It has to be done in LaTeX and the R code, output, graphics etc must be embedded using Sweave.

Consider the Hilbert matrix Hn of order n whose elements are hij 1 i+j-1 i, j = 1,2,...,n. Consider the vector b with elements b; = D)=1 G/(i+j - 1)), i = 1,2,..., n. Let x be the vector with elements i =i, i = 1,2,...,n. (a) Check that H x = b. (b) For n = 1,2,.. ,20, write a program that solves H, z = b. Use the Gaussian elimination method as done in class (including swapping entries in the index vector instead of actually swapping rows in the matrix). (c) Due to machine errors, the output of your program is x + Ax for some Ax. Let Ab be such that H, (x + Ax) = b + Ab. Create a table with the values of ||Ab||2/|bl|2 and ||Ax||2/||||2 for n = 1,2...,20. Use this to give lower bounds on the condition member of H, n = 1,2,..., 20