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E7 – Prof. Alam Fall 2019, UC Berkeley Homework Assignment 11 Interpolation, Root Finding, Numerical Differentiation and Integration This assignment will introduce you to interpolation, root finding and numerical differentia- tion and integration in MATLAB. The due time is 11:59am 4 Dec 2019. NO LATE HOMEWORK WILL BE AC- CEPTED. Please upload the following files through the bCourses website: • The function files myInterp.m, interpthermo.m, refineimage.m, damping.m MyDer.m and mySimpson.m. Directions to upload files can be found here. 1 Interpolating Data from an Experiment You are doing an experiment that finds the relation between two quantities t and d. t is an independent variable which you vary and you measure d for each such value of t in order to obtain some readings. In this example, your independent variable t is stored in the variable t and your dependent variable displacement is stored in the variable d. Now that you have a set of t and d data points, you want to interpolate a curve to fit the data. Write a function myInterp with the function declaration line func t i on D = myInterp ( t , d ,T, opt ion ) The inputs are: • t: an n-by-1 column vector containing the t data points. • d: an n-by-1 column vector containing the d data points. • T: an m-by-1 column vector containing the t instances for which an interpolation is desired. In general, m ≥ n. 1 https://guides.instructure.com/m/4212/l/54353-how-do-i-upload-a-file-to-my-assignment-submission E7 – Prof. Alam Fall 2019, UC Berkeley • option: a string, containing either ’nearest’, ’linear’, ’cubic’, ’regression’, or ’cubicpoly’. The output is: • D: an m-by-1 column vector containing the interpolated d data. Use the MATLAB built-in function interp1 to do the interpolation for the options ’nearest’, ’linear’, and ’cubic’. Additionally, if the content of option is invalid, generate an error ’Option selection is incorrect.’ using the error function. For the options ’regression’ and ’cubicpoly’, perform regression of the data values on a straight line and a cubic polynomial respectively. That means, you should solve for the coefficients of the line by representing the problem in the Ax = b form, and evaluate T on the equation representing the line. Use the backslash operator to solve for the coefficients. You can check the following cases to test your code: >> t = [ 0 6 9 21 36 42 57 6 0 ] ’ ; >> d = [ 1 .0 1 .2214 1 .3499 1 .4235 1 .4567 1 .5217 1 .6063 1 . 6 7 0 8 ] ’ ; >> D1 = myInterp ( t , d , l i n s p a c e (0 , 60 , 100 ) ’ , ’ nearest ’ ) ; >> D2 = myInterp ( t , d , l i n s p a c e (0 , 60 , 100 ) ’ , ’ l i n e a r ’ ) ; >> D3 = myInterp ( t , d , l i n s p a c e (0 , 60 , 100 ) ’ , ’ cubic ’ ) ; >> D3 = myInterp ( t , d , l i n s p a c e (0 , 60 , 100 ) ’ , ’ r e g r e s s i o n ’ ) ; >> D3 = myInterp ( t , d , l i n s p a c e (0 , 60 , 100 ) ’ , ’ cubicpoly ’ ) ; You can generate a graph of your data points (t versus d)) as well as the interpolated line (T versus D) to visually check the output of your function. 2 Thermodynamic Property Tables Consider a thermodynamic system that operates with superheated water vapour. In analysis, it is common to refer to tables that contain properties of the operating fluid. In this problem, you are going to work with a cell array that stores data for internal energy as a function of temperature. You are going to use linear interpolation to find internal energy values for temperatures located in between those in the table. (a) Create a function interpthermo which takes in two input arguments T1 and newT. T1 is an input cell array of properties. newT is a double array of new temperatures. Your function should call a subfunction interpthermo2 with the appropriate arguments and 2 E7 – Prof. Alam Fall 2019, UC Berkeley create a new cell array T2 which contains properties at new temperatures in addition to those already in T1. (b) Create a subfunction interpthermo2 which takes in two input arguments T1 and newT. The input T1 is a cell array of properties stored in ascending order of temperatures and newT is a temperature which does not exist in the cell array T1. The output argument is T2, a cell array with the new interpolated internal energy for temperature newT with the same names for properties, also stored in ascending order of temperatures. You should make use of the function interp1 in your computations. You can use the following example to check your code: >> T = {80 2 4 8 7 . 3 ; 120 2 5 4 4 . 7 ; 160 2 6 0 2 . 7 ; 200 2 6 6 1 . 4 ; 240 2721} 5 x 2 c e l l array T = 80 2487 .3 120 2544 .7 160 2602 .7 200 2661 .4 240 2721 >> Tnew = interpthermo (T, [ 9 0 123 185 215 ] ) ans = 9 x 2 c e l l array 80 2487.3 90 2501.7 120 2544 .7 123 2549 160 2602 .7 185 2639 .4 200 2661 .4 215 2683 .8 240 2721 3 E7 – Prof. Alam Fall 2019, UC Berkeley 3 Image Interpolation In this problem you will use interpolation to refine a image. From bcourses, you can download imdata.mat (class double), which is a 2-D array data storing a image that corresponds to a sample image. You can show this color image with the following code load imdata . mat f i g u r e imagesc ( data ) If you want to display the image in black, white, and gray, just add the following command. colormap gray Write a function with the following header: f unc t i on re f ined imagedata = re f i n e image ( imagedata ) where input is the name of the image data file (string), and output is refined image data, which is generated by halving the grid size of the image. In the function, use built in function interp2 to create interpolated image data. Use the ”spline” method to do the 2-D interpolation. You can use the commands below to check your code: load imdata . mat f i g u r e (1 ) imagesc ( data ) colormap gray a x i s image a x i s o f f r e f ined imagedata = re f i n e image ( data ) f i g u r e (2 ) imagesc ( re f ined imagedata ) ; colormap gray shading i n t e r p a x i s image a x i s o f f 4 Root Finding The spring shock system in automobiles is an example of a real mechanical system which involves the deflection of nonlinear springs that would demonstrate a damping effect over 4 E7 – Prof. Alam Fall 2019, UC Berkeley time (see behavior response in Figure 1 below). The resistance force of the spring F (N), with respect to time t (s), is given by the following equation F (t) = k1(cos (πt))(e − t k2π ) where k1 and k2 are defined as spring constants (N/s) Figure 1: Damping effect with spring constants k1 = 1000 N/s and k2 = 0.6 We will use the above equation to describe the force on a car’s shock system following a speed bump encountered at t = 0 seconds. We want to find out a time instance near t = 5.5 seconds when the resistance force F is equal to zero. In order to do that, we need to find the roots of F (t). You will find the roots using three different methods. In part (a), you will use the MATLAB built-in function fzero. In part (b), you will use the Newton-Raphson method. In part (c), you will use the bisection method. Write a function damping with the function declaration line func t i on [ S ] = damping ( k1 , k2 , i n i t i a l ) The inputs are: • k1, k2: the spring constants defined above. • initial: the time instance of interest, near which we want to find a root. 5 E7 – Prof. Alam Fall 2019, UC Berkeley The output is: • S: a 1-by-1 structure array. Its fieldnames and content will be described in detail in the following subproblems. In your function, represent F (t) as a function handle stored in variable F. Also, create a variable tol = 1e-6 which represents the value of the tolerance required for the bisection method and the Newton-Raphson method. Note that the tolerance is defined such that if the absolute value of the difference between the actual function value F(t) and the expected function value (0, in this case) is ≤ tol, then the algorithm should be stopped. (a) Use fzero to determine the root of F with initial as the guess location. Store the root to S in the fieldname fzero root. (b) Within damping.m, write a subfunction myNewton with function declaration line func t i on newton root = myNewton( f , df , x0 , t o l ) which will recursively calculate the root nearest to x0 given the function handle f, the function handle of the derivative of f (which you need to calculate on your own) df, the initial guess x0, and the tolerance tol. Store the root found using Newton-Raphson method to S in the fieldname newton. (c) Within damping.m, write a subfunction bisection with function declaration line func t i on b i s e c t r o o t = b i s e c t i o n ( f , a , b , t o l ) which will recursively calculate the root located between a and b given the function handle f, the endpoints of the range a and b, and the tolerance tol. Use a range of ±0.5 (make sure to check the signs of the function evaluated at the endpoints) and store the root found using bisection method to S in the fieldname bisect root. The expected