best polynomial approximation

Best Polynomials Approximation of Degree One at C[a, b] Spaceby Method of Least Parallelogram

Dadang Amir Hamzah, S.Si., M.Si.

INSTITUT TEKNOLOGI BANDUNG2013

Dadang Amir Hamzah (ITB) KNM XVI UNPAD 25 February 2013 1 / 26

Outline

1 IntroductionIntroduction and Problem

2 Main ResultMethod of The Least Parallelogram

3 Reference


Outline



3 Reference


Introduction

Approximation of function is the technique to replace a functionwith another simpler function, such as Taylor series, step functionor polynomials.

Before we pose the approximation problem we should decidethree main component, first function space F which its elementwill be approximated, second the function class A which we use toapproximate, and the third the Norm which measure theapproximation error.In this presentation we use F = C[a, b] space which is the set ofcontinuous function at closed interval [a, b], then the class ofapproximation function A = P1 which is class of polynomial ofdegree one, and the norm we use is maximum norm which is thenorm defined at C[a, b] space that is

‖f‖ = ‖f‖∞ = maxx∈[a,b]

|f(x)|; (1)


Introduction

Approximation of function is the technique to replace a functionwith another simpler function, such as Taylor series, step functionor polynomials.Before we pose the approximation problem we should decidethree main component, first function space F which its elementwill be approximated, second the function class A which we use toapproximate, and the third the Norm which measure theapproximation error.

In this presentation we use F = C[a, b] space which is the set ofcontinuous function at closed interval [a, b], then the class ofapproximation function A = P1 which is class of polynomial ofdegree one, and the norm we use is maximum norm which is thenorm defined at C[a, b] space that is

‖f‖ = ‖f‖∞ = maxx∈[a,b]

|f(x)|; (1)


Introduction

Approximation of function is the technique to replace a functionwith another simpler function, such as Taylor series, step functionor polynomials.Before we pose the approximation problem we should decidethree main component, first function space F which its elementwill be approximated, second the function class A which we use toapproximate, and the third the Norm which measure theapproximation error.In this presentation we use F = C[a, b] space which is the set ofcontinuous function at closed interval [a, b], then the class ofapproximation function A = P1 which is class of polynomial ofdegree one, and the norm we use is maximum norm which is thenorm defined at C[a, b] space that is

‖f‖ = ‖f‖∞ = maxx∈[a,b]

|f(x)|; (1)


Best Polynomial Approximation Problem

Suppose given f ∈ C[a, b]. Find p1 ∈ P1 such that

‖f − p1‖ = minx∈P1

‖f − p1‖∞

Such p1 ∈ P1 called best polynomial approximation of f ∈ C[a, b].Because of its property that minimize for every maximum possibilitypolynomial p1 above also called minimax polynomial.


Best Polynomial Problem in C[a, b] Space

Theorem (Existence and Uniquenes)Suppose given f ∈ C[a, b]. There is a pn ∈ Pn such that‖f − pn‖∞ = minq∈Pn ‖f − q‖∞.


Example 1

Suppose given increasing function f ∈ C[a, b]. Find p0 ∈ P0 forf ∈ C[a, b].Let p0(x) ≡ c0, we seek c0 ∈ R such that‖f − p0‖∞ = maxx∈[0,1] |f(x)− c0| minimum.Because f is increasing, f(x)− c0 minimum at x = a andmaximum at x = b, in result |f(x)− c0| maximum at one of endpoint of [a, b] that is

E(c0) = maxx∈[0,1]

|f(x)− c0| = max{|f(a)− c0|, |f(b)− c0|}.


Example 1

Then

E(c0) =

{f(b)− c0 ,if c0 < 1

2(f(a) + f(b)),c0 − f(a) ,if c0 ≥ 1

2(f(a) + f(b)).

c0

E(c0)

f(b)12(f(b) + f(a))f(a)

f(a)

12(f(b)− f(a))

f(b)

f(b)− f(a)


Example 1

The minimax polynomial of f is

p0(x) ≡1

2(f(a) + f(b)



How to construct minimax polynomial p1?Before we answer this problem lets see the case for f nonmonotone, that is f either increasing and decreasing at [a, b].There will always be ξ, η ∈ [a, b] Such that f(ξ) maksimum andf(η) minimum.By the same argument as example 1 easily we can get minimaxpolynomial for f at [a, b] that is

p0(x) ≡1

2(f(η) + f(ξ)), x ∈ [a, b].

Let the Error approximation

f(x)− p0(x) =1

2(f(x)− f(ξ)) + 1

2(f(x)− f(η))

From this we get f(x)− p0(x) positif for x = ξ and negative forx = η.This gives us notion that f(x)− p0(x) has 2 opposites sign at itsmaximum and minimum of f . Is this valid for every case?



Teorema (De la Vallee Poussin)Suppose given f ∈ C[a, b] andr ∈ Pn. Then suppose there is n+ 2point x0 < x1 < . . . < xn+1 in [a, b]. if f(xi)− r(xi) andf(xi+1)− r(xi+1) has opposites sign, for i = 0, 1, 2, . . . , n then

minq∈Pn

‖f − q‖∞ ≥ mini=0,1,...,n+1

|f(xi)− r(xi)|. (2)



Teorema (Chebyshev Oscillation)Suppose given arbitrary f ∈ C[a, b]. A polynomial r ∈ Pn is minimaxpolynomial for f if and only if there is n+ 2 chronological pointsxi, i = 0, 1, 2, . . . , n+ 1 such that a ≤ x0 < x1 < x2 < . . . < xn+1 ≤ b

|f(xi)− r(xi)| = ‖f − r‖∞, i = 0, 1, . . . , n+ 1,

dan

f(xi)− r(xi) = −[f(xi+1)− r(xi+1)], i = 0, 1, 2, . . . , n.


Outline



3 Reference


Method of The Least Parallelogram

DefinitionSuppose given f ∈ C[a, b]. A parallelogram J which is bordered byg1, g2 line and g1 > g2 is called contain f if

g1 − f ≤ g1 − g2, dan f − g2 ≤ g1 − g2.

Definition (The Least Parallelogram)Suppose given f ∈ C[a, b]. A parallelogram J with height tJ = g1 − g2is called the least parallelogram for f if

1 J contain f .2 If there is another parallelogram say J∗ with height tJ∗ which also

contain f then tJ∗ ≤ tJ .


TheoremSuppose given f ∈ C[a, b]. If J is the least parallelogram for f thenthere must be one of border line of J passing through two point of fand another border line of J passing through one point of f .

Proof.Misalkan J adalah jajargenjang terkecil bagi f . Berdasarkanpengamatan geometris, haruslah terdapat dua titik yang terletakpada kurva f yang masing-masing dilewati oleh batas J .Andaikan masing-masing batas J hanya melewati satu titik padakurva f .Dengan mengubah gradien dari batas J akan didapatjajargenjang baru J ′ yang masih memuat f tetapi tinggi J ′ lebihkecil dari tinggi J .Kontradiksi dengan J jajargenjang terkecil.


TheoremSuppose given f ∈ C[a, b]. If J is the least parallelogram of f then themedian line of J is the minimax polynomial for f .

Proof.Misalkan J adalah jajargenjang terkecil bagi f .Perhatikan garis tengah J .Menurut Lema sebelumnya ada tiga titik pada kurva f sedemikiansehingga jarak garis tengah J terhadap f di tiga titik tersebutbernilai sama, dengan selisih dari garis tengah J dan f , dua kaliberubah tanda.Menurut teorema Osilasi Chebyshev garis tengah J adalahpolinom minimaks berderajat satu bagi f .


TheoremSuppose given f ∈ C[a, b]. if p is the minimax polynomial of f with‖f − p‖∞ = E then the parallelogram of J which is bordered by p+ Eand p− E is the least parallelogram of f .

Proof.Misalkan J adalah jajargenjang yang memuat f dengan garisbatas p+ E dan p− E.Misalkan J∗ 6= J adalah jajargenjang lain yang memuat f .Dengan menggunakan Lema dapat ditunjukkan bahwa tinggi J∗

selalu lebih besar dari tinggi J .


TheoremSuppose given f ∈ C[a.b]. A parallelogram J which contain f is calledthe least parallelogram of f if and only if there are two points at f suchthat one of border line of J passing through two point of f and anotherborder line of J passing through one point of f .


Example 2

Misal diberikan f(x) = ex dengan x ∈ [−1, 1]. Akan dicari polinomminimaks p1 ∈ P1 untuk f pada [0, 1].Fungsi f merupakan fungsi monoton naik cekung pada [−1, 1].Gradien garis yang melalui (−1, f(−1)) dan (1, f(1)) adalah 1.175.Kemudian titik d yang memenuhi f ′(d) = 1.175 adalah d = 0.161.


x

f(x)

g1

g2p1

Sehingga Jajargenjang Terkecil yang memuat f dinyatakan olehpersamaan garis

g1 ≡ p1(x) = 1.175x+ 1.543

g2 ≡ p1(x) = 1.175x+ 0.985

dengan tinggi t = 0.557

Dengan demikian didapat p1(x) = 1.175x+ 1.264.


Example 3

Misal diberikan f(x) = x3 pada selang [−1, 1]. Polinom p1 =34x

adalah polinom minimaks bagi f dengan ‖f − p1‖∞ = 14 .

x

f(x)g1

g2

p1


Outline



3 Reference


Reference

D.S. Mitrinovic, J.E. Pcaric, dan A.M. Fink. Classical and New Inequalities in Analysis,Kluwer Academic Publisher, London 1993.

E. Suli, D. Mayers, An Introduction to Numerical Analysis, Cambridge University Press,New York 2003.

E. Kreyszig, Introduction to Functional Analysis an Applications, John Wiley and SonsPress, Canada 1978.

G. Mastroianni, G. V. Milovanovic, Interpoation Processes: Basic Theory and Applications,Springer Monoghraph in Mathematics, Berlin 2008.

G.M.M. Phillips, P. J. Taylor, Theory and Applications of Numerical Analysis, ElsevierScience and Technology Books, Scotland 1996.

I. Kornyei, On Remez Algorithm, Annales Universitatis Scientiarum Budepestinensis deRolando Eotvos Nominatae Sectio Computatorica volume 004, Budapest 1983.

Mason. J. C, Handscomb. D. C, Chebyshev Polynomials, CRC Press, New York 2003.

S. A. Tawfik, Minimax Approximation and Remez Algorithm, Faculty of Engineering CairoUniversity, Egypt 2005.


Terima Kasih


best polynomial approximation

Education

function class

function space f

simpler function

class ofapproximation

set ofcontinuous function

problem2 main resultmethod

maximum norm

class of polynomial