Question II $(12$ Marks$)$

You are tasked with designing a computer program for obtaining polynomial models of up to order $15$ using least squares approximation. This would typically require the solution of an overdetermined system $Ax=b.$ You have access to the following numerical library functions that you can use. For you application accuracy both accuracy and CPU cost are important, however more emphasis is given to accuracy.

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

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

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

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

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

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

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

h$)$ HouseholderQR$($A$)$ Returns $Q$ and $R$ such that $A=Q**R($uses the Householder reflections$)$

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

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

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

A$)$ Identify the key properties of this problems that you need to consider for your design, and choice of algorithm to be used. $(2$ Marks$)$