2) Write a program that uses Newtons method and double-precision oating point arithmetic to obtain all the zeros of x7 28x6 + 322x5 1,960x4 +6 ,769x3 13,132x2 + 13 ,068x5,040 The exact zeros are 1,2,3,4,5,6, and 7. Starting with initial guesses of 0.9, 1.9, 2.9, 3.9, 4,9, 5.9, and 6.9, and using the termination condition |xi xi1|< 107|xi|, have your main program print a table of values for your solutions and the actual function value to 8 places after the decimal point as well as the number of iterations necessary to obtain that solution. Evaluate the polynomial and its derivative using Horners method.

3. In problem 2, change the coecient of the x2 term to 13,133 and repeat your process (that is, after a change of one unit in the fth place of one coecient is made.) What zeros do you now nd? What is the dierence in the solutions due the this small perturbation of one coecient? Since coecients of polynomials of high degree are often found experimentally, what does that tell you about using this method when there is some doubt about the accuracy of the coecients?

4. A modication of the regula falsi method, called the secant method, retains the use of secants throughout, but may give up the bracketing of the root. Secant method: Given a function f(x) and two pointsx1,x0 for n =0 ,1,2,...,until satised do: calculate xn+1 = f(xn)xn1f(xn1)xn f(xn)f(xn1) The function f(x) = 4sin x ex has a zero on the interval [0, 0.5] Find this zero correct to foursignicant digits using the secant method