LU Factorization Without Pivoting Calculator

Calculator for performing an LU factorization without pivoting of a matrix.

Using

• Here, LU factorization refers to the factorization of a matrix A into a unit lower triangular matrix L and an upper triangular matrix U.
• Input a matrix as you would for a 2D array in JavaScript. E.g., to factorize the identity matrix of size 3 input [[1,0,0],[0,1,0],[0,0,1]].
• If a matrix has infinitely many LU factorizations, then elements that can be any complex number will be shown as α₁, α₂, ..., α_n, where n is the total number of such elements.
• If a matrix has no possible factorization, then an approximation will be made by performing a factorization on a copy of the matrix with the elements that prevent the factorization from materializing incremented by ε.

Credits

• Nerdamer is used to evaluate the math
• MathJax is used to display the math
• Approach to obtain all LU factorizations comes from Froilán M. Dopico, Charles R. Johnson, and Juan M. Molera, Multiple LU factorizations of a singular matrix, Linear Algebra and its Applications. 419 (2006), no. 1, 24–36, DOI 10.1016/j.laa.2006.03.043.
• Approach to approximate LU comes from Ly Jacky Nhiayi and Tuyetdong Phan-Yamada, Examining Possible LU Decompositions, North American GeoGebra Journal. 9 (2021), no. 1, 1–7.