Write a program implementing the Lanczos method. Test your method

on a random symmetric matrix of order $n$ having eigenvalues $1,2,$dots,$n.$ To generate

such a matrix, first generate a random $nn$ matrix $B$ with random entries uniformly

distributed in $[0,1).$ Let the $B=QR$ be the $QR$ factorization of $B.$ Now take

$A=QD{Q}^T,$ where $D$ is a diagonal matrix with diagonal entries $1,2,$dots,$n.$ The

Lanczos algorithm needs to find eigenvalues of a tridiagonal matrix $-$ use matlab library

routine to find these eigenvalues. For the purpose of this exercise, run the Lanczos

iteration for a full $n$ iterations.

To see graphically how the Ritz values behave, construct a plot with the iteration

number on the vertical axis and the Ritz value at each iteration on the horizontal axis.

Plot each pair $(,k)$ where $$ is a Ritz value in iteration $k$ as a discrete point. Try

several values of $n=30,40,50.$

solve correctly