − fk2 = [fn(x) − f(x)] dx ; a is easy to solve if we represent fn(x) as a linear combination of orthogonal polynomials, n X fn(x) = cjpj(x): j=0 Each polynomial pj(x) is of degree j, and the set of polynomials p0(x); p1(x); : : : ; pn(x) are orthog- onal with respect to the inner product Z b hf; gi = f(x)g(x) dx: a That is, Z b hpk; pji = pk(x)pj(x) dx = 0; k 6= j: a Given this sequence of orthogonal polynomials, the coefficients cj in the linear combination used to compute fn(x) are given by hpj; fi cj = ; cj = 0; 1; : : : ; n: hpj; pji Now, we focus on the task of finding such a sequence of orthogonal polynomials. Recall the process known as Gram-Schmidt orthogonalization for obtaining a set of orthogonal vectors p1; p2;:::; pn from a set of linearly independent vectors a1; a2;:::; an: p1 = a1 p1 ⋅ a2 p2 = a2 − p1 p1 ⋅ p1 1 . n−1 X pj ⋅ an p = a − p : n n p ⋅ p j j=0 j j By normalizing each vector pj, we obtain a unit vector 1 qj = pj; jpjj n and a set of orthonormal vectors fqjgj=1, in that they are orthogonal (qk ⋅ qj = 0 for k 6= j), and unit vectors (qj ⋅ qj = 1). We can use a similar process to compute a set of orthogonal polynomials. For simplicitly, we will require that all polynomials in the set be monic; that is, their leading (highest-degree) coefficient must be equal 1. We then define p0(x) = 1. Then, because p1(x) is supposed to be of degree 1, it must have the form p1(x) = x − 1 for some constant 1. To ensure that p1(x) is orthogonal to p0(x), we compute their inner product, and obtain 0 = hp0; p1i = h1; x − 1i; so we must have h1; xi = : 1 h1; 1i For j > 1, we start by setting pj(x) = xpj−1(x), since pj should be of degree one greater than that of pj−1, and this satisfies the requirement that pj be monic. Then, we need to subtract polynomials of lower degree to ensure that pj is orthogonal to pi, for i < j. To that end, we apply Gram-Schmidt orthogonalization and obtain j−1 X hpi; xpj−1i p (x) = xp (x) − p (x): j j−1 hp ; p i i i=0 i i However, by the definition of the inner product, hpi; xpj−1i = hxpi; pj−1i. Furthermore, because xpi is of degree i + 1, and pj−1 is orthogonal to all polynomials of degree less than j, it follows that hpi; xpj−1i = 0 whenever i < j − 1. We have shown that sequences of orthogonal polynomials satisfy a three-term recurrence relation 2 pj(x) = (x − j)pj−1(x) − j−1pj−2(x); j > 1; 2 where the recursion coefficients j and j−1 are defined to be hpj−1; xpj−1i j = ; j > 1; hpj−1; pj−1i 2 2 2 hpj−1; xpji hxpj−1; pji hpj; pji kpjk2 j = = = = 2 ; j ≥ 1: hpj−1; pj−1i hpj−1; pj−1i hpj−1; pj−1i kpj−1k2 Note that hxpj−1; pji = hpj; pji because xpj−1 differs from pj by a polynomial of degree at most j − 1, which is orthogonal to pj. The recurrence relation is also valid for j = 1, provided that we define pj−1(x) ≡ 0, and 1 is defined as above. That is, hp0; xp0i p1(x) = (x − 1)p0(x); 1 = : hp0; p0i If we also define the recursion coefficient 0 by 2 0 = hp0; p0i; and then define pj(x) qj(x) = ; 0 1 ⋅ ⋅ ⋅ j then the polynomials q0; q1; : : : ; qn are also orthogonal, and hpj; pji hpj−1; pj−1i hp0; p0i 1 hqj; qji = 2 2 2 = hpj; pji ⋅ ⋅ ⋅ = 1: 0 1 ⋅ ⋅ ⋅ j hpj; pji hp1; p1i hp0; p0i That is, these polynomials are orthonormal. If we consider the inner product Z 1 hf; gi = f(x)g(x) dx; −1 then a sequence of orthogonal polynomials, with respect to this inner product, can be defined as follows: L0(x) = 1; L1(x) = x; 2j + 1 j L (x) = xL (x) − L (x); j = 1; 2;::: j+1 j + 1 j j + 1 j−1 These are known as the Legendre polynomials. One of their most important applications is in the construction of Gaussian quadrature rules. Specifically, the roots of Ln(x), for n ≥ 1, are the nodes of a Gaussian quadrature rule for the interval [−1; 1]. However, they can also be used to easily compute continuous least-squares polynomial approximations, as the following example shows. Example We will use Legendre polynomials to approximate f(x) = cos x on [−=2; =2] by a quadratic polynomial. First, we note that the first three Legendre polynomials, which are the ones of degree 0, 1 and 2, are 1 L (x) = 1;L (x) = x; L (x) = (3x2 − 1): 0 1 2 2 3 However, it is not practical to use these polynomials directly to approximate f(x), because they are orthogonal with respect to the inner product defined on the interval [−1; 1], and we wish to approximate f(x) on [−=2; =2]. To obtain orthogonal polynomials on [−=2; =2], we replace x by 2t/, where t belongs to [−=2; =2], in the Legendre polynomials, which yields 2t 1 12 L~ (t) = 1; L~ (t) = ; L~ (t) = t2 − 1 : 0 1 2 2 2 Then, we can express our quadratic approximation f2(x) of f(x) by the linear combination f2(x) = c0L~0(x) + c1L~1(x) + c2L~2(x); where hf; L~ji cj = ; j = 0; 1; 2: hL~j; L~ji Computing these inner products yields Z =2 hf; L~0i = cos t dt −=2 = 2; Z =2 2t hf; L~1i = cos t dt −=2 = 0; Z =2 ~ 1 12 2 hf; L2i = 2 t − 1 cos t dt −=2 2 2 = (2 − 12); 2 Z =2 hL~0; L~0i = 1 dt −=2 = ; Z =2 2t2 hL~1; L~1i = dt −=2 8 = ; 3 Z =2 2 ~ ~ 1 12 2 hL2; L2i = 2 t − 1 dt −=2 2 = : 5 4 It follows that 2 2 5 10 c = ; c = 0; c = (2 − 12) = (2 − 12); 0 1 2 2 3 and therefore 2 5 12 f (x) = + (2 − 12) x2 − 1 ≈ 0:98016 − 0:4177x2: 2 3 2 This approximation is shown in Figure 1. 2 Figure 1: Graph of cos x (solid blue curve) and its continuous least-squares quadratic approximation (red dashed curve) on [−=2; =2] It is possible to compute sequences of orthogonal polynomials with respect to other inner prod- ucts. A generalization of the inner product that we have been using is defined by Z b hf; gi = f(x)g(x)w(x) dx; a where w(x) is a weight function. To be a weight function, it is required that w(x) ≥ 0 on (a; b), and that w(x) 6= 0 on any subinterval of (a; b). So far, we have only considered the case of w(x) ≡ 1. 5 Another weight function of interest is 1 w(x) = p ; −1 < x < 1: 1 − x2 A sequence of polynomials that is orthogonal with respect to this weight function, and the associated inner product Z 1 1 hf; gi = f(x)g(x)p dx 2 −1 1 − x is the sequence of Chebyshev polynomials C0(x) = 1; C1(x) = x; Cj+1(x) = 2xCj(x) − Cj−1(x); j = 1; 2;::: which can also be defined by −1 Cj(x) = cos(j cos x); −1 ≤ x ≤ 1: It is interesting to note that if we let x = cos , then Z 1 −1 1 hf; Cji = f(x) cos(j cos x)p dx 2 −1 1 − x Z = f(cos ) cos j d: 0 In later lectures, we will investigate continuous and discrete least-squares approximation of functions by linear combinations of trigonometric polynomials such as cos j or sin j, which will reveal one of the most useful applications of Chebyshev polynomials. 6.