Question II

You are tasked with designing a computer program for obtaining polynomial models using least squares approximation. This would typically require the solution of an overdetermined system $Ax=b,$ where the matrix $A$ is a Vandermonde Matrix known for being ill$-$conditioned. You have access to a standard numerical library $($such as Matlab for example$)$ with the following methods available to you that you can use without

a$)$ Fundamental matrix and vector operations $($multiplication$,$ addition etc$)$

b$)$ Norm$(A,n)$ Returns the Ln norm of A

c$)[L,U,P]=$PLU$(A)$ Returns $L,U$ and $P$ such that $P**A=L**U($Partial pivoting$)$

d$)[L,U]=LU(A)$ Returns $L$ and $U$ such that $L**U=A($No pivoting$)$

e$)At=$transp$(A)$ Returns the transpose of $A$

f$)L=$Chol$(A)$ Returns $L$ such that $A=L^{**}{L}^T($Uses Cholesky decomposition $-$A must be symmetric positive definite$).$

g$)[Q,R]=MGSqr(A)$ Returns $Q$ and $R$ such that $A=Q**R($uses the modified Gram$-$Schmidt algorithm$)$

h$)[Q,R]=$ Householder $QR(A)$ Returns $Q$ and $R$ such that $A=Q**R($uses the Householder reflections$)$

i$)[Q,R]=$ Givens $QR(A)$ Returns $Q$ and $R$ such that $A=Q**R($uses Givens Rotations$)$

j$)y=$FwdSub$(L,b)$ Returns y such that $L^{**}y=b($L must be lower triangular$).$

k$)x=$BwdSub$(U,y)$ Returns $x$ such that $U^{**}x=y($U must be upper triangular$).$

You are tasked with solving this problem using Tikhonov Regularization. Write a Matlab program that computes $x,$ given A and $b.$ You can use any of the functions outlined above if needed. Do not worry about minor syntax errors as long as the program is clear as a pseudocode.

There are number of possible implementations or answers to this problem. Explain the rationale behind the approach you have used.