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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 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.
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 .
An alternative way to eliminate taking square roots in the decomposition is to compute the LDL decomposition =, then solving = for y, and finally solving =. For linear systems that can be put into symmetric form, the Cholesky decomposition (or its LDL variant) is the method of choice, for superior efficiency and numerical stability.
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.
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 ]
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
The Bareiss algorithm for an LU decomposition is stable. [6] An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the ...