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 (see matrix decomposition). 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. The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938.[1] To quote: "It appears that Gauss and Doolittle applied the method
[of elimination] only to symmetric equations. More recent authors, for example, Aitken, Banachiewicz, Dwyer, and Crout … have emphasized the use of the method, or variations of it, in connection with non-symmetric problems … Banachiewicz … saw the point … that the basic problem is really one of matrix factorization, or “decomposition” as he called it."[2]
It is also sometimes referred to as LR decomposition (factors into left and right triangular matrices).
^Schwarzenberg-Czerny, A. (1995). "On matrix factorization and efficient least squares solution". Astronomy and Astrophysics Supplement Series. 110: 405. Bibcode:1995A&AS..110..405S.
^Dwyer, Paul S. (1951). Linear Computations. New York: Wiley.
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix...
A\mathbf {x} =\mathbf {b} } , the matrix A can be decomposed via the LUdecomposition. The LUdecomposition factorizes a matrix into a lower triangular matrix...
In linear algebra, a Block LUdecomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular...
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easier to solve, they are very important in numerical analysis. By the LUdecomposition algorithm, an invertible matrix may be written as the product of a...
are referred to as decomposition methods. Examples include the LUdecomposition, the QR decomposition or the Cholesky decomposition (for positive definite...
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a true LUdecomposition of the original matrix. The argument applies also for the determinant, since it results from the block LUdecomposition that det...
result of a multiplication Matrix decompositions: LUdecomposition — lower triangular times upper triangular QR decomposition — orthogonal matrix times triangular...
) {\displaystyle O(n^{3})} operations by forming the LUdecomposition A = L U {\displaystyle A=LU} (typically via Gaussian elimination or similar methods)...
decomposed (i.e. factored) in O ( n 2 ) {\displaystyle O(n^{2})} time. The Bareiss algorithm for an LUdecomposition is stable. An LUdecomposition gives...
matrix P is called a similarity matrix, and is necessarily invertible. LUdecomposition splits a matrix into a matrix product of an upper triangular matrix...
{\displaystyle M^{\frac {1}{2}}} for any such decomposition, or specifically for the Cholesky decomposition, or any decomposition of the form M = B B {\displaystyle...
algorithm was preceded by the LR algorithm, which uses the LUdecomposition instead of the QR decomposition. The QR algorithm is more stable, so the LR algorithm...
Jordan–Chevalley decomposition of an element in algebraic group as a product of semisimple and unipotent elements The Bruhat decomposition G = BWB of a semisimple...
computations (matrix product, solving systems of linear equations, LUdecomposition, etc.) for banded matrices. Early applications include computing greatest...
e.g. Newton-Raphson method) system of linear equations (using e.g. LUdecomposition) ordinary differential equations (using e.g. Runge–Kutta methods) integration...
proves the desired equality. Moreover, one gets the LUdecomposition of V as V = L U − 1 {\displaystyle V=LU^{-1}} . The third proof is based on the fact that...
The Doolittle method may refer to: The Doolittle algorithm for LUdecomposition in numerical analysis and linear algebra The most common method of rearing...
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