Chapter 8 Approximation Theory Chebyshev Polynomials and Economization

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series 8. 3 Chebyshev Polynomials and Economization of Power Series The general least squares approximation problem is to find a generalized polynomial P(x) such that E = (P – y, P – y) = || P – y ||2 is minimized. Minimize || P – y || -- the minimax problem Takeyou it easy. It’s not so Didn’t say it’s a very difficult we consider difficultifproblem? polynomials only. 1/11

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series v 1. 0 Find a polynomial Pn(x) of degree n such that || Pn f || is minimized. Definition: If P(x 0) – f (x 0) = || P f || , x 0 is called a ( ) deviation point. It is not easy to construct the polynomial from nowhere. However, we can examine the features of the polynomial: If f C[a, b] and f is not a polynomial of degree n, then there exists a unique polynomial Pn(x) such that || Pn f || is minimized. Pn(x) exists, and must have both + and – deviation points. Pn(x) f(x) (Chebyshev Theorem) Pn(x) minimizeshas || Pat f || n+1 Proots. n n(x) has at least n+2 alternating + and – deviation points with respect to f. That is, there exists a set of points a t 1 <…< tn+2 b such that Pn(tk) – f (tk) = (– 1)k || Pn f || . The set { tk } is called the Chebyshev alternating sequence. 2/11

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series y y = f ( x) + En y = Pn ( x) y = f ( x) En Pn(x) is an interpolating polynomial of f(x) x 0 v 2. 0 3/11 Determine the interpolating points { x 0, …, xn } such that Pn(x) minimizes the remainder

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series v 2. 1 Find { x 1, …, xn } such that ||wn|| is minimized on [ 1, 1], where wn( x) = n ( x x ) i =1 i Notice that wn(x) = xn – Pn– 1(x). The problem becomes to … Find a polynomial Pn– 1(x) such that || xn – Pn– 1(x) || is v 3. 0 minimized on [ 1, 1]. From Chebyshev theorem we know that Pn 1(x) has n+1 deviation points with respect to xn , that is, wn(x) obtains its maximum and minimum values alternatively on n+1 points. 4/11

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series Chebyshev polynomials Consider the n + 1 extreme values of cos(n ) on [ 0, ]. cos(n ) assumes its maximum value 1 and minimum value 1 alternatively at points . And there exist coefficients a 0, …, an such that Let x = cos( ), then x [ 1 , 1 ]. Tn(x) = cos( n · arc cos x ) is called the Chebyshev polynomial. More about Tn: Tn(x) assumes its maximum value 1 and minimum value 1 alternatively at Tn(x) has n roots 5/11 . That is, 1

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series Tn(x) has the recurrence relation: T 0(x) = 1, T 1(x) = x, Tn+1(x) = 2 x Tn(x) – Tn– 1(x). Tn(x) is a polynomial of degree n with leading coefficient 2 n 1. { T 0(x), T 1(x), … } are orthogonal on [ 1 , 1 ] with respect to the weight function That is, OKOK, I think it’s enough for us… What’s our target again? Find a polynomial Pn– 1(x) such that || xn – Pn– 1(x) || v 3. 0 is minimized on [ 1, 1]. wn(x) = xn – Pn– 1(x) = Tn(x) / 2 n 1 6/11

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series v 2. 1 Find { x 1, …, xn } such that ||wn|| is minimized on [ 1, 1], where wn( x) = n ( x x ) i =1 i { x 1, …, xn } are the n roots of Tn(x). v 2. 0 = { monic polynomials of degree n } Determine the interpolating points { x 0, …, xn } such that Pn(x) minimizes the remainder Take the n+1 roots of Tn+1(x) as the interpolating points { x 0, …, xn }. Then the interpolating polynomial Pn(x) of f(x) assumes the minimum upper bound of the absolute error 7/11

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series Example: Find the best approximating polynomial of f (x) = ex on [0, 1] such that the absolute error is no larger than 0. 5 10 4. Solution: Determine n: Make a change of the variable n= 4 Find the roots of T 5(t): Make a change of the variable: Compute L 4(x) with interpolating points x 0, …, x 4. 8/11

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series Given Pn(x) f (x), economization of power series is to reduce the degree of polynomial with a minimal loss of accuracy. Consider approximating an arbitrary n-th degree polynomial Pn(x) = anxn + an– 1 xn– 1 + … + a 1 x + a 0 with a polynomial Pn– 1(x) by removing an n-th degree polynomial Qn(x) that has the coefficient an for xn. Then max | f ( x ) Pn 1 ( x ) | max | f ( x ) Pn ( x ) | + max | Qn ( x ) | [ 1, 1] [ 1 , 1 ] To minimize the loss of accuracy, Qn(x) must be The loss of accuracy. 9/11

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series Example: The 4 -th order Taylor polynomial for f (x) = ex on [ 1, 1] is The upper bound of truncation error is Please reduce the degree of the approximating polynomial to 2. Solution: If we simply take 10/11 , then the error is

Chapter 8 Approximation Theory -- Chebyshev Polynomials and Economization of Power Series Note: A change of variable is needed for a general interval [a, b]. That is, let x = [(b – a)t + (a + b)]/2, then find the polynomial Pn(t) for f (t) on [ 1, 1] and finally obtain Pn(x). Another method is to write each term of xk as a linear combination of T 0(x), …, Tk(x). For example, x = T 1(x) and x 3 = [T 3(x) + 3 T 1(x)] / 4. Then simply remove the Chebyshev functions from the original polynomial. HW: p. 517 #3, 7, 9 11/11