LU Algorithm

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. LU decomposition was introduced by Polish mathematician Tadeusz Banachiewicz in 1938.

Steps and Performance

LU decomposition is basically a modified form of Gaussian elimination. We transform the matrix A into an upper triangular matrix U by eliminating the entries below the main diagonal. The Doolittle algorithm does the elimination column-by-column, starting from the left, by multiplying A to the left with atomic lower triangular matrices. It results in a unit lower triangular matrix and an upper triangular matrix. The Crout algorithm is slightly different and constructs a lower triangular matrix and a unit upper triangular matrix.