MATH 251 Homework, Fall 2011

Matlab Assignment 1 Due Date: November 4, 2011 Important Instructions • Students are allowed to work in groups of 2 ONLY. • ONLY ONE student from each group should submit the assignment. • Assignments should be uploaded on Moodle. • The name of the zipped folder to be uploaded should consist of the IDs of the group members. Example: 200600000-200600001.zip • The uploaded zipped folder should include one folder for each exercise, named Ex1, Ex2,...containing Matlab files and word documents for test cases. • Extra points would be added for checking special cases in the exercises: (ex: validity of the input) • You are not allowed to use non trivial built-in functions (i.e. functions that do the job for you)! • Your Assignment will be graded ZERO: – In all cases of PLAGIARISM (BOTH PROVIDERS AND CHEATERS). – In case your MATLAB Quiz grade is less then half of the Assignment grade. • If you need assistance pass by MATLAB-CLINIC during the GA’s office hours (or by appointment) 1 Exercise 1 : Conversion Methods 1. Write a MATLAB function: function [E8 , F8] = Convert2to8(E2, F2) which takes as input two binary vectors E2 and F2 that are respectively the integral and fractional parts of a positive binary number b, converts them to octals and outputs the results as 2 vectors E8 and F8 that are respectively the integral and fractional parts of a positive octal number o. 2. Write a MATLAB function: function [E10 F10] = Convert8to10(E8, F8) which takes as input two octal vectors E8 and F8 that represent respectively the integral and fractional parts of a positive octal number o, converts them to base 10 and outputs the results as 2 decimal numbers, E10 and F10 that represent respectively the integral and fractional parts of the positive decimal number d using Nested Polynomial Evaluation. At the end, this function should also display d as a decimal number. 3. Test each one of the 2 functions above for 3 different test cases and save the results in a word document.(consider different lengths for all input vectors). Exercise 2 : Successors and Rounding Procedures Let x = +mx × 10ex be a positive decimal number in F(10, p, −20, +20), written in normalized floating point form, with −20 ≤ ex < +20, and p < 15. 1. Write a MATLAB function : function [my, ey] = GetSuccessor(mx, ex, p) which takes as inputs: • mx : the mantissa of x in standard normalised floating point notation • ex : the exponent of x • p : the precision of the floating point system to which x belongs Let y be the successor of x in F(10, p, −20, +20). This function should output: • my : the mantissa of y displayed with a precision p (the non significant digits of the fractional part need not be displayed) HINT : first compute my, then use num2str(my,p) to output my in the required format • ey : the exponent of y 2. Let m = +m1.m2m3...mp be a positive decimal number whose integral part is m1, and whose fractional part is 0.m2m3...mp. Write a MATLAB function : function [m] = ConvertVectortoDecimal(M) which takes as input a vector M of length p whose ith component is the decimal digit mi, for i = 1, ..., p, and whose output is the decimal number m represented by M. Use ” format long g” to display m in double precision, discarding the non significant zeros of the fractional part . 3. Write a MATLAB function : function [mz, ez] = Round(Mx, ex, n, t) which takes as inputs: • Mx : a vector of length p whose components represent the mantissa mx of the decimal number x ∈ F(10, p, −20, +20) • ex : the exponent of x 2 • n : a positive integer less then or equal to p (n ≤ p) , representing the precision required to reach • t : a parameter taking the values 1 or 2 This function should compute z: the representative of x in F(10, n, −20, +20) by rounding x to the closest if t = 1 or by chopping if t = 2, and output • mz : the mantissa of z displayed with a precision n. HINT : first compute mz, then use num2str(my,n) to output mz in the required format (the non significant zeros of the fractional part will be discarded) • ez: the exponent of z At the end your function should also display z in normalized floating point representation in F(10, n, −20, +20). 4. Test each one of the 2 functions above for 3 different test cases and save the results in a word document. Remark: Call for previous functions when needed. Exercise 3 : Root finding methods The aim of this exercise is to approximate π by computing the root of f(x) = sin(x) in the interval (3, 4), based on Newton’s method. For that purpose: 1. Write 2 MATLAB functions function[sinx]= mysin(x, p) function[cosx]= mycos(x, p) that both input: • a variable x representing some angle in radians • a precision p Using Taylor’s series expansion, these functions should compute respectively the sine and cosine of x, up to some tolerance Tol = 0.5 ∗ 10(−p+1), and output respectively: • the values of sinx or cosx in F(10, p, −20, +20) HINT : first compute sinx and cosx, then use num2str(. , p) Note that: sin(x) = x − x3 3! + x5 5! − x7 7! ... cos(x) = 1 − x2 2! + x4 4! − x6 6! + ... N.B. Do not use the MATLAB built in function for the factorials. Test each of the functions above for x = π/3, π/4 and π/6 with p = 14 and save your results in a word document. 2. Write a MATLAB function [root,k]= myNewton(f, df, a, b, p, kmax) that takes as inputs: 3 • a function f and its derivative df (as function handles) • 2 real numbers a and b, where (a, b) is the interval locating the root of f • a precision p • a maximum number of iterations kmax Based on Newton’s method, this function should output: • root : the approximation to the root of f up to p decimal figures HINT : first compute root, then use num2str(root , p) • k : the number of iterations used to reach the required precision p Tol = 0.5 ∗ 10(−p+1) is the relative error bound to the computed root Test your function for 2 different functions f and save your results in a word document. 3. Write a MATLAB function [ mypi, errpi, k]= mypiNewton( p, kmax) that takes as inputs p and kmax as defined in the previous question. Applying Newton’s method on the interval (3, 4) and using the functions mysin and mycos programmed in part 1, this function should output: • mypi : the approximation to π up to p decimal figures • errpi : the absolute error |mypi − π| where π is considered in double precision • k: the number of iterations used in Newton’s method to reach the precision p HINT: Note that after calling the functions myNewton, mysin and mycos , their outputs should be converted back to numbers using the command str2num(.) Test this function for kmax = 20 and successively for p = 2, 3, 7, 10, 15. Save your numerical results in a word document. 4