This method is a modified version of LU factorization method to solve linear equations. This method is known as the square root method.
DERIVATION
Since this is a positive definite hence,
Now, according to Cholesky, there exists a lower triangular matrix L such that
Then
Hence
The above method can be written where A can be decomposed into upper triangular matrix as:
Input:
A matrix
OUTPUT:
Lower triangular matrix and its transpose
PROCESS:
Step 1: [taking inputs from user]
Read n [the order of the matrix]
for i = 0 to n - 1 repeat
for j = 0 to n - 1 repeat
Read arr[i][j])
[End of ‘for’ loop]
[End of ‘for’ loop]
Step 2: [Cholesky method]
for i = 0 to n - 1 repeat
for j = 0 to n - 1 repeat
Set l[i][j] ← 0
[End of ‘for’ loop]
[End of ‘for’ loop]
for i = 0 to n - 1 repeat
for j = 0 to i repeat
Set s ← 0
if j = I then
for k = 0 to j - 1 repeat
Set s ← s + (l[j][k] × l[j][k])
[End of ‘for’ loop]
Set l[j][j] ← square root of(arr[j][j] - s)
else
for k = 0 to j - 1 repeat
Set s ← s + (l[i][k] × l[j][k])
[End of ‘for’ loop]
Set l[i][j] ← (arr[i][j] - s) / l[j][j]
[End of ‘if’]
[End of ‘for’ loop]
[End of ‘for’ loop]
for i = 0 to n - 1 repeat
for j = 0 to n - 1 repeat
print l[i][j])
[End of ‘for’ loop]
Move to next line
[End of ‘for’ loop]
for i = 0 to n - 1 repeat
for j = 0 to n - 1 repeat
print l[j][i])
[End of ‘for’ loop]
Move to next line
[End of ‘for’ loop]
[End of Cholesky method]
ADVANTAGES
1. It solves the equations faster than the LU decomposition method of solving linear equations.
2. It is very useful to solve linear equations with symmetric positive and definite matrices.
DISADVANTAGES
1. It can decompose only the symmetric positive definite matrices.
APPLICATIONS
1. It is commonly used in the Monte Carlo method to solve simulating systems with multiple correlated variables.
2. It is used to solve linear equations.
3. It can also be used in Kalman filters and matrix inversion.
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