Test2BSolutions - TEST #ZB (November 13, 2008) PGE 310...

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Unformatted text preview: TEST #ZB (November 13, 2008) PGE 310 Name: Solutions Signature: UT EID: NOTES: 1. You have 1 hour and 15 minutes. Points will be deducted for tests turned in after 12: 15 2. Write your name on all pages of the test 3. SHOW ALL WORK. Partial (or any credit) cannot be given for problems without work shown . Qualitative Questions (30 points) (a) I want to solve some nonlinear equation f(x) = 0 using x=1 and x=2 as guess values. If the first 3 iterations of the Bisection and False-Position Method are given below, what would I get for “x” in the FIRST iteration of the Secant method? , Iteration x_new (Bisection) x_new (False Position) 1 1.5 1.6914 2 1.75 1.7272 g 3 ' 1.875 1.7317 1.6914: The first iteration of False Position is the same as the first iteration of Secant (b) Consider the non-linear function below; f(x)=0. Using an initial guess of x = -0.5, On the curve, show GRAPHICALLY the first step of the Newton- Raphson method. X (c) I have to solve a system of equations Ax=b several times (maybe hundreds). The matrix A never changes, but the vector b changes every time. Should I use Gauss Elimination or LU decomposition? Briefly explain (and be specific) LU decomposition. If you have already created L and U in LU decomposition, you can backward substitute right away for d and then x. If you use Gauss Elimination, you will have to create an upper triangular matrix many times and thus increasing your computation (by a very large amount if you have a . large A! I). (d) Consider the following system of equations. As written, is convergence guaranteed for an iterative method like I acobi/ Gauss Seidel. If not, rewrite the equations so that convergence is guaranteed. My} BMW ‘5x1 +2352 +353 :15 1+ {5 not dé‘afioaaly (imminent x—6x +3x =9 WWW/w whit/us; 1/ mmmw x1 +3x2 —8x3 =7 (e) Briefly explain the disadvantage of measuring error in curve fitting by the sum of the absolute values of the residuals as opposed to the“sum of the squares” of the residuals. By summing the absolute value of the errors, you may have multiple solutions for a data set. (i) Experimental data is gathered for the growth of certain bacteria. A theoretical model for bacteria grth is given by the equation below. Rewrite in such a way that linear least squares regression can be used. _ (1/1): P — Poe 1n(P) = 1n(Po) + (l/r)t where ln(Po) = intercept = a0 (l/r) = slope = a1 2. Solve using 2 iterations of the secant method and guesses x1=1 and xz = 2 ( 15 points) f(x)==2x2 +336“2 —-6=O Seam-f Me—HWA: XkH: XK- ((ka " (UM) Iferanio/x '3 $<x25:;(4)+_§,g:ma H flaw—am 3-6?! x = 9- ”W"W“‘>x\ HS 3 943w(fl) [::::] 3137 XL! : Hqg_(‘.3‘137)(\.l‘15*<l3 :Eflg7flfifla : LQQG‘ o 1 3. Solve using Gauss Elimination. SHOW ALL WORK (15 points) x2+3x3 =11 x1+2x2 +4x3 =17 23c1+3x2 =8 1 3 XI; 56¢?ka Rid—’Rza a ‘1 X .2: I7 :7 ' 3 X; b C) 4. Solve using ONE iteration of the multidimensional Newton-Raphson method. Use initial guesses of x = y = 1 (15 points) 3x+ln(y)=—3 1533M My +3 y+2y3=—4 g: wag»: ‘7’ 5. Write a MATLAB function file call PolynomialFit.m that went sent input vectors x and y of data, the coefficients of a 2"d order polynomial that best fits the data, a0, a1, and a2, are returned Assume that there already exists another function file in your directory called “GaussElim.m” that when sent a matrix A and vector b, the solution to the system of equations, x, is returned. Do not use built-in functions max, average, sum, or MATLAB’s backslash operator. (25 points) function [a0,a1,a2] = PolynomialFit(x,y) sumx = 0; sumxy = 0; sumx2 = 0; sumx3 = 0; sumx4 = O; sumy = 0; sumx2y = 0; n = length(x); for i = 1m sumx = sumx + x(i); sumx2 = sumx2 + (x(i))’\2; sumx3 = sumx3 + (x(i))"3; sumx4 = sumx4 + (x(i))"4; sumy = sumy + y(i); sumxy = sumxy + x(i)*y(i); sumx2y = sumx2y + (x(i)"2)*y(i); end A(1,l) = n; A(1,2) = sumx; A(1,3) = sumx2; A(2,1) = sumx; A(2,2) = sumx2; A(2,3) = sumx3; A(3,1) = sumx2; A(3,2) = sumx3; A(3,3) = sumx4; b(1,l) = sumy; b(2,1) = sumxy; b(3,l) = sumx2y; x = GaussElim(A,b) a0 = x(l); a1 = x(2); a2 = x(3); ...
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Test2BSolutions - TEST #ZB (November 13, 2008) PGE 310...

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