Project 1

1. Project 1 Write a program that will use the existing Fixed Point Algorithm (FPA) and use it to find a root of a function instead of a fixed point. The modification is based on the idea that if h(x) = f(x) + x, a fixed point xp of the function h(x) will be a root of the function f(x). Use the exact same parameters as the fixed point algorithm we did in class except the output will be a root. Project 2 Write a program that takes as input a polynomial p(x) in any form and writes the polynomial in Horner’s form. The idea here is that you can get the coefficient of the term xn by calculating the nth derivative plugging in 0 and dividing by n! (i.e. an = dnp(0)/dxn) Project 3 Write a program that takes as input a square matrix (of any size) and produces as output a picture with all of the Gerschgorin circles to estimate the Eigenvalues. An easy way to calculate radii is to sum up the absolute values of all entries in a row and subtract the absolute value of the diagonal entry.

2. Project 4 Write a program that will use an accelerated 8 trapezoid estimate of an integral. The user inputs a function f(x) and interval [a,b] and the output is a graphic displaying the combined area of the eight trapezoids along with a message saying what its estimate for the integral is. Project 5 Write a program that takes as input a square matrix M of any size. It calculates the Eigenvalues for M and displays this in a graphic as points. With each of these projects not only do you need to produce a program that works. I want you to develop a plan of how you will explain how it works, demonstrate it on tests data, determine any input values for which it will not work (if possible).