91Ó°ÊÓ

Use the Fast Fourier Transform Algorithm to compute the trigonometric interpolating polynomial of degree 4 on \([-\pi, \pi]\) for the following functions. a. \(f(x)=\pi(x-\pi)\) b. \(\quad f(x)=|x|\) c. \(\quad f(x)=\cos \pi x-2 \sin \pi x\) d. \(\quad f(x)=x \cos x^{2}+e^{x} \cos e^{x}\)

Find the least squares polynomials of degrees 1,2 , and 3 for the data in the following table. Compute the error \(E\) in each case. Graph the data and the polynomials.+ $$ \begin{array}{lllllll} \hline x_{i} & 1.0 & 1.1 & 1.3 & 1.5 & 1.9 & 2.1 \\ y_{i} & 1.84 & 1.96 & 2.21 & 2.45 & 2.94 & 3.18 \\ \hline \end{array} $$

Find the continuous least squares trigonometric polynomial \(S_{3}(x)\) for \(f(x)=e^{x}\) on \([-\pi, \pi]\).

Find the general continuous least squares trigonometric polynomial \(S_{n}(x)\) for \(f(x)=e^{x}\) on \([-\pi, \pi]\).

a. Determine the trigonometric interpolating polynomial \(S_{4}(x)\) of degree 4 for \(f(x)=x^{2} \sin x\) on the interval \([0,1]\). b. Compute \(\int_{0}^{1} S_{4}(x) d x\). c. Compare the integral in part (b) to \(\int_{0}^{1} x^{2} \sin x d x\).

Find the least squares polynomials of degrees 1,2, and 3 for the data in the following table. Compute the error \(E\) in each case. Graph the data and the polynomials. $$ \begin{array}{lllllll} \hline x_{i} & 0 & 0.15 & 0.31 & 0.5 & 0.6 & 0.75 \\ y_{i} & 1.0 & 1.004 & 1.031 & 1.117 & 1.223 & 1.422 \\ \hline \end{array} $$

Determine the Padé approximation of degree 6 with \(n=m=3\) for \(f(x)=\sin x\). Compare the results at \(x_{i}=0.1 i\), for \(i=0,1, \ldots, 5\), with the exact results and with the results of the sixth Maclaurin polynomial.

Given the data: $$\begin{array}{llllllllll} \hline x_{i} & 4.0 & 4.2 & 4.5 & 4.7 & 5.1 & 5.5 & 5.9 & 6.3 & 6.8 & 7.1 \\ y_{i} & 102.56 & 113.18 & 130.11 & 142.05 & 167.53 & 195.14 & 224.87 & 256.73 & 299.50 & 326.72 \\ \hline \end{array}$$ a. Construct the least squares polynomial of degree 1, and compute the error. b. Construct the least squares polynomial of degree 2, and compute the error. c. Construct the least squares polynomial of degree 3, and compute the error. d. Construct the least squares approximation of the form \(b e^{a x}\), and compute the error. e. Construct the least squares approximation of the form \(b x^{a}\), and compute the error.

Find the general continuous least squares trigonometric polynomial \(S_{n}(x)\) for $$ f(x)= \begin{cases}0, & \text { if }-\pi

Use the zeros of \(\tilde{T}_{3}\) and transformations of the given interval to construct an interpolating polynomial of degree 2 for the following functions. a. \(\quad f(x)=\frac{1}{x}, \quad[1,3]\) b. \(\quad f(x)=e^{-x}, \quad[0,2]\) c. \(\quad f(x)=\frac{1}{2} \cos x+\frac{1}{3} \sin 2 x, \quad[0,1]\) d. \(\quad f(x)=x \ln x, \quad[1,3]\)