Question

Question 524 marks he aim of this question is to assess your understanding of various methods for solving systems of non-linear equations This question concerns the system of non-linear equations f(x) 0 given by i(x)4 62 121 +6 0, f2(x)152 31 -i10 0, where x = [x1,x2)? (a) By sketching an appropriate graph, either by hand or using Maxima, show that there are two roots of the system. Give approximate values for the roots to one decimal place. (b) Write down the Jacobi iteration function g for this system of equations, and show by analysis that the Jacobi method will converge to the root nearest the origin for any initial approximation x(0) in the region

Answer 1

Please see the answer of part (a) in the attached
Part (b) Jacobian of the function g is J = [partialdiffrential f_1/ partialdiffrential x_1 partialdiffrential f_2/ partialdiffrential x_1 partialdiffrential f_1/ partialdiffrential x_2 partialdiffrential f_2/ partialdiffrential x_2] then J = [-12 -12x_1^3 - 2x_1 12x_2^2 - 6 -15]This Jacobian converge to the root x_1 = 0.2657 & x_2 = 0.6609 which lies in the range R = {x: 0 lessthanorequalto x_1 lessthaniorequalto 1, 0 lessthanoreqaualto x_2 lessthanorequalto 1} for this answer we were use the newtons Rathason method ffor two variable.
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