Question

Please step by step solution!!!

Cholesky factorization versus QR factorization. In this problem we compare the accuracy of the two methods for solving a least-squares problem minimize Ar - b We take b10- 1-10-k ke A10 0 10k for k 6, k- 7 and k 8. (a) Write the normal equations, and solve them analytically (i.e., on paper, without using MATLAB). (b) Solve the least-squares problem in MATLAB, for k = 6. K-7 and k = 8, using the recommended method x - A\b. This method is based on the QR factorization. (c) Repeat part (b), using the Cholesky factorization method. i. e., X- (A, *A)\(A, *b). (We assume that MATLAB recognizes that A' A is symmetric positive definite, and uses the Cholesky factorization to solve AT Ax 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.

Answer 1

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Answer:

Explanation:

CODE:

%% 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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