29 01 2025 E TS

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Calcolo NumericoA.A. 2023 – 2024 Prof: C. de Falco Exam Numerical Analysis - 2025.01.29 ©2007 – 2025This text is licensed to the public under the Creative Commons Attribution-NonCommercial-NoDerivs2.5 License (http://creativecommons.org/licenses/by-nc- nd/2.5/). Excercise 1 (7 points) Consider the nonlinear system below whose unknown is the vectorx= [x 1,x 2]: { sin (x1) = cos (x2) x2−x1 = 2π a) After reformulating the system in the form f(x) = [f 1(x),f 2(x)]T =0. Write aMatlabfunction implementingfsuch that the input is a column vector of size 2 and the output is also a column vector of size 2. b) Write aMatlabfunction implementing the computation of the Jacobian matrixJ(x) off(x). c) What kind of convergence test is implementet in the providedMatlabfunctionnewtonsys.m? In what circumstances this kind of stopping criterion is reliable? d) Use the providedMatlabfunctionnewtonsys.mto solve the system. Set the tolerancetol=1e-6, the maximum number of iterations to 1000 and the initial guess vectorx (0) = [1/10,1/10 + 2π]T . Repeat the computation with the different initial guessx (0) = [1/10,1/10 + 4π]T . Provide the results in the comments to your code. e) Comment on the results in view of the theory. 1 Excercise 2 (8 points) Consider the following Cauchy problem: { y′= 10 (e−y−1)t∈(0,10] y(0) = 1 (1) a) Using the provided functionheun.m, solve the problem (1) with a steph= 2 5 and with a step h= 1 10. In the comments to the uploaded code, write the value of the numerical solution, obtained with each of the two values ofh, evaluated att= 4. b) Plot the numerical solutions obtained (for both values ofh) and upload the plot in.pngformat c) Knowing that, in the interval of values ofythat contains the solution to (1), the following holds : ∂f ∂y<0; min t,y ∂f ∂y=−10,…