Laboratory 1 Solving Non-linear Equations

Due Date: September 19

The goal of this laboratory is to implement the Bisection and Newton's method algorithms for finding the root of a function. In addition, the convergence rate of each algorithm will be compared.

Instructions: Write a brief summary desribing both algorithms and why they work. Encode each algorithm in the programming language of your choice. Your programs should be well documented. The function whose root you are finding should be programmed in as a procedure or subroutine.

The bisection program should be able to receive as inputs (either from the terminal or a file) the left and right end-points of an interval that contains the root along with a tolerance value. Your routine should output an approximate value of the root to within the stated tolerance along with the number of iterations that the bisection algorithm executed.

Your Newton's method program should receive as inputs (again, either from the terminal or a file) the initial approximate root (the x_0) along with a tolerance value. Your program should return an approximate root to within the stated tolerance along with the number of iterations that Newton's algorithm executed.

In addition, do the following.

• Execute both algorithms with the following function f(x)=x^3+x-1 with tolerance 10^{-3} and 10^{-5}. Compare the number of iterations required for each algorithm.
• Use your programs to solve problem 18 on page 67 assuming that the engineer invests $500 per month for 360 months and wishes to retire with $600,000. Use 10^{-4} and 10^{-6} as your tolerances. Again, compare the number of iterations required for each algorithm. In this problem, remember that i in this problem is the interest rate per month and that the interest is compounded monthly.

Click here for a C-code solution.

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