Chapter 3
Polynomial or Rational
Approximations
Using a finite number of additions, subtractions, multiplications, and compar-
isons, the only functions of one variable that one can compute are piecewise
polynomials. If we add division to the set of available operations, the only func-
tions one can compute are piecewise rational functions. Therefore it is natural to try
to approximate the elementary functions by polynomials or rational functions.
The questions that immediately spring to mind are:
• How can we compute such polynomial or rational approximations?
• What is the best way (in terms of accuracy and/or speed) to evaluate a
polynomial or a rational function?
• The final error will be the sum of two errors: the approximation error (i.e., the
“distance” between the function being approximated and the polynomial
or rational function), and the evaluation error due to the fact that the poly-
nomial or rational function are evaluated in finite precision floating-point
arithmetic. Can we compute tight bounds on these errors?
Throughout this chapter we denote by P
n
the set of the polynomials of
degree less than or equal to n with real coefficients, and by R
p,q
the set of the
rational functions with real coefficients whose numerator and denominator have
degrees less than or equal to p and q, respectively.
Let us focus first on the problem of building polynomial approximations.
Of course, it is crucial to compute the coefficients of such approximations using
a precision significantly higher than the “target precision” (i.e., the precision of
the final result). We want to approximate a function f by an element p
∗
of P
n
on
an interval [a, b]. The methods presented in this chapter can be applied to any
continuous function f (they are not limited to the elementary functions). Two
kinds of approximations are considered here: the approximations that minimize
the “average error,” called least squares approximations, and the approximations
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