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Polynomial Minimax Approximation ofPiecewise Linear Functiott5 23
Table 2.3.. The Data Palrs Derining a
Piecewise Linear Function
x
y
-I
o
o
0.6
+1
0.6
Table 2.4.
Chebyshev and Power Series
Coefficients for n = 4
"0 ~ 0.409014
a,~ 0.30
a,= -0.127324
aJ= O.
a4 = 0.0254648
From Vlach, 1967. ©1967 IEEE.
bo~ 0.561803
b, = 0.30
b 2 = - 0.458366
b,= o.
b 4 = 0.203718
o ,
,
+1
Figure 2.8.
Degree-4 approximation to a three-poinl function. [From Vlach, 1967. © 1967 IEEE.]
Example 2.5. Consider the data given in Table 2.3. Input these into Program
B2-4 and ask for polynomial degrees 4 through 6. The program will display all
seven coefficients for the Chebyshev expansion first. then the degree-4 power
series coefficients. These are shown in Table 2.4, and the graph of either of
these representations is plotted in Figure 2.8. Note that the program also lists
the approximating function values and the errors at each sample point in x.
Program B2-4 documentation also indicates two variations on the coding,
also given by Vlach (1969). These make the data input and calculations more
efficient when strictly even or odd functions of x are approximated. The data
samples must be on the closed (end points included) interval 0" x" + 1. Run