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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.
Also known as: UTV decomposition, ULV decomposition, URV decomposition. Applicable to: m-by-n matrix A. Decomposition: =, where T is a triangular matrix, and U and V are unitary matrices. Comment: Similar to the singular value decomposition and to the Schur decomposition.
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.
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 LU decomposition is an excellent general-purpose linear equation solver. The biggest disadvantage is that it fails to take advantage of coefficient matrix to be a sparse matrix. The LU decomposition of a sparse matrix is usually not sparse, thus, for a large system of equations, LU decomposition may require a prohibitive amount of memory ...
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 ...