91Ó°ÊÓ

Write an Octave function that implements Horner's method. The first line of your function should be function [p,pprime] \(=\) horner \((\mathrm{x} 0, \mathrm{c})\) where \(c\) is an array containing the coefficients of the polynomial, \(x 0\) is the number at which to evaluate it, \(\mathrm{p}\) is the value of the polynomial at \(\mathrm{x} 0,\) and pprime is the value of the derivative of the polynomial at \(\mathrm{x} 0 .\) This way, the values \(\mathrm{p}\) and pprime are returned by the function in an array. The function is called like this: $$ [\mathrm{y}, \mathrm{yy}]=\text { horner }(-2,[5,4,3,2,1]), $$ setting y to the value of the polynomial and yy to the value of its derivative. Test your code well by comparing outputs of your function to hand/calculator computations.

Use Horner's method to calculate \(g(-2)\) and \(g^{\prime}(-2)\) where \(g(x)=4 x^{4}-5 x^{3}+6 x-7\). Do not use a computer.

Using Horner's method as an aid, and not using your calculator, find the first iteration of Newton's method for the function \(f(x)=2 x^{3}-10 x+1\) using \(x_{0}=2\).

Compute \(a_{0}, a_{1},\) and \(a_{2}\) of Aitken's delta-squared method for the sequence in problem 2 on page \(29 .\) Since the sequence has an undefined term at \(n=1\), start the sequence \(\left\langle\frac{n+1}{n-1}\right\rangle\) with \(n=2 .\) In other words, consider the sequence in problem 2 on page 29 to be \(3,2, \frac{5}{3}, \frac{3}{2}, \frac{7}{5} \ldots\) so \(p_{0}=3, p_{1}=2, p_{2}=\frac{5}{3},\) and so on.

Suppose you are using the secant method with \(x_{0}=1\) and \(x_{1}=1.1\) to find a root of \(f(x)\). (a) Find \(x_{2}\) given that \(f(1)=0.3\) and \(f(1.1)=0.23\). (b) Create a sketch (graph) that illustrates the calculation. HINT: \(x_{2}\) will be located where the line through \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(x_{1}, f\left(x_{1}\right)\right)\) crosses the \(x\) axis.