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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 ...
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.
When P is an identity matrix, the LUP decomposition reduces to the LU decomposition. Comments: The LUP and LU decompositions are useful in solving an n -by- n system of linear equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } .
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.
The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables. The covariance matrix is decomposed to give the lower-triangular L. Applying this to a vector of uncorrelated observations in a sample u produces a sample vector Lu with the covariance properties of the system being ...
One can then generalize this procedure; the ILU(k) preconditioner of a matrix A is the incomplete LU factorization with the sparsity pattern of the matrix A k+1. More accurate ILU preconditioners require more memory, to such an extent that eventually the running time of the algorithm increases even though the total number of iterations decreases.
The Doolittle algorithm for LU decomposition in numerical analysis and linear algebra; The most common method of rearing queen bees This page was last edited on 18 ...
LU decomposition — write a matrix as a product of an upper- and a lower-triangular matrix Crout matrix decomposition; LU reduction — a special parallelized version of a LU decomposition algorithm; Block LU decomposition; Cholesky decomposition — for solving a system with a positive definite matrix Minimum degree algorithm