(^214) 7. Approximations and Inequalities
two functions can be defined in many ways, and we will content ourselves
here with a very brief sampler of approximation theory.
Exercises
- Least squares. Suppose we have a function f whose values are known
for certain points ei within an interval: f (ei) = bi. Typically, bi may
be the experimentally observed values of one variable when another
variable is given the value ei (eg. indices of refraction correspond-
ing to temperatures). It often happens that the points (ei, bi) in the
Cartesian plane fall roughly along a straight line, so that it is rea-
sonable to make a linear approximation p(t) = mt + k to f(t). The
coefficients m and k are to be chosen to make p “fit” as closely as
possible to f at the data points (ai, bi).
The criterion used for least squares approximation is that m and k
should be chosen to minimize
2 Ip(ui) - f (ui)12 = k(mai + k - bi)2
i=l i=l
(a) Consider the particular case, f(0) = 1, f(1) = 3, f(2) = 4. Show
that, according to the criterion, m and k should be chosen to
minimize(k - 1)2 + (m + k - 3)2 + (2m + k - 4)2= 5m2 + 6mk + 3k2 - 22m - 16k + 26.
To carry out the minimization, fix k and complete the square
for the resulting quadratic in m. Deduce a relationship between
m and k for the minimum to occur. Now carry out the same
procedure reversing the roles of m and k which will yield the
minimizing values. Verify that these are m = 312 and k = 716.
Plot the points (0, l), (1,3), (2,4) and the line y = mc+k. Judg-
ing with your eye, do you think the line obtained is reasonable?
(b) Experiment with some other examples.
(c) What happens when r = 2?
(d) What happens when the points (ai, bi) turn out to be collinear?- Alternation. In Exercise 1, we sought a line of closest fit on the basis
of a finite number of values of the function to be approximated. Here,
we consider a different setting in which we actually have an expression
for f(t) for every t in the interval [a, b]. For any approximant p(t) we
let the quantity
maxIIf (t) - p(t)1 : a F t L bl