Lecture Notes Course à MA 190 Numerical Mathematics, First ...

Chapter 6On interpolation withpolynomials6.1 Spaces of polynomials6.1.1 On polynomials as approximating functionsIn this chapter we will discuss some linear spaces often encountered in numericalwork. As written earlier, we need to approximate a given function f with anexpression g, which may be evaluated with a finite number of arithmetic operationsand logical choices. Sometimes we want to perform further operations, likeintegration or derivation on g and it is an advantage, if these latter operationscan be carried out easily, or even analytically. As an example, assume that wehave constructed g such thatf(t) ≈ g(t), −1 ≤ t ≤ 1 .We seek an approximation for the integral of f and make the approximation∫ 1−1f(t) dt ≈∫ 1−1g(t) dt.Hence it is desirable that the integral on the right hand side may be evaluatedmore easily than that on the left hand side. A common choice it to construct apolynomial g to approximate f. In the sequel we will present some systematicmethods for doing this. We will need to describe some useful properties ofpolynomials. We begin withExample 6.1.1 Let E 5 be the linear space of polynomials of degree < 5. Wewant to give some examples of bases to this space which has the dimension 5.Let now {t 1 , t 2 , t 3 , t 4 , t 5 } be 5 distinct real points such thatt 1 < t 2 < t 3 < t 4 < t 5 .40