Understanding MATLAB and Python for Computational Problems

(a) [ 4 marks ] (i) What is the value of after the following commands are executed in MATLAB or Python (with defined to be the relative machine precision for )? (ii) What are the elements of after the following commands are executed in MATLAB or Python (with the appropriate import commands)? MATLAB: Python: (b) [ 3 marks ] A technician claims that the amount of energy used in a chemical reaction (in appropriate units) is and that the measurement was made to decimal places. (i) Give the correctly rounded value for . (ii) Give an estimate of the absolute error in . MATLAB: Python: MATLAB: Python: MATLAB: Python: MATLAB: Python: MATLAB: Python: MATLAB: Python: MATLAB: Python: MATLAB: Python: MATLAB: Python: MATLAB: Python: MATLAB: Python:

(iii) Give an estimate of the relative error in . (c) [ 3 marks ] You are asked to calculate the expression when and is much smaller in magnitude than . (i) Is this expression good or not good for implementation on a computer? (ii) Find a mathematically equivalent, but numerical preferable, expression for . (Select the original expression if it is already the preferable one.) This expression risks a potential catastrophic cancellation. This expression is good for implementation on a computer.

(a) [ 4 marks ] The computational complexities of some common operations with by matrices are given in the table below. Operations Flops Matrix-matrix multiplication Matrix-vector multiplication LU factorization Cholesky factorization Back/forward substitution Tridiagonal solve You have a GHz workstation with cores where each core can do ﬂoating point operations per clock cycle. Estimate how long it will take to solve the by linear system where is symmetric and positive definite and . (b) [ 4 marks ] Estimate the size of the largest by matrix that can be stored in GB RAM using double precision ﬂoating point arithmetic. Assume that GB = bytes. (c) [ 2 marks ] Suppose is an invertible matrix with no special structure and . A programmer claims that the best strategy to solve two linear systems and is to first calculate the inverse and then compute and . Justify or refute this claim. 0.039 seconds 18 minutes 8.9 minutes 7.6E-6 seconds FALSE. Better to compute the factorization of the matrix once, then use forward and back substitutions to solve for . TRUE. Computing the inverse of the matrix is the most efficient way.

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You are given the results of the following MATLAB commands and some related spy plots corresponding to a matrix with real number entries. NOTE: Related (not necessarily the same) Python commands are (a) [ 2 marks ] You are told that has the Cholesky factorization . Tick ALL statements that apply.

(b) [ 2 marks ] Is the matrix positive definite? Tick ALL answers that apply. (c) [ 2 marks ] What are the steps to solve using the Cholesky factorization ? (d) [ 4 marks ] The elements of the coefficient matrix are known exactly and the elements of the right-hand-side vector are known to significant figures. From the above MATLAB results it can be deduced that the -norm condition number of the matrix is . (i) Give an estimate on the relative error in the computed solution to . (ii) How many significant figures do you have in the computed solution? The matrix is lower triangular. The matrix is upper triangular. All the diagonal elements of are positive. All the diagonal elements of are equal to . All the diagonal elements of are nonzero. No, since is not symmetric because . Yes, since is symmetric and its Cholesky factorization exists. Yes, since all eigenvalues of are positive. Not enough information, since we cannot deduce that is symmetric from its spy plot. Yes, since is symmetric and all its eigenvalues are positive. Solve for by back substitution, and then solve for by forward substitution. Solve for by back substitution, and then solve for by forward substitution. Solve for by forward substitution, and then solve for by back substitution. Solve for by forward substitution, and then solve for by back substitution.

You are given the results of the following MATLAB commands and some related spy plots corresponding to a matrix with real number entries. NOTE: Related (not necessarily the same) Python commands are (a) [ 2 marks ] You are told that has an factorization . Tick ALL statements that apply.

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(b) [ 2 marks ] Is an identity matrix this example? Tick ALL answers that apply. (c) [ 2 marks ] What are the steps to solve using the factorization ? (d) [ 4 marks ] From the above MATLAB results it can be deduced that the -norm condition number of the matrix is . You want to guarantee significant figures in the computed solution to . The elements of the coefficient matrix are known exactly. (i) Estimate the relative error in the right-hand-side vector that would be required to guarantee this. (ii) How many significant figures do you need in the right-hand-side vector ? All the diagonal elements of are equal to . The matrix is upper triangular. The matrix is obtained by swapping columns of the identity matrix. The matrix is obtained by swapping rows of the identity matrix. All the diagonal elements of are equal to . The matrix is lower triangular. No, from the spy plot of we see that is not even a diagonal matrix. Yes, is always the identity matrix. Not enough information, since we cannot tell the entries of from its spy plot. No, will never be the identity matrix. Solve for by back substitution, and then solve for by forward substitution. Solve for by forward substitution, and then solve for by back substitution. Solve for by forward substitution, and then solve for by back substitution. Solve for by back substitution, and then solve for by forward substitution. The desired accuracy on is impossible to achieve. The desired accuracy on is impossible to achieve.

You do NOT need MATLAB or Python to answer this question. Consider the data values measured at the times for given in the table below. 1 2 3 4 5 0 0.5 1.0 1.5 2.0 0.2 3.1 4.9 7.2 8.1 The data, which is in column vectors and , produces the approximation obtained with the following MATLAB commands The data and the approximation are plotted in the figure below. NOTE: Related (not necessarily the same) Python commands are (a) [ 2 marks ] What are the values of and for this example? (b) [ 2 marks ] It is claimed that the solution to the linear system is given by , where . Is this correct for this example? 5 3

(c) [ 2 marks ] What do the MATLAB commands above calculate? (d) [ 2 marks ] Give the formula for the approximation obtained. Hint: You should enter a Maple expression such as 1 + 2*x + 3*x^2. Remember to type the multiplication symbol * whenever appropriate. Click on the preview button to double check your answer. Type exp for the exponential function if needed. For this answer box, just use the numbers from the given MATLAB output rather than running your own calculation. (But keep in mind that in a real calculation there are generally more digits than shown by the default MATLAB output format.) (e) [ 2 marks ] For a different matrix , the results of the following MATLAB commands are NOTE: Related (not necessarily the same) Python commands are Explain why ? YES. This gives one of the many possible solutions. NO. The inverse of exists but the solution is not given by . NO. The inverse of does not exist. YES. The inverse of exists and the unique solution is given by . The exact solution to the linear system . The least squares approximation to the data using the exponential model . The least squares approximation to the data using the linear model . The least squares approximation to the data using the quadratic model . 0.22 + 5.98*x -x^2 Since , and is not an identity matrix. This is due to the limited floating point precision.

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You will need to use MATLAB or Python to answer this question. We want to find the interpolating polynomial for the data points below: This problem can be formulated as a linear system , with being a column vector of the data values and with being an -by- matrix depending on the values . The matrix has condition number . Use this information as a consistency check to be sure that you have defined correctly. (a) [ 2 marks ] What are the values of and ? and (b) [ 7 marks ] Provide the approximation you obtained. Give your coefficients to at least 3 significant figures. Enter 0 for terms that do not appear. (c) [ 1 mark ] Use your approximation to estimate the data value at . Give your answer to at least 3 significant figures. 5 5 5.4847307 -7.517792 3.2512343 -0.506354 0.0257812 0 0 1.9796338

You are given the results of the following MATLAB commands and some related spy plots corresponding to a matrix with real number entries. NOTE: Related (not necessarily the same) Python commands are

(a) [ 3 marks ] Is the matrix symmetric? Give reasons for your answers. (b) [ 3 marks ] Is the matrix positive definite? Give reasons for your answers. True False Equation Editor Styles Font Size Since relative error given by symchk = norm(A-A’,1)/norm(A,1) = 5.5511e-16 is small and close to eps, matrix A is symmetric. Words: 20 True False Equation Editor Styles Font Size A matrix is positive definite if it has a Cholesky factorisation . Words: 12

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(c) [ 2 marks ] What are the steps to solve using the Cholesky factorization ? (d) [ 2 marks ] What does the MATLAB command aim to achieve. Is it successful on this example? Equation Editor Styles Font Size Step 1. Cholesky factorisation A = R' R Step 2. Forward substitution - solve R' y for x Step 3. Back substitution - solve Rx = y Words: 27 Equation Editor Styles Font Size MATLAB symrcm (symmetric reverse Cuthill-McKee permutation): reorder columns and rows of a matrix to move non-zero elements closer to the diagonal, to yield a banded matrix, so that the corresponding Cholesky factorisation will be banded too (fill-in occurs within bands) In this example, p = symrcm(A) did work as indicated in the spyplot of A(p,p) since the non-zero elements are now much closer to the diagonal as compared to the non-zero elements in the spyplot of A Words: 77

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