Reference no: EM133168951

CSCI 4650 Numerical Analysis - University of Colorado

Question 1. Solve the equations x⁵ - x - 1 = 0 and ln(x) = sin(x) using (a) the bisection method (b) the fixed point method (c) Newton's method (both roots have multiplicity 1) (d) the Secant Method and (e) the built in function fsolve in scipy.optimize. Note that the last part will require a bit of independent Googling as it is something that we didn't cover in class. I think this is GREAT practice for the real world. If you do need help on this however please come to office hours of send me an email! For each function include a table listing your iterations and the approximation of the root, except for (e). Also include a very brief comparison of the methods for these problems.

Fun math fact (unrelated to the work you need to do)! One can show, using a field of math called abstract algebra, that although there are equations to find the roots of polynomials of degrees 2, 3, and 4, there can't be one for 5. This is known as the Abel-Ruffini theorem. Practically, this means that the root you find of x5 x 1 can't actually be written down exactly using "common" operations (think roots, addition, division, etc.). Since this is already the case for polynomials, hopefully this gives you a good idea that in general finding roots not computationally can be very hard or even outright impossible.

Question 2. How many iterations are required of the bisection method, starting with an interval of [ a, b] to approximate a root with accuracy at least 10-6?

Question 3. Consider√f (x) = x/2 + 1/x, g(x) = (2x)/3 + 2/(3x), h(x) = (3x)/4 + 1/(2x). Each of these has a fixed point of √2. Which will converge the fastest using the fixed point method?

Question 4. Find the fixed point iteration produced by Newton's Method on f (x) = x³ - A, where A is an an arbitrary number (a parameter). That is, find a general formula for calculating ³√A.

Question 5. What is the multiplicity of the root at 0 of f (x) = x² sin(x²)? Write down the fixed point iteration produced by modified Newton's method for this problem.

Question 6. A fisher wants to set the net at a water depth where the temperate is 10C. By dropping a line with a thermometer attached, they find that the temperature is 8 degrees at a depth of 9 meters, and 15 degrees at a depth of 5 meters. Use the Secant Method to determine a best estimate for the depth at which the temperature is 10.

Extra Credit: The Mandelbrot set is one of the most interesting and beautiful objects in mathematics. Do a bit of research into this set and discuss how it is generated and what topic it relates to from this chapter.