Economizing a Power Series

• We begin a search for better power series representations of functions by using Chebyshev polynomials to economize a Maclaurin series. This example will give a modification of the Maclaurin series that produces a fifth-degree polynomial whose errors are only slightly greater than those of a sixth-degree Maclaurin series. We start with a Maclaurin series for $e^x$:
  
  $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+\frac{x^6}{720}+\ldots$$
  
  

• if we would like to use a truncated series to approximate $e^x$ on the interval $[0, 1]$ with a precision of $0.00l$, we will have to retain terms through that in $x^6$, because the error after the term in $x^5$ will be more than $1/720$. Suppose we subtract
  
  $$\left(\frac{1}{720}\right)\left(\frac{T_6}{32}\right)$$
  
  
  from the truncated series. We note from Eqn. 1 that this will exactly cancel the $x^6$ term and at the same time make adjustments in other coefficients of the Maclaurin series. Because the maximum value of $T_6$ on the interval $[0, 1]$ is unity, this will change the sum of the truncated series by only
  
  $$\left(\frac{1}{720}\right)\left(\frac{1}{32}\right)<0.00005$$
  
  
  which is small with respect to our required precision of $0.001$. Performing the calculations, we have
  
  $$\begin{array}{ll} e^x \approx & 1+x+\frac{x^2}{2}+\frac{x^3}... ...499219x^2+\frac{x^3}{6}+0.043750x^4+\frac{x^5}{120} \end{array}$$
  
  

• The resulting fifth-degree polynomial approximates $e^x$ on $[0, 1]$ nearly as well as the sixth-degree Maclaurin series: its maximum error (at $x = 1$) is $0.000270$, compared to $0.000226$ for the Maclaurin polynomial. We economize in that we get about the same precision with a lower-degree polynomial.
• By subtracting $\frac{1}{120}\frac{T_5}{16}$ we can economize further, getting a fourth-degree polynomial that is almost as good as the economized fifth-degree one.
• So that we have found a fourth-degree power series that meets an error criterion that requires us to use two additional terms of the original Maclaurin series.
  • Because of the relative ease with which they can be developed, such economized power series are frequently used for approximations to functions
  • are much more efficient than power series of the same degree obtained by truncating a Taylor or Maclaurin series.
• Table 1 compares the errors of these power series.
  • Observe that even the economized polynomial of degree-4 is more accurate than a fifth-degree Maclaurin series
  • Also notice that near $x=0$, the economized polynomials are less accurate
  
  

  Table 1: Comparison of economized series with Maclaurin series.

  
  
  
• We can get the economized series with MATLAB by employing our M-file for the Chebyshev series. We must start with $x$ as a symbolic variable, then get the Maclaurin series and subtract the proper multiple of the Chebyshev series:

  >> syms x
  >> ts=taylor(exp(x),7)
  1+x+l/2*x2+1/6*x^3+1/24*x^4+1/l20*x^5+1/720*x^6
  >> cs=Tch(6);
  >> es=ts-cs/factorial(6)/2^5
  es=23041/23040+x+639/l280*x^2+1/6*x^3+7/160*x^4+l/l20*x^5
  >> vpa(es,7)