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CS 3200-001 Spring 2024 Instructor: Martin Berzins Duke Nguyen u1445624 February 13, 2024 Assignment #2 – Written Report (Due date : February 11, 2024 by 11:59PM) What to turn in: For the assignments, we expect both SOURCE CODE and a written REPORT be uploaded as a zip or tarball file to Canvas. • Source code for all programs that you write, thoroughly documented. o Include a README file describing how to compile and run your code. • Your report should be in PDF format and should stand on its own. o It should describe the methods used. o It should explain your results and contain figures. o It should also answer any questions asked above. o It should cite any sources used for information, including source code. o It should list all of your collaborators. This homework is due on February 11 th at 11:59 pm. If you don't understand these directions, please feel free to email the TAs or come see one of the TAs or the instructor during office hours well in advance of the due date. 1

Problem 1: Question 5.10 of Cleve Moler’s book (50 points) I initially sort a set of (x, y) data points in ascending order of their x-values and then to create a scatter plot of these sorted points. This line sorts the rows of the matrix yy based on the values in its second column. The matrix yy is assumed to contain data points, with each row representing a point. The second column likely contains the x-values of these points. After this line, yy will be rearranged so that its rows are in ascending order of the x-values. % Sort the data based on x values sortrows(yy, 2); sorted_x = yy(:, 2); sorted_y = yy(:, 1); % Plot the sorted data figure; plot(sorted_x, sorted_y, '*' ); xlabel( 'x' ); ylabel( 'y' ); title( 'Sorted and Plotted Data' ); grid on ; 2 (a) (10 points) As your first experiment, use the code to read this data file and sort it and plot it . You should see something like the figure of sample output below.

The first line sorts the rows of the matrix yy based on the values in its second column. The matrix yy is assumed to contain data points, with each row representing a point. The second column likely contains the x-values of these points. After this line, yy will be rearranged so that its rows are in ascending order of the x-values. The next two lines extract the second and first column of the now-sorted matrix yy and stores it in the variable sorted_x and sorted_y respectively. Those columns represent the x and y-values of the data points, now in sorted order. Use the plot method, a graph is obtained as below: Figure 1.1: A graph plotting the given data after being sorted 3

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Using the formula above, the sorted data points are now scaled in the range [-1,1] by using the below snipper of codes. % Linear scaling for x and y x_min = min(sorted_x); x_max = max(sorted_x); y_min = min(sorted_y); y_max = max(sorted_y); scaled_y = (2 * sorted_y) / (y_max - y_min) - (y_max + y_min) / (y_max - y_min); scaled_x = (2 * sorted_x) / (x_max - x_min) - (x_max + x_min) / (x_max - x_min); Hence, we will obtain an updated version of the graph. Figure 1.2: A graph plotting the same data set after being sorted and scaled 4 (b) (10 points) Use a linear scaling to put both x and y values in the range [-1,1] such as y i = 2 ( y max − y min ) y i − ( y max + y min ) ( y max − y min )

The below code is structured to execute polynomial fitting at varying degrees, specifically for degrees 5, 10, 15, 20, and 25. It caters to a dataset containing 82 points, with the data already scaled to fit within the range of -1 to 1. The process involves creating a Vandermonde matrix for each polynomial degree, which is then used to perform least squares fitting on the scaled data. This fitting process computes the polynomial coefficients that best approximate the scaled data in a least squares sense. For each polynomial degree, it evaluates the fitted polynomial over a collection of 1001 points in range from -1 to 1. This evaluation helps visualize how well the polynomial fits the data across this range. The results are plotted, showing both the original data points and the corresponding polynomial fit, thus providing a clear visual representation of the fitting quality. % Degree of the polynomial m_values = [5, 10, 15, 20, 25]; n = 82; % Number of data points % Evaluate the polynomial at 1001 points xplot = linspace(-1, 1, 1001); % Loop over each degree m for m = m_values 5 (c) (20 points) Write a program that uses the Monomial Vandermode matrix approximation to produce a least squares approximation to work with this data set. In this case n = 82 as there are 82 data points and m is the degree of the polynomial with m + 1 terms. Use values of m equal to 5, 10, 15, 20, 25. Plot the values of the polynomial by evaluating the Vandemonde polynomial at (say) 1001 points and use plots to show how the different polynomials behave. Contrast the visual appearance of the polynomial in each of the five cases m = 5, 10, 15, 20, 25. Use the supplied regression code on the canvas page as a model if you find that helpful. An easier option is to use the matlab functions polyfit and polyval . If you use these you will not be able to estimate the condition numbers of the matrices. Instead MATLAB will tell you that the matrix is ill-conditioned.

% Create Vandermonde matrix for fitting A = ones(n, m+1); for j = 2:m+1 A(:, j) = scaled_x.^(j-1); end % Perform least squares fitting B = A' * A; c = A' * scaled_y; a = B \ c; % Polynomial coefficients % Create Vandermonde matrix for evaluation Aplot = ones(length(xplot), m+1); for j = 2:m+1 Aplot(:, j) = xplot.^(j-1); end % Evaluate the polynomial yFit = Aplot * a; % Plot the data and the polynomial figure; plot(scaled_x, scaled_y, '*' , xplot, yFit, '-' ); xlabel( 'x' ); ylabel( 'y' ); title([ 'Polynomial Fit with m = ' , num2str(m)]); grid on ; legend( 'true values' , 'best fit line' ) NKK end The code block above performs polynomial fitting using the Vandermonde matrix and least squares method for different polynomial degrees (5, 10, 15, 20, and 25) on a set of 82 data points that have been scaled to the range [-1, 1]. For each specified degree, the code constructs a Vandermonde matrix based on the scaled x-values, solves the least squares problem to determine the polynomial coefficients, and then evaluates these polynomials at 1001 evenly spaced points within [-1, 1]. Each resulting polynomial is plotted against the scaled data, allowing for visual comparison of the fit quality across different polynomial degrees. This procedure illustrates how increasing the polynomial degree affects the fit to the data, highlighting the trade-off between fitting the data closely and potentially overfitting, especially with higher degrees. The graphs obtained for each degree of polynomial regression are as follow: 6

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(The graphs will be appeared in the next page) Figure 2.1: A graph plotting the same data using Vandermore Regression at polynomial degree 5. This fit appears to be quite smooth and possibly underfitting the data, as it does not capture all the variations in the dataset. This is expected with lower-degree polynomials, which have fewer coefficients and hence less flexibility to fit complex patterns. 7

Figure 2.2: A graph plotting the same data using Vandermore Regression at polynomial degree 10. The polynomial of degree 10 seems to fit the data better than m = 5, showing more flexibility and better adaptation to the curvature of the data points. It captures more of the data's structure without appearing to overfit. 8

Figure 2.3: A graph plotting the same data using Vandermore Regression at polynomial degree 15. The fit for m = 15 continues the trend of increasing conformity to the data points. The polynomial starts to indicate slight overfitting tendencies, as it begins to exhibit more fluctuations to reach individual data points. 9

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Figure 2.4: A graph plotting the same data using Vandermore Regression at polynomial degree 20. At this degree, the polynomial is fitting very closely to the data points. While the overall trend is followed, there is a visible indication that the polynomial might be capturing noise as well as the underlying data structure. 10

Figure 2.5: A graph plotting the same data using Vandermore Regression at polynomial degree 25. The polynomial of degree 25 shows clear signs of overfitting, particularly towards the ends of the range. The oscillations and deviations from the general trend at the extremities are typical signs of a polynomial that is too complex for the data, which can lead to poor predictive performance on new, unseen data. 11

Add the following code snippet in the for loop in part (c) before end of loop. From here, we can retrieve the least squared error E for each corresponding polynomial degree m . yFit = A * a; % Calculate the least squares error E = sum((scaled_y - yFit).^2); printf( 'Least Squares Error for m = %d: %f\n' , m, E); After running the above code, we are able to get the results as follows: Least Squares Error for m = 5: 0.985991 Least Squares Error for m = 10: 0.125133 Least Squares Error for m = 15: 0.090016 Least Squares Error for m = 20: 0.082654 Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 5.068625e-19. Least Squares Error for m = 25: 0.081382 The least squares errors for polynomial fits with degrees ranging from 5 to 25 show a trend of decreasing error as the polynomial degree increases, suggesting tighter fits with higher degrees. However, the marginal error reduction from a 20th to a 25th-degree polynomial, alongside a warning of numerical instability indicated by a very small RCOND value, points to the diminishing benefits and increasing risks of overfitting and numerical issues with higher- degree polynomials. This implies that while higher degrees can offer marginally better fits, they come with the cost of potential overfitting and unreliable results due to ill-conditioning of the Vandermonde matrix. A polynomial degree between 10 and 15 seems to provide a reasonable balance between fitting accuracy and numerical stability, avoiding the pitfalls of excessive complexity. 12 (d) (10 points) In each of the five cases calculate the least squares error E = ∑ i = 1 82 ( y i − p ( x i ) ) 2 where y i are the data values at points x i and p ( x i ) are the calculated values of the least squares polynomial. Compare the errors with the cases in which MATLAB has problems with the equations as evidenced by a condition number error message. Note that the distance between the least squares polynomial and the data may not change all that much as the polynomial degree is varied from 5 to 25.

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(End of Student’s answer for Problem 1) 13

Problem 2: Truss Structure and Solving Matrix Equation (50 points) 14

15 In this problem, you will build your own bridge and see for yourself why solving systems of equations with accuracy and precision is important. You must build a bridge using this tool that looks like any of the following: Feel free to use your imagination and creativity to build interesting structures. It is highly recommended that you play around with the tool to get comfortable using it. Think about where the forces on the bridge could be acting on. A quick google search on trusses and types of ‘supports’ and ‘forces’ used in bridges will get you up to speed on how to declare them in the simulator. Feel free to experiment and create!

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In this Problem, I am going to use the pre-built truss number 3 for calculation: Figure 3.1: Pre-built Truss Structure #3 16 Once you are done building your bridge, click on ‘Show matrix eqns’ and you should be able to see all matrices which give you information about everything. It’s in the form Ax = b. ‘A’ tells you about how members, supports, and nodes are connected in 2D space. ‘B’ tells you the forces acting on nodes [+/- tell the directions], and the ‘x’ tells you the forces acting on different members, supports, etc. When building real bridges, one can approximate A and B to calculate ‘x’. It is very important that engineers know how much force each member and support experiences so that an appropriate material in the required dimension is used. [In real world, engineers use reasonably good approximations to account for all conditions while declaring both A and b, however realworld conditions can be very hard to quantify accurately.] Now, copy matrix A and b. Assume that you do not know ‘x’. Implement a MATLAB program to compute ‘x’ using any technique of your choice in MATLAB. Write a detailed explanation using the ideas explored in the question to explain your results here and uncertainties that could be involved in them. Please include your constructed bridge pictures in the report as well. Speculate and write about the number of equations that one would have to solve while building a real-world bridge, and the challenges involved in. [Note: if you don’t feel comfortable using the tool and creating your own structure, select pre-built truss number ‘3’ and solve for it.]

Using the pre-built truss, I am able to import a .xls spreadsheet containing two spreadsheets with matrix A and b respectively, both are in the matrix_data.xls . For the coding part, I was using the code snippet below: % Read the input matrix A = readmatrix( 'matrix_data.xls' , 'Sheet' , 'matrixA' , 'Range' , 'A1:R18' ); b = readmatrix( 'matrix_data.xls' , 'Sheet' , 'matrixB' , 'Range' , 'A1:A18' ); % Check if matrix A is square and invertible if rank(A) == min(size(A)) % If A is square and full rank, we can solve the system directly x = A\b; else % If A is not square or not full rank, we need to use a least squares solution x = pinv(A)*b; end % Output the result disp(x); The code snippet solves the linear system Ax = b by first reading matrices A and b from an Excel file using the readmatrix function. It then checks if A is a square and full-rank matrix, which indicates that a unique solution can be found. If A is square and full-rank, the code uses MATLAB's backslash operator \ to solve the system directly, which is an efficient method that leverages matrix factorization techniques. If A is not square or not full-rank, indicating the system is underdetermined or overdetermined, the code computes the pseudoinverse of A with pinv(A) and finds a least squares solution, which minimizes the residuals between the actual and predicted values. The solution x is then displayed using disp(x), providing either the exact or an approximate solution to the system of equations defined by the matrices from the Excel file. In computational mathematics, the precision of numerical calculations is finite, leading to rounding errors. This is a consequence of the finite representation of numbers in computer memory and the operations performed on them. When solving systems of linear equations, as in the case of bridge force analysis, these rounding errors can accumulate, resulting in small discrepancies between expected and actual results. In engineering practice, a slight deviation in the magnitude of computed forces is typically acceptable, as designs incorporate safety factors to account for such uncertainties. The accuracy of solutions to linear systems also depends on the condition number of the matrix A. A matrix with a high condition number is considered ill-conditioned, meaning that 17

small changes in the matrix or the vector b can cause large changes in the solution x. In engineering computations, it is essential to check the condition number to assess the stability and reliability of the solution, especially when the system is sensitive to perturbations. Engineering models, by necessity, simplify the complex behaviors of real-world structures into manageable calculations. These simplifications can include assuming linear material properties, idealized connections, and static loading conditions. Such assumptions are standard in preliminary design stages but can diverge from actual conditions. Real structures may behave non-linearly, have varying material properties, and be subjected to dynamic loads that are not captured in simplified static models. Material properties used in calculations are typically derived from standardized tests and may not precisely reflect the properties of the materials used in construction, which can exhibit variability. The strength, stiffness, and durability of construction materials can differ due to factors like manufacturing inconsistencies, treatment variations, and the impact of environmental conditions. Engineers must consider these variations to ensure that structures can withstand unforeseen stresses over their service life. The loads applied to a bridge in a model, represented by vector b, are estimates based on anticipated usage scenarios. However, actual loads can vary significantly due to factors such as traffic volume, vehicle weights, and environmental conditions. These uncertainties can be mitigated by applying conservative load estimates and incorporating dynamic load analysis to better approximate the range of conditions the bridge will experience. When scaling up from a model to a real-world bridge, the number of equations increases dramatically due to the addition of more members, supports, and load cases. A real bridge is a three-dimensional structure with complex load paths, and its analysis may involve solving a large system of simultaneous equations that capture the interactions between all elements of the bridge under various loading conditions. Finally, the real-world calculation of forces in bridge structures faces numerous challenges beyond the scale and complexity of the equations. These include accounting for non- 18

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linear behavior under extreme loads, time-dependent effects such as creep and shrinkage, and the impact of dynamic and transient forces like wind, earthquakes, and vehicular traffic. Moreover, long-term effects such as corrosion, fatigue, and wear must be factored into the design to ensure durability and safety throughout the bridge's operational lifespan. Advanced modeling techniques such as finite element analysis are employed to address these challenges, coupled with empirical data and safety factors to ensure a robust design. (End of Student’s answer for Problem 2) 19

REFERENCES 1. MATLAB code for reading and processing data from a text file: a. MathWorks. (n.d.). fopen [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/fopen.html b. MathWorks. (n.d.). feof [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/feof.html c. MathWorks. (n.d.). fgetl [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/fgetl.html d. MathWorks. (n.d.). sscanf [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/sscanf.html e. MathWorks. (n.d.). fclose [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/fclose.html 2. MATLAB code for plotting, scaling, and polynomial fitting: a. MathWorks. (n.d.). plot [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/plot.html b. MathWorks. (n.d.). linspace [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/linspace.html c. MathWorks. (n.d.). legend [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/legend.html 3. MATLAB code for matrix operations and solving systems of equations: a. MathWorks. (n.d.). readmatrix [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/readmatrix.html b. MathWorks. (n.d.). rank [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/rank.html c. MathWorks. (n.d.). pinv [Function documentation]. MATLAB Documentation. Retrieved from https://www.mathworks.com/help/matlab/ref/pinv.html 20

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