Numerical Analysis Project 1: [Root Finding Methods] The goal is to find the points at which ex/5 = sin(x) on the interval [0, 10]. 1. [Visual Inspection] Use matlab to plot the functions ex/5 and sin(x). From this make your first approximations to the points where they are equal. 2. [Bisection Method] Write a short matlab program which will implement the bisection method, given initial intervals [ak , bk ]. Use the results from Problem 1 to implement the program, and find the approximations to the points where f(x) = ex/5 sin(x) = 0. Let the midpoint method run until f(x) < 107 . Count the number of steps that the midpoint method takes for each point. 3. [Newton's Method] Write a short matlab program which will implement Newton's method, given an initial starting point, such as bk ak from above. Let the midpoint method run until f(x) < 107 . Count the number of steps that the Newton's method takes for each point. Compare the speed to Newton's method to that of the midpoint method. 4. [Newton's Method Part 2] a) Use Newton's method to compute the solution to (x3)4 sin(x) using x0 = 2. Note its convergence rate. b) Use the altered Newton's method to compute this and note it's convergence rate. [Interpolation and Approximation Methods] Compare and Contrast the following Interpolation/Approximation Methods. 4. [Lagrange Interpolation] Interpolate f(x) = 1 1 + x2 at evenly spaced points on the interval [5, 5], with Lagrange polynomials of order n = 5, 10, 20. Does the approximation get better? 5. [Piecewise Linear Interpolation] Interpolate f(x) = 1 1 + x2 at evenly spaced points on the interval [5, 5] using piecewise linear interpolation with the same points as in Problem 4. Does the approximation get better with more points? 6. [Raised Cosine Interpolation] Interpolate f(x) = 1 1 + x2 at evenly spaced points on the interval [5, 5] using a raised cosine basis function and the same points as in Problem 4. Does the approximation get better with more points? Does this seem better than the result from Problem 4 and 5? 7. [Least Squares Approximation] Approximate f(x) = 1 1 + x2 using least squares approximation with polynomials of order n = 5, 10, 20 on [5, 5]. Does the approximation get better higher order polynomials? Does this seem better than the result from Problem 4, 5, and 6? 1 Numerical Analysis Project 1: Bonus 5 pts 1. Square Root Calculator: Write a program which will eventually converge at a quadratic rate and calculate the square root of any number, using only addition, multiplication, and division. Test this program on numbers from 10-10000. Explore options for picking a starting point... perhaps for Newton's method to finish. Give a succinct and detailed report of your explorations. Demonstrate that it converges at a quadratic rate. 2. Altered Newton's Method: Write a program which will alter Newton's method when a multiple zero is involved. Use the examples of (x4)2 sin(x) and (x4)3 sin(x) as test cases. The algorithm should be able to identify the level of the zero ( p = 2 or 3 in the previous examples ), and alter its approach for a quadratic convergence rate. Demonstrate that it converges quadratically in both cases. 2