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Unformatted text preview: TEST #2A (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 FalsePosition 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 1.6667 2 1.75 1.72727 3 1 .875 1 .73 17 1.6667: The ﬁrst iteration of False Position is the same as the ﬁrst iteration of
Secant (b) Consider the nonlinear 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! l). ((1) Consider the following system of equations. As written, is convergence
guaranteed for an iterative method like J acobi/Gauss Seidel. If not, rewrite the equations so that convergence is guaranteed. § § x1+3x2——6x3=4 )Il/ Bl 41,; _4x1+2x2+x3:6 1+ my? l : x1‘5x3 . . . 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. _ (l/z‘)t P — Poe . ln(P) = ln(PO) + (l/r)t where ln(Po) = intercept = a0
(1/1) = slope = a1 2. Solve using 2 iterations of the secant method and guesses x1=1 and x; = 2 (15 points) f(x)=2x3+x'1/2—4=0 f‘fzmﬂoa 1: 15cm: M + F; J1: 0.707
«CW: 96 Ni :4 Ya: ;_ ‘9»*707<3~“U
9.70%”! 23A 9:“ \X: 3+; [email protected]}3’,gé"/
.ﬂy’zﬁ" MW W533» 15W — 12.707 Seam/ﬂ Meﬁxcxli '
Xm: X‘: ‘3“ (XIX Xk~f>
‘HX’J ~ $(xkﬂ) ‘W.WMWW.W_V“W.__W_a w._m._wwwwwmw...M.u‘.m.ﬂm.wm._v......_.m.,.m “we. 3. Solve using Gauss Elimination. SHOW ALL WORK (15 points)
x2 + 3x3 = 10
x1+2x2 +4x3 =16
2xl + 3x2 = 7 \ ; Limo 9L4; \ ; W/é
OI 3 IO :7 013110
O! 8&5 0 O”5\"\5 4. Solve using ONE iteration of the multidimensional NewtonRaphson method.
Use initial guesses of x = y = 1 (15 points) 2x+21n(y) = —3 5. Write a MATLAB function ﬁle call PolynomialFitm that went sent input vectors
x and y of data, the coefﬁcients of a 2nd 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 “GaussElimm” that when sent a matrix A and vector b, the solution to the system
of equations, x, is returned. Do not use builtin functions max, average, sum, or
MATLAB’s backslash operator. (25 points) function [a0,al,a2] = PolynomialFit(x,y) sumx = 0; sumxy = 0; sumx2 = O; sumx3 = O; sumx4 = 0; sumy = O;
sumx2y = 0; n = length(x); for i = 1:11
sumx = sumx + x(i);_
sumx2 = sumx2 + (x(i))"2;
sumx3 = sumx3 + (x(i))"3;
sumx4 = sumx4 + (x(i))"4; l
sumy = sumy + y(i); sumxy = sumxy + x(i)*y(i);
sumx2y = sumx2y + (x(i)’\2)*y(i); end A(l,1) = n; A(1,2) = sumx;'A(l,3) = sumx2; A(2,l) = sumx; A(2,2) = sumx2;
A(2,3) = sumx3; A(3,1) = sumx2; A(3,2) = sumx3; A(3,3) = sumx4; l' b(1,l) = sumy; b(2,l) = sumxy; b(3,1) = sumx2y; x = GaussElim(A,b) E
a0 = x(l); l
a1 = x(2); a2 = x(3); ...
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 Spring '06
 Klaus
 Triangular matrix, Secant method, sumx

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