Question

The Cholesky factorization one

3. Consider the linear system Ax = b, where 6.25 -1 0.5 2.12 3.6 and [ 7.51 b= -8.68 [ -0.24 Write a MATLAB program for LU-factorization with a unit lower triangular L (meaning that the diagonal entries should be equal to one). Then write a program for the Cholesky factorization. WARNING: avoid using MATLAB shortcuts. The programming should be done "from scratch"

Answer 1

ANSWER:

• The solution consists of 3 code files-
  1. lu_fact.m - manual LU Factorization function
  2. cholesky_fact.m - manual Cholesky function
  3. main.m - calling and testing above functions
• I have provided code in both text and image format so you can easily copy the code as well as check for correct indentation
• I have provided the output image of the code so you can easily cross-check for the correct output of the code.
• Have a nice and healthy day!!

CODE TEXT

1. lu_fact.m

% function for lu factorization

function [L,U]=lu_fact(A)

% fetching n

n=rank(A);

% defining L and U

L=zeros(n); % lower matrix

U=zeros(n); % upper matrix

% factorizing matrix into L and U

for i=1:n

% working on U

for j=i:n

% defining sum of multiplication of elements

sum=0;

for k=1:i

sum=sum+(L(i,k)*U(k,j));

end

% subtracting sum from A

U(i,j)=A(i,j) -sum;

end

% working on L

for j=i:n

if i==j

L(i,i)=1; % for diagonal

else

sum=0;

for k =1:i

sum = sum + (L(j,k) * U(k,i));

end

% subtracting sum from A

L(j,i)=(A(j,i)-sum)/U(i,i);

end

end

end

end

2. cholesky_fact.m

% function cholesky

function [L] = cholesky_fact(A)

% fetching n

n = rank(A);

% initializing L

L = zeros(n+1);

% factorizing a matrix

for i = 1:n

for j=1:i+1

sum = 0;

% suming for diagonals

if (j==i)

for k=1:j

sum = sum + L(j,k)^2;

end

L(j,j)=sqrt(A(j,j)-sum);

else % for remaining elements

for k=1:j

sum = sum + L(i,k) *L(j,k);

end

if L(j,j)>0

L(i,j) = (A(i,j) - sum)/L(j,j);

end

end

end

end

% removing 0 rows and cols

L=L(1:n,1:n);

end

3. main.m

% Initializing A, given in question

A=[6.25 -1 0.5;-1 5 2.12;0.5 2.12 3.6];

% finding LU Factorization using hand written function lu_fact

disp('LU Factorization');

[L,U]=lu_fact(A)

disp('Multiplying L*U gives');

disp(L*U);

% finding Cholesky factorization using hand written function

disp('Cholesky Factorization');

L=cholesky_fact(A);

disp("Multiplying L*L' gives");

disp(L*L');

CODE IMAGES

a. lu_fact.m

lu_fact. mx cholesky_fact. mx main. mx + 1 % function for lu factorization 2 function [LU]=lu_fact(A) % fetching n n=rank(A); % defining L and U Luzeros(n); % lower matrix U=zeros(n); % upper matrix % factorizing matrix into L and U for i=1:n % working on U for j=i:n % defining sum of multiplication of elements sum=0; for k=1:1 sum=sum+(L(i,k)*U(k,j)); end % subtracting sum from A U(i, j)=A(i,j) -sum; end % working on L for j=i:n if i==j L(i,i)=1; % for diagonal else sum=0; for k=1:1 sum = sum + (L(j,k) * Uk,i)); end LLL LL LLLLLL LLLLLLL

end % subtracting sum from A L(j,i)=(A(j,i)-sum)/U(i,i); end end end end

b. cholesky_fact.m

lu_fact.m x cholesky_fact.mx main.mx % function cholesky 2 function [L] = cholesky_fact(A) % fetching n n = rank(A); % initializing L L = zeros(n+1); % factorizing a matrix for i = 1:n for j=1:1+1 sum = 0; % suming for diagonals if (j==i) for k=1:j sum = sum + L(j,k)^2; end L(jj)=sqrt(A(jj)-sum); else % for remaining elements for k=1:j sum = sum + L(i,k) *L(j,k); end if L(j,j)>0 l(i,j) = (A(i,j) - sum)/L(1,1); end end end end % removing Orows and cols L=L(1:n, 1:n); 29 - L end

c. main.m

lu_fact. mx cholesky_fact. mx main. mx + % Initializing A, given in question 2 - A=[6.25 -1 0.5; -1 5 2.12;0.5 2.12 3.6); 8 - % finding LU Factorization using hand written function lu_fact disp('LU Factorization'); [L,U]=lu_fact(A) disp('Multiplying L*U gives'); disp(L*U); % finding Cholesky factorization using hand written function disp('Cholesky Factorization'); L-cholesky_fact(A); disp("Multiplying L*L' gives"); disp(L*L'); 10 - 11 - 12 -

OUTPUT IMAGE

LU Factorization L = 1.0000 -0.1600 0.0800 1.0000 0.4545 1.0000 U = 6.2500 0 0 -1.0000 4.8400 2 0.5000 2.2000 .5600 Multiplying L*U gives 6.2500 -1.0000 -1.0000 5.0000 0.5000 2.1200 0.5000 2.1200 3.6000 cholesky Factorization Multiplying L*L' gives 6.2500 -1.0000 -1.0000 5.0000 0.5000 2.1200 0.5000 2.1200 3.6000