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Unformatted text preview: C v Port: PE. 5;: Curve Fitting and Interpolation 105 Polynomial Approximation to a Function 100 —~ +
80 Suppose y = f (21:) where f (at) is an unknown function. However, Suppose we
have N + 1 pairs of values (56],, yk), k = O, 2, ..., N. An approximation to f(:1:) is
the N m order polynomial, pN(:c) that passes through the N + 1 points. This can
be very useful. For example derivatives or integrals of f can be approximated
by the corresponding derivatives or integrals of pN. Also, pN is an interpolating
function for f. One obvious way to determine the required N + 1 coefﬁcients, (2, for pN is to
write the N + 1 equations: N .
Z 3:20, : yk, k = 0, 1, ..., N
i=0 This is equivalent to the matrix equation, ch = y There is another way to determine an approximating (interpolating) polynomial
that does not require solution of a matrix equation. To introduce it, suppose we
seek the polynomial that passes through just two points ($0,110), (331,3;1). It is
easy to show that the polynomial is given by: p1(w)=<x_$l)yo+($$O)yi $0 " 17,1 331 — C’30 p is the linear combination of two order—1 polynomials L and can be written as: 101 (113) = L1,0($)yo + L1,1($)yi 106 L17! LP?— T he polynomials LN,k(m) are called Lagrange Polynomials. The polynomial rep
resentation can be extended to the case of N + 1 points as: N
PN(€E) = Z LN,kf($k)
k=0 The Lagrange Polynomials, L N,k(x) are polynomials of order N and have the
following properties: 1 forj=k
LN’k(xj) : { 0 forjaék The polynomials that have these properties are: N x _ :1:
LN,k($) = ll 1 j=o,#k wk * 539‘ 107 H L: ,€379)3!'31.§§§£é3‘E§jEE:¥__":%;_Q§§)_,_WW._ L. k W: W:
_ 043*“, J; Q); 3 6,:Etél: If},,.frf_?(1:1)walﬂ w W _ 21’1“”3(373????3:iaﬂ‘é’éﬂi} I V .W. . w» ? . W V . , 108 CW”. «mm. .. ,7“WW7V~~.W_~_F_..._M.._. mm“. Mame1' IL;— Numerical Differentiation Numerical Differentiation is used when: 1. A functional form is so complicated that it is more convenient to do
numerical integration, 2. when we have a table of values of [rich f (3:1)] and we wish to ﬁnd df/drr
for some given value(s) of m. Examples of situations for which derivatives are needed include: . . . . . . _ _ i“; _ Q?
1. Quantities given in terms of derivatives. 1) — dt, u — 635' 2. Mathematical procedures requiring derivatives: 0 A function y = f(ac) is to be approximated by 3) 2 ﬁx) and f
contains constants to be determined which minimize the error in t2he
ﬁt of the function to N points at xi. Error : xiv [ f (331) — f (332)] 0 Finding the roots of y = f (as) In other words, ﬁnd the values of a:
such that f(:c) : 0. Two principal methods for obtaining numerical estimates of f’ ($3) when we
have a set (table) of pairs of values lay, f(:rz) E fi] , i = 1, 2, ..., N are: 1. Develop relatively simple formulae that provide estimates of the deriva—
tive in terms of values of fi and mi, 2. Determine an analytic function g(:c) Which is a good approximation to
f (x) and differentiate g(a:) analytically. We will consider the ﬁrst method here. The second is in the category of
functional estimation or approximation. 109 ...
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 Spring '07
 HenrikSchmidt

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