V(0.1) V( 0.5) 0.6 V(0.5) V( 0.5)

Transcription

1 In-class exams are closed book, no calculators, except for one 8.5"x11" page, written in any density (student may bring a magnifier). Students are bound by the University of Florida honor code. Exam papers are likely to have more than one version. Providing a solution to a version different from the one you get is typically evidence of cheating. Fourth in-term exam About a quarter of the problems will be from the assigned homework. You need to be know the following Matlab functions polyfit, polyval, spline, interpl, interp, trapz, quad, diff, gradient, ode45. Sample problems: 1. (Based on problem 15.9) Given the following table of current and voltages i V a. Write Matlab code to fit the all data with a single polynomial and evaluate the voltage for i=0.1. >> i=[ ] i = e e e e e e+000 >> v=[ ] v = e e e e e e+00 >> p=polyfit(i,v,5) p = e e e e e e-014 >> v01=polyval(p,0.1) v01 =.340e+000 b. Calculate by hand the value obtained for the same voltage from linear interpolation. V(0.1) V( 0.5) 0.6 V(0.5) V( 0.5)

2 . Write Matlab code to fit the data in Problem 1using splines with the end slopes specified to be the slopes that correspond to the two data point closest to each end. The slope at the left end is ( )/1=540.5, and it is the same slope at the other end. >> vsp01=spline(i,[540.5 v 540.5],0.1) vsp01 = 3.587e (Based on 17.8) Determine the distance travelled from the following data: t v a. Write Matlab code to do with trapezoidal rule >> t=[ ] t = e e e e e e e+000 Columns 8 through e e e+001 >> v=[ ] v = e e e e e e e+000 Columns 8 through e e e+000 >> dist=trapz(t,v) dist = 6.045e+001 b. Write Matlab code to fit with cubic polynomial and calculate the integral from the polynomial. >> p=polyfit(t,v,3) p = e e e e+000 >> tt=linspace(1,10); >> vv=polyval(p,tt);

3 >> dist3=trapz(tt,vv) dist3 = e Calculate by hand the distance traveled between t=6 and t=8, by Trapezoidal rule, Simpson s rule, and Trapezoidal rule combined with Richardson extrapolation. 1 (6) 4 (7) (8) 1 (8 4 7) Simpson s rule, Eq. 17. d v v v Trapezoidal rule over the entire interval will give us 8+7=15. Trapezoidal rule using the mid-point at 7 will give d 1 v(6) v(7) v(8) 1 (8 1 7) 13.5 Then the Richardson extrapolation, Eq I 4 I( h ) I( h1) Calculate by hand the integral answer. 3 x dx 0 using -point Gaussian integration and compare to exact 1 The two integration points for the interval (-1,1) are Since our interval is translated by , the two points become 0.433, The integrals is then The exact integral is 4 x Write Matlab code to estimate the accelerations in Problem 3. Also, calculate by hand the acceleration at t=6 and t=7 using three-point difference equations. difft = e e e e e e e- 001 Columns 8 through e e-001 >> diffv=diff(v) diffv = e e e e e e e+000

4 Columns 8 through e+000 >> acc=diffv./difft acc = e e e e e e e+000 Columns 8 through e+000 >> accp=[acc,acc(9)] accp = e e e e e e e+000 Columns 8 through e e+000 For t=7, we can use the central difference equation a(7) v(8) v(6) / (6 8.5) / 1.5 For t=6, we can use the 3-point forward difference equation, nd equation in Figure 19.3 v(8) 4 v(7) 3 v(6) a(6) (Based on 0.1) Consider the following initial value problem dy yx y y dx 1.1 (0) 1 a. Perform one step of the explicit Euler method with x 1 y'(0) 1.1, y(1) y(0) xy'(0) b. Perform one step of the implicit Euler method with the same step. y(1) y(0) xy'(1) y(0) 1 y(1) 1.1 y(1) y(0) 0.1 y(1) y(1) y(0) / c. Perform one step of Heun s method with the same step. Can use Eq. 0.17, or reason as follows. With the prediction of explicit Euler, y(1)=-0.1. So the derivative at x=1 is predicted to be dy yx y dx So we will use the average of the this derivative and the derivative at x=0, that is ( )/=

5 Then y(1) dydx = d. Write Matlab code to obtain the solution up to x= using one of the ode routines built into >> [x,y]=ode45(dydx,[0 1],1); >> y(length(y)) ans = e-001 Third in-term exam Students whose last names start with A-R will take the exam in the regular classroom, 0 FLG. Students whose last names begin in S-Z will take it in CSE E-0. The exam covers chapters 8-14, and most of the problems will be taken (though simplified) from the Problems section at the end of the chapters. About a quarter of the problems will be from the assigned homework. You need to be know the Matlab matrix operations in Chapter 8 and the following Matlab functions lu, chol,inv, norm, cond, fminsearch, hist, rand, randn, polyfit, polyval, regress. Sample problems: (Simplified 8.4): Given the matrices A 1, B 3 1, perform all the 3 1 multiplications that can be performed between the two.. (Combined 9. and 9.4): Given the system of equations 4x18x 4 x 6x 34 1 a) Write Matlab code to solve graphically. (b) Solve by Gauss elimination (c) Write Matlab code to solve using Matlab left division. (d) Solve by Cramer s rule 3. (Simplified 10.4): Use LU factorization to solve the following system of equations. Show also how to perform the solution in Matlab using the factorization. x1x 7 6x 5x 19 1

6 (Problem 10.14) Compute the Cholesky factorization of Show Matlab code using the corresponding Matlab functions. 5. Calculate the inverse of the matrix in Problem 4, show also the Matlab command that will do that. 6. (Simplified 11.9) Calculate the Frobenius and infinity norms of the matrix in Problem 3. Show also the Matlab commands for doing that, as well as the Matlab command that will calculate the condition number of the matrix. 7. Do one iteration of the Gauss-Seidel solution of the equations in Problem 3, using the vector (,) as initial guess. 8. (Simplified 1.8) Do one iteration of Newton-Raphson for solution of the following system starting with (1,1) x 5 y y1 x 9. (Modified 13.9) : The concentration of bacteria E. coli in a swimming pool after a storm was measured as function of time and given in the following table. T (hr) C (CFU/100ml) (a) Write the Matlab code to fit a linear polynomial to the data and use it to estimate the concentration at time t=0. (b) Because the concentration of bacteria cannot be negative, a linear fit is not likely to do well to predict future concentrations (it will eventually become negative). Suggest a relation between concentration and time that also has two unknowns parameters and that will not go negative, and write the Matlab code to implement it and use it to estimate the concentration at t= (Simplified 14.10) In Problem 9 assume that there are two varieties of E. coli, each decaying exponentially with time. Write Matlab code to estimate their concentrations at t=0, first with regression, and then by minimizing the residuals using fminsearch. Exam 3 with solution: Second in-term exam All students take the exam in the regular classroom. The exam covers chapters 5-7, and most of the problems will be taken from the Problems section at the end of the chapters. About a quarter of the problems will be from the assigned homework. The Matlab functions that you need to know are fzero, optimset, roots, polyval, fminbnd, contour, fminsearch, and fmincon. The function meshgrid is also useful. Sample problems.

7 1. (Problem 5.6b,c) Determine the root of f(x)=-14-0x+19x -3x 3 in the interval (-1,1) (b) Do one iteration of bisection, and (c) one iteration of false position.. Write Matlab code to find the root using fzero. 3. (Problem 6.3mod). Determine the lowest root of f(x)=x 3-6x +11x-6 (a). Do one iteration of Newton s method using x=0 as starting point; (b) Do one iteration of the secant method using the two points x=-1 and x=0; (c) Write a possible fixed point iteration and do one iteration starting at x=0; (d) write Matlab code to calculate the roots using the roots function; (e) write Matlab code to calculate the lowest root using fzero. 4. (Problem 6.8mod). Find the root of the equation e x =6-10x (a) Write a possible fixed point iteration and assess its convergence potential for x in (0,1); (b) perform one iteration of the fixed point iteration; (c) perform one iteration of Newton s method starting at x=1. Use e= (Problem 7.7mod). Find the maximum of the function f(x)=4x-x +x 3-0.5x 4 known to be in the interval (0,) (a) Indicate at what value of x you will evaluate the function for the golden search method; (b) Using parabolic interpolation with the points x=0, x=1, x=. (c) Do one iteration of Newton s method starting at x=1; (d) write Matlab code to solve the problem using fminbnd. 6. (Problem 7.19mod) Write Matlab code to use fminsearch to calculate the minimum of f(x,y)=y -.5xy-1.75y+1.5x starting at (0,0). Write Matlab code to solve the same problem graphically using contours. Exam and solution. First in-term exam Students whose last names start with A-R will take the exam in the regular classroom, 0 FLG. Students whose last names begin in S-Z will take it in CSE E-0. The exam covers chapters 1-4, and the problems will be taken entirely from the Problems section at the end of the chapters. About a quarter or a third of the problems will be from the assigned homework. Sample problems: 1, Problem.4 is an example of a problem that can be given as is a short problem without changes, because the solution involves a single Matlab statement. Problems.,.11, 3.4, 3.11 are examples of problems that can be given as long problem without changes because the solutions require only five Matlab statements or less per problem. 3. Problem.15b is an example of a problem that can be shortened by asking only as q[4::10]; sum(q); 4. Problem 3.8 is an example of a problem that can be shortened to 8 statements by giving everybody below 80 a C grade. It can be further shortened by giving you the first and last statements. (function grade = lettergrade(score), and end). 5. Problem 4.1 is an example of a problem that can be converted into a short problem by leaving out the first couple of digits. That is what is the base 10 number corresponding to binary. 6. Problem 4.8 is an example of a problem that cannot be given on the exam because it requires a calculator.

8 7. Problem 4.15 is an example that can be simplified so that it can be done without a calculator, by simplifying f(x)=x and changing the step size to h=1. Exam-1 solution.