Homework Set 6

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University of Florida *

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3344

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Mechanical Engineering

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Oct 30, 2023

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Problem Set 6 Problems Completed: 11.1, 11.6, 11.9, 11.13, 12.2, 12.8, 12.15

Problem 11.1 Determine the matrix inverse for the following system: 10 x 1 + 2 x 2 − x 3 = − 27 − 3 x 1 − 6 x 2 +2 x 3 = − 61.5 x 1 + x 2 +5 x 3 = − 21.5 Check your results by verifying that [ A ][ A ] -1 = [ I ] . Do not use a pivoting strategy. Solution: ▪ MATLAB Code: see file: EGM3344_HW6_1.m ▪ Output:

Problem 11.6 Determine || A || f , || A || 1 , || A || ∞ for 8 2 10 A 1 3 15 1 6 9 −     =     −   − Before determining the norms, scale the matrix by making the maximum element in each row equal to one. Solution: Norms: 1/9920, 2.8000, 2

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Problem 11.9 Besides the Hilbert matrix, there are other matrices that are inherently ill-conditioned. One such case is the Vandermonde matrix, which has the following form: 2 1 1 2 2 2 2 3 3 1 A 1 1 x x x x x x     =       (a) Determine the condition number based on the row-sum norm for the case where x 1 = 4 , x 2 = 2 , and x 3 = 7. (b) Use MATLAB to compute the spectral and Frobenius condition numbers. Solution: (a) Cond[A] = 323

(b) ▪ MATLAB Code: see file: EGM3344_HW6_119.m ▪ Output:

Problem 11.13 (a) Determine the matrix inverse and condition number for the following matrix: 1 2 3 A 4 5 6 7 8 9     =       (b) Repeat (a) but change a 33 slightly to 9.1. Solution: (a) ▪ MATLAB Code: see file: EGM3344_HW6_3a.m ▪ Output: (b) ▪ MATLAB Code: see file: EGM3344_HW6_3b.m ▪ Output:

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Problem 12.2 (a) Use the Gauss-Seidel method to solve the following system until the percent relative error falls below εs = 5%: (b) Repeat (a) but use overrelaxation with λ = 1 . 2 . Solution: (a) ▪ MATLAB Code: see file: EGM3344_HW6_122a.m ▪ Output: after 6 iterations, (b) ▪ MATLAB Code: see file: EGM3344_HW6_122b.m ▪ Output: after 4 iterations,

Problem 12.8 Determine the solution of the simultaneous nonlinear equations 2 0.75 x y x + + = − 2 5 y xy x + = Use the Newton-Raphson method and employ initial guesses of x = y = 1.2. Solution: ▪ MATLAB Code: see file: EGM3344_HW6_128.m ▪ Output:

Problem 12.15 Determine the roots of the following simultaneous nonlinear equations using (a) fixed-point iteration, (b) the Newton-Raphson method, and (c) the fsolve function: y = − x 2 + x + 0.75 y + 5 xy = x 2 Employ initial guesses of x = y = 1.2 and discuss the results. Solution: ▪ MATLAB Code: see file: EGM3344_HW6_1215.m ▪ Output:

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