Question 5 only need to solve with maxima software. no need of simple calculation, only on maxima

{ 24 marks The 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 f1(x) = 3x1 2x32 + 2x2 + 1 = 0; f2(x) = x41 + x21 + 5x2 + 2 = 0; where x = [x1; x2]T and f = [f1; f2]T . (a) By sketching an appropriate graph, either by hand or using Maxima, show that there are two roots of the system. From your graph, estimate values for the roots correct to one decimal place. [3] (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 R = fx : 0:5 6 x1 6 0; 0:5 6 x2 6 0g: [8] (c) Carry out three iterations of the Jacobi method starting with x(0) = [0; 0]T . You should calculate each iterate correct to six decimal places. [3] (d) Use an appropriate Maxima worksheet, starting with x(0) = [0; 0]T , to obtain a solution accurate to six decimal places, by: (i) the Jacobi method; (ii) the Gauss{Seidel method; (iii) the Newton{Raphson method; (iv) the Newton{Raphson method with reuse of the Jacobian when k(r)k 6 0:1; (v) the add-Nx method with N = J(x(0)). Compare the eciencies of the ve methods. [5] (e) Use an appropriate Maxima worksheet and your results from part (a) to obtain the other root, accurate to six decimal places. [3] (f) Suppose that the second equation is rewritten as f2(x) = x41 + x21 + cx2 + 2 = 0; where the parameter c is subject to small errors about the value c = 5. Determine the absolute and relative conditioning, with respect to small changes in c, of the problem of nding the root closest to the origin. [2] Your solutions to parts (d) and (e) should include suitable extracts from Maxima worksheets, showing the problem input and the key results, in addition to an interpretation of the relevant Maxima output. If you use Maxima for parts (a) and (f ), then your solutions to these parts should also include suitable extracts with accompanying interpretations. page 6