HW_5___MCEN_3030___Computational_Methods___Fall_2022

MCEN 3030 Computational Methods - HW 5 - Fall 2022 Prof. Robert MacCurdy Submission Deadline: October 14th 2022, Friday @ 11.59pm Submission Guidelines: Follow the homework format and submission instructions in the Syllabus and on the course Canvas site! Read all questions carefully and answer all questions. When in doubt, provide more information rather than less. Question 1 - More Norms For the following vectors and matrices ( NOTE: you may use MATLAB builtin functions for Question 1): V 1 = − 2 . 6 2 . 8 6 . 4 V 2 = 5 . 6 4 . 8 6 . 2 M 1 =   2 . 3 2 . 2 7 . 3 3 . 9 7 . 9 − 3 . 3 7 . 7 3 . 5 − 0 . 5   M 2 =   − 1 . 6 2 . 8 9 . 2 2 . 5 7 . 9 − 1 . 3 6 . 4 3 . 3 − 0 . 5   (a) Calculate the difference between these two vectors, V 1 and V 2 , and quantify the magnitude using the following norm measures. Show these norms in a table and comment on your results. i. The L1 norm. ii. The L2 norm. iii. The Infinity norm. (b) Calculate the difference between these two matrices, M 1 and M 2 , and quantify the magnitude using the following norm measures. Show these norms in a table and comment on your results. i. The L1 norm. ii. The L2 norm. iii. The Frobenius norm. iv. The Infinity norm. Question 2 - Gauss-Seidel Method (a) Write a program to solve an NxN system of equations of the form Ax=b in MATLAB, using your own implementation of the Gauss-Seidel algorithm. Use a stopping criterion such that the answer is correct up to 3 significant digits, and ensure that you use a maximum iterations count in your loop. In your program, you should also include code to check for the Gauss-Seidel convergence criteria and show appropriate outputs based on this check. (b) Use the code you just wrote to compute the solution, and then check your solution by computing the (more precise) solution using the backslash operator and show what the error between the true solution and the Gauss-Seidel method is. ( NOTE: if necessary, you may change the value of a 33 to -10.5) A =     − 10 . 6 2 . 8 1 . 4 − 4 . 2 2 . 7 7 . 9 − 1 . 3 3 . 8 6 . 7 3 . 1 − 1 . 5 − 2 . 9 4 . 4 − 1 . 6 − 0 . 9 7 . 1     b =     3 4 − 2 11     Question 3 - 1D Regression You are given the following data of the population (X) of many different cities (in a country that shall remain nameless), and the number of people from each city who prefer Vegemite (Y) to other spreadable “food” products. 1