detailed solution plz Cholesky factorization versus QR factorization. In this pr

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detailed solution plz

Cholesky factorization versus QR factorization. In this problem we compare the accuracy of the two methods for solving a least-squares problem minimize |A-bll We take 10 b = | 1 + 10-k 1- 10-k A 10-k 0 10-k for k = 6, k 7 and k=8 (a) Write the norma equations, and solve them analytically (i.e., on paper, without (b) Solve the least-squares problem in MATLAB, for k 6, k 7 and k 8, using the (c) Repeat part (b), using the Cholesky factorization method, i.e., x (A'*A)CA'b using MATLAB) recommended method x - A. This method is based on the QR factorization (We assume that MATLAB recognizes that A A is symmetric positive definite, and uses the Cholesky factorization to solve ATAx ATb). Compare the results of this method with the results of parts (a) and (b) Remark. Type format long to make MATLAB display more than five digits

Explanation / Answer

SAVE THE FOLLOWING FUNCTION AND SCRIPT FILE IN MATLAB-

%% Cholesky Factorization

function [F]=cholesky(A,option)

if ~isequal(A,A'),

error('Input Matrix is not Symmetric');

end

if isPositiveDefinite(A),

[m,n]=size(A);

L=zeros(m,m);%Initialize to all zeros

row=1;col=1;

j=1;

for i=1:m,

a11=sqrt(A(1,1));

L(row,col)=a11;

if(m~=1), %Reached the last partition

L21=A(j+1:m,1)/a11;

L(row+1:end,col)=L21;

A=(A(j+1:m,j+1:m)-L21*L21');

[m,n]=size(A);

row=row+1;

col=col+1;

end

end

switch nargin

case 2

if strcmpi(option,'upper'),F=L';

else

if strcmpi(option,'lower'),F=L;

else error('Invalid option');

end

end

case 1

F=L;

otherwise

error('Not enough input arguments')

end

else

error('Given Matrix A is NOT Positive definite');

end

end

%% QR Factorization

A=sym(wilkinson(4));

R=qr(A)'

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