Assignment 11 - University of California, Berkeley Fall...

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University of California, Berkeley Department of Mechanical Engineering Fall Semester 2008 Instructors: M. Frenklach, R. Horowitz E7, Assignment 11 Assigned: Saturday, November 22, 2008 Due: 12:00 pm (noon), Wednesday, December 3 , 2008 This assignment is an introduction to solving nonlinear equations in MATLAB as well as an introduction to probability and statistics. As before, turn in the hard copy of your published file to the drop boxes in Etcheverry 1109 and upload the soft copies of your script and your functions (the M-files) to bspace. Do not forget to name your main M-file as lastname_firstname_SID_lab11.m NOTE: Do not forget to display the contents of your user-defined functions using the command type. MATLAB commands * introduced in this assignment: rand, randn, hist 1. a) Write a MATLAB function called my_bisection that will perform the bisection algorithm (as described in class) to find the real roots of a continuous real mathematical function f(x) . Specifically, the function will have 6 input arguments: fh : a function handle to a prescribed mathematical function x1 : the lower guess x2 : the upper guess tol : the desired tolerance value (a positive small number). tol_type : a character variable that specifies the iteration algorithm’s termination criterion: 1 1 1 || tol_type 'a' tol_type 'r' t o l tol_type 'f' |() |t o l kk k xx x k fx + + + ⇒− = =⇒ = ⇒≤ tol Here x k is the k-th iteration bisection midpoint. max_iter : the maximum number of iterations allowed * Please refer to MATLAB help to learn how to use the functions introduced in this assignment. Assignment 11 E7 1
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University of California, Berkeley Department of Mechanical Engineering Fall Semester 2008 Instructors: M. Frenklach, R. Horowitz Here, max_iter is the maximum number of iterations to perform until the termination criterion specified by tol and tol_type is met. The outputs of your function my_bisection are: root : the final estimate of the root obtained by the algorithm rhist : a vector containing the sequence of midpoints obtained by the method niter : number of iterations performed error : the final error calculated by the method specified by tol_type Here, rhist contains the sequence of midpoints that are calculated by your bisection algorithm. Hence, length(rhist) should return a number equal to niter . If either x1 or x2 are roots, then niter = 0 . K , , 2 1 x x Note that the error at the k- th iteration is a function of the estimates and . Since this poses a problem when we have computed only one estimate ( k = 1), we will define the errors at the first iteration to be = x1 ( the lower guess). x x 1 x x 0 x Test your function my_bisection by finding the roots of the following functions. You should create the functions as anonymous functions in MATLAB. Specifically, you should obtain the roots using the 3 different errors specified by tol_type and compare it to the exact result.
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Assignment 11 - University of California, Berkeley Fall...

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