“LU decomposition”的概念、定义、翻译、参考文献-科学参考

LU decomposition

Mathematics

A square matrix A has an LU decomposition if A = LU, where L is a lower triangular matrix with all diagonal entries equalling 1 and U is an upper triangular matrix. The system of equations Ax = b can then be solved separately as Ly = b and then Ux = y, which is computationally more efficient. The system Ly = b efficiently yields y₁,y₂,…,y_n in order and is known as forward substitution. The system Ux = y likewise yields x_n,…,x₂,x₁ in that order and is known as backward substitution. Not all matrices A have LU decompositions, but there is always a permutation matrix P such that PA has an LU decomposition; P effectively just reorders the equations in the linear system.

Computer

A method used in numerical linear algebra in order to solve a set of linear equations,
$A x = b$
where A is a square matrix and b is a column vector. In this method, a lower triangular matrix L and an upper triangular matrix U are sought such that
$L U = A$
For definiteness, the diagonal elements of L may be taken to be 1. The elements of successive rows of U and L may easily be calculated from the defining equations.
Once L and U have been determined, so that
$L U x = b,$
the equation
$L y = b$
is solved by forward substitution. Thereafter the equation
$U x = y$
is solved for x by backward substitution. x is then the solution to the original problem.
A variant of the method, the method of LDU decomposition, seeks lower and upper triangular matrices with unit diagonal and a diagonal matrix D, such that
$A = L D U$
If the matrix A is symmetric and positive definite, there is an advantage in finding a lower triangular matrix L such that
$A = L L^{T}$
(see transpose). This method is known as Cholesky decomposition; the diagonal elements of L are not, in general, unity.

单词	LU decomposition
释义	LU decomposition Mathematics A square matrix A has an LU decomposition if A = LU, where L is a lower triangular matrix with all diagonal entries equalling 1 and U is an upper triangular matrix. The system of equations Ax = b can then be solved separately as Ly = b and then Ux = y, which is computationally more efficient. The system Ly = b efficiently yields y₁,y₂,…,y_n in order and is known as forward substitution. The system Ux = y likewise yields x_n,…,x₂,x₁ in that order and is known as backward substitution. Not all matrices A have LU decompositions, but there is always a permutation matrix P such that PA has an LU decomposition; P effectively just reorders the equations in the linear system. Computer A method used in numerical linear algebra in order to solve a set of linear equations, $A x = b$ where A is a square matrix and b is a column vector. In this method, a lower triangular matrix L and an upper triangular matrix U are sought such that $L U = A$ For definiteness, the diagonal elements of L may be taken to be 1. The elements of successive rows of U and L may easily be calculated from the defining equations. Once L and U have been determined, so that $L U x = b,$ the equation $L y = b$ is solved by forward substitution. Thereafter the equation $U x = y$ is solved for x by backward substitution. x is then the solution to the original problem. A variant of the method, the method of LDU decomposition, seeks lower and upper triangular matrices with unit diagonal and a diagonal matrix D, such that $A = L D U$ If the matrix A is symmetric and positive definite, there is an advantage in finding a lower triangular matrix L such that $A = L L^{T}$ (see transpose). This method is known as Cholesky decomposition; the diagonal elements of L are not, in general, unity.
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