The document describes implementing L-U decomposition without pivoting to solve systems of linear equations. It includes: 1) An objective to write an M-file that performs L-U decomposition without pivoting for solving systems of linear equations, and to write both script and function files. 2) An explanation that L-U decomposition decomposes a matrix into lower and upper triangular matrices, making systems with triangular matrices easier to solve. 3) Code examples of a script and function file that perform L-U decomposition on a sample matrix without pivoting.

1. PRACTICAL
Name- Saloni Singhal
M.Sc. (Statistics) II-Sem.
Roll No: 2046398
Course- MATH-409 L
Numerical Analysis Lab
Submitted To: Dr. S.C. Pandey

2. OBJECTIVE
• To write an m-file to implement L-U
Decomposition (without pivoting) for solving
system of linear equations.
• To write both script and function files for the
same program.

3. Theory
• The system of equation is based on the
observation that system of linear equation
involving triangular Coefficient Matrix are
easier to deal with.
• The composition is the product of lower
triangular Matrix and an upper triangular
matrix

4. Script File
% defines matrix A
A = [2 1 5; 3 5 2;2 1 4]
[m,n]=size(A);
if m~=n
fprintf("not square matrix")
exit
end
L=eye(n);
U=A;
ref=[U L];
%decomposition
for i=1:m-1
for j=i+1:m
a=ref(j,i)/ref(i,i); %back substitution
ref(j,:)=ref(j,:)-a*ref(i,:);
end
end
%representation
U=ref(1:n,1:n)
L=ref(1:n,n+1:2*n);
L=inv(L)
%verification
[c]=L*U

5. Function
File
function [L,U] =decomposition(A)
[m,n]=size(A);
if m~=n
fprintf("not square matrix")
exit
end
L=eye(n); U=A;
ref=[U L];
%decomposition
for i=1:m-1
for j=i+1:m
a=ref(j,i)/ref(i,i); %back substitution
ref(j,:)=ref(j,:)-a*ref(i,:);
end
end
%representation
U=ref(1:n,1:n)
L=ref(1:n,n+1:2*n);
L=inv(L)
%verification
[c]=L*U
end

6. Cases/Example

7. Time Complexity
7
For the given program it is calculated as:
For i=1 T(j)=4n(m-1)
For i=2 T(j)=4n(m-2)… on summation
T(n)= 4n(m-1) +(m-2)4n+ 4n(m-3)…..1
T(n)=4n(m2 – m(m-1)/2 ),
since in square matrix m=n T(n3+n2)/2=O(n3)
Big O notation is the most common metric for calculating time complexity. It
describes the execution time of a task in relation to the number of steps
required to complete it.
Initially form i loop
of range 1:n-1, we
take n operations,
then 4 operation
inside the loop for
j. ‘O’ takes its
dominant value

8. Conclusion
• It is more efficient as it requires gaussian elimination to be
performed once
• It is better for solving repeatedly and number of equation with
same on the left hand side.
• It does not require augmented matrix.
• Cholesky decomposition is faster by a factor of 2 and take
lesser memory and space

9. Caveats
• Using L-U Decomposition can be
unstable in algorithm, we may run
into blown up errors with poorly
managed flops and operations.
• The original matrix is lost
• Sometimes it is impossible to write
a matrix in the form of L-U
composition if the leading matrix
don't have a non-zero determinant

10. References
• https://learn.lboro.ac.uk/archive/olmp/olmp_resourc
es/pages/workbooks_1_50_jan2008/Workbook30/3
0_3_lu_decmp.pdf
• Lecture Notes of Prof. S.C. Pandey Sir
• MATLAB Documetation