(Continuation) Pad´e rational approximation is the best approximation of a function by a...

(Continuation) Pad´e rational approximation is the best approximation of a function by a rational function of a given order. Often it gives a better approximation of the function than truncating its Taylor series, and it may work even when the Taylor series does not converge! Consequently, the Pad´e rational approximations are frequently used in computer calculations such as for the basic function sin x as discussed in Computer Exercise 2.2.17. Rather than using high-order polynomials, we use ratios of low-order polynomials. These are called rational approximations. Let

(Solving systems of linear equations numerically is discussed in Chapter 2.) Finally, we evaluate these m + 1 equations for a₀, a₁, . . . , a_m.

Note that a _j = 0 for j > m and b _j = 0 for j > k. Also, if k = 0, then R_m,0 is a truncated Maclaurin series for f . Moreover, the Pad´e approximations may contain singularities.

a. Determine the rational functions R_1,1(x) and R_2,2(x). Produce and compare computer plots for f (x) = e^x , R_1,1, and R_2,2. Do these low-order rational functions approximate the exponential function e^x satisfactorily within [−1, 1]? Howdo they compare to the truncated Maclaurin polynomials of the preceding problem?

b. Repeat using R_2,2(x) and R_3,1(x) for the function g(x) = ln(1 + x).

Information on the life and work of the French mathematician Herni Eug`ene Pad´e (1863–1953) can be found in Wood [1999]. This reference also has examples and exercises similar to these. Further examples of Pad´e approximation can be seen.