Using MATLab for this problem, Given a vector x ? R n with elements such that x_i + x_j 0 for all i and j, the Cauchy matrix C(x) is the n-by-n matrix with entries, C_ij =1/[x_i+x_j]

Write a Matlab function which returns the Cauchy matrix for a random input vector x = rand(n,1). This can be achieved in one line with repmat. For n = 4, 8, 12, 16, do the following. Compute the corresponding C(x).

Define z_exact = [1, 1, . . . , 1]^T = ones(n,1), and then compute b = Cz_exact. Using Matlabs backslash, numerically solve the equation Cz=b, thereby producing a computed solution z_c. In exact arithmetic z_c = z_exact of course, but in IEEE double precision z_c will not equal z_exact. The quantity is the forward error. Construct a table which, for each n, collects the relative forward error, relative backward error, magnification factor, and (infinity-norm) condition number of C