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24 Some Fundamental Numerical Methods
the modified programs with three data pairs to approximate a constant and a
45-degree line for the even and odd cases, respectively.
2.4.5. Piecewise Linear Function Approximation Summary. Arbitrary real
functions of real variables can be expressed as a linear weighted sum of
Chebyshev polynomials of the first kind. The coefficients are determined by
an integral formula, but for piecewise linear functions the Chebyshev coefficients
are found by an algebraic formula. Chebyshev polynomials have many
amazing characteristics, one being the minimax error property (see Hamming,
1973. for a commentary).
A linear (weighted) summation of Chebyshev polynomials is easily restated
as a power series polynomial in real, independent variable x. Since these
approximations are found on the normalized interval - I .; x'; + I, a simple
linear mapping is required in the usual case where given functions are defined
otherwise. ft is not commonly observed, but the approximation described in
this section has close connections with Fourier series approximations, which
are more familiar to electrical engineers. Many other closed-form approximations
are related to the method described (see Ralston, 1965, p. 286).
2.5. Rational Polynomial LSE Approximation 01 Complex Functions
There are many applications in electrical engineering for complex curve
fitting, i.e., finding a complex function of a complex variable such as frequency.
Examples include modeling an antenna impedance versus frequency
for interpolation or for synthesis of an equivalent network; the latter might be
used as a "dummy load" in place of the real antenna. Another example is
approximation of a higher-order-system transfer function by a lower-order one
over a limited frequency range.
A rational polynomial in complex (Laplace) frequency s has more approximating
power than an ordinary polynomial in s and can be an intermediate
step to synthesizing an equivalent network. Such rational polynomials take the
form
(2.47)
where s=jw will be used interchangeably. The relationship between functions
of complex s and real w is rooted in the concept of analytic continuation,
which is described in most network synthesis textbooks (see Van Valkenburg,
1960). Although the method to be presented will generally assume that s= jw,
it also applies to the less general real-variable case S2= _w 2 for approximating
even functions.
The kind of problem to be solved is shown in Figure 2.9; it was given by
E. C Levy (1959), who published the algorithm to be described in this section.