Python coding:

Consider the linear system $Ax=y,$ where

$A=\left[\begin{array}{ccccc}1& \frac{1}{2}& \frac{1}{3}& \text{c}\text{d}\text{o}\text{t}\text{s}& \frac{1}{10}\\ \frac{1}{2}& \frac{1}{3}& \frac{1}{4}& \text{c}\text{d}\text{o}\text{t}\text{s}& \frac{1}{11}\\ \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& ?\\ \frac{1}{10}& \frac{1}{11}& \frac{1}{12}& \text{c}\text{d}\text{o}\text{t}\text{s}& \frac{1}{19}\end{array}\right]$

and

$x=(55,44,33,44,22,12,10,15,18,20{)}^T.$

Suppose we perturb $y$ as follows

$y^{}=y+\left[\begin{array}{c}\\ 0\\ \text{v}\text{d}\text{o}\text{t}\text{s}\\ 0\end{array}\right],=0\mathrm{.}1.$

Define $x^{}$ such that $A{x}^{}=y^{}.$ Use Tikhonov regularization to solve $A{x}^{}=y^{},$ where the regularization

parameter $$ is chosen from ${10}^z,$ where $z$ is linearly$-$spaced between $-9$ and $-1$ with $1000$ partitions. Let

$x_{}^{}$ be the solution of the Tikhonov regularization, given $.$ Plot $||x_{}^{}-x|{|}_2$ as a function of $.$