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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. The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938. [1]
The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems () = and = require fewer additions and multiplications to solve, compared with the original system =, though one might require significantly more digits in inexact arithmetic such as floating point.
In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix (P). It was developed by Prescott Durand Crout. [1] The Crout matrix decomposition algorithm differs slightly from the ...
is called an incomplete LU decomposition (with respect to the sparsity pattern ). The sparsity pattern of L and U is often chosen to be the same as the sparsity pattern of the original matrix A . If the underlying matrix structure can be referenced by pointers instead of copied, the only extra memory required is for the entries of L and U .
It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also includes routines to implement the associated matrix factorizations such as LU , QR , Cholesky and Schur decomposition . [ 2 ]
A frontal solver is an approach to solving sparse linear systems which is used extensively in finite element analysis. [1] Algorithms of this kind are variants of Gauss elimination that automatically avoids a large number of operations involving zero terms due to the fact that the matrix is only sparse. [2]
In numerical analysis, Stone's method, also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations.The method uses an incomplete LU decomposition, which approximates the exact LU decomposition, to get an iterative solution of the problem.
In linear algebra, a Block LU decomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U.