There are many algorithms for computing the LU decomposition of the matrix A. All algorithms derive a matrix L and a matrix U that satisfy Equation 37. Most algorithms also permit L and U to occupy the same amount of space as A. This implies that either L or U is computed as a unit triangular matrix so that explicit storage is not required for its diagonal (which is all ones).
There are two basic approaches to arriving at an LU decomposition:
Simulate Gaussian elimination by using row operations to zero elements in A until an upper triangular matrix exists. Save the multipliers produced at each stage of the elimination procedure as L.
Use the definition of matrix multiplication to solve Equation 37 directly for the elements of L and U.
Discussions of the subject by Fox [5], Golub and van Van Loan [8], Duff et al. [4], and Press et al. [14] are complementary in many respects. Taken as a group, these works provide a good sense of perspective concerning the problem.