How to perform LU factorization of a full rank matrix in FORTRAN

\(LU\) factorization is a method for decomposing a matrix into the product of two matrices, a lower triangular matrix \(L\) and an upper triangular matrix \(U\). The \(LU\) factorization of a matrix \(A\) can be written as follows:
\(A=LU\)
where \(L\) is a lower triangular matrix with all diagonal elements equal to \(1\), and \(U\) is an upper triangular matrix.
The \(LU\) factorization of a full rank matrix can be used to solve systems of linear equations, to compute the determinant of a matrix, and to invert a matrix. To perform \(LU\) factorization, we can use the following steps:

1. Initialize \(L\) to the identity matrix.
2. For each row \(i\), from top to bottom, do the following:
• Subtract the product of row \(i\) and the first column of \(U\) from all rows below row \(i\).
• Divide row \(i\) by the diagonal element of \(U\).
3. The resulting matrices \(L\) and \(U\) are the \(LU\) factorization of \(A\).

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