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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]