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Review of Linear Algebra
Published in Mohammad Monir Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, 2019
Let A be a symmetric positive definite matrix. Then A can be decomposed as A=LLT where L is a lower triangular matrix and LT islower triangular the transpose of L. This is known as Cholesky decomposition. If A is a symmetric positive semi-definite matrix, this decomposition is also possible. If A is a Hermitian positive definite (or semidefinite) matrix,Hermitian positive definite then we can write the Cholesky decomposition of A asCholesky decompositionA=LL∗ where L∗ denotes the conjugate transpose of L.conjugate transpose
Linear Algebra
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
The Gauss–Seidel iterative method decomposes the matrix into an upper triangular matrix, U and a lower triangular matrix, L. These are not the same upper and lower triangular matrices that form the LU decomposition. With Gauss–Seidel, the upper triangular matrix is the entries of the matrix A above the main diagonal. The lower triangular matrix is the entries of the original matrix below the main diagonal and including the main diagonal. Like with the Jacobi iterative process, the undefined entries of both U and L are set to zero. The iterated equation for Gauss–Seidel is conceptually identical to the equation for Jacobi’s method, so that, x(n+1)=L-1(b-Ux(n)). $$ \begin{aligned} \mathbf {x}^{(n + 1)} = L^{-1}(\mathbf {b} - U\mathbf {x}^{(n)}) \text{.} \end{aligned} $$
Numerical Solution of Systems of Equations
Published in Ramin S. Esfandiari, Numerical Methods for Engineers and Scientists Using MATLAB®, 2017
LU factorization (or decomposition) of a matrix An×n means expressing the matrix as A = LU, where Ln×n is a lower triangular matrix and Un×n is upper triangular. Subsequently, the original system Ax = b is rewritten as [LU]x=b⇒L[Ux]=b⇒LUxy=b
A study and implementation of large-scale log-determinant computation to cloud
Published in International Journal of Computers and Applications, 2021
Md. Alamgir Hossain, Jannatul Ferdush, Marjia Khatun
Matrix inversion can be performed using several approaches which include gaussian elimination, Newton's method, cayley-hamilton method, blockwise inversion and others. Using a divide-and-conquer technique, matrix inversion can be accomplished by computing the inverses of two half-sized matrices and performing six half-sized matrix multiplications. Hence, the time complexity of performing matrix inversion is equivalent to that of computing matrix multiplication. In [22], mohassel proposes an algorithm for outsourcing matrix inversion that uses his solution for privacy-preserving matrix multiplication as a building block. The technique is used in this paper for outsourcing is same as the matrix multiplication outsourcing technique. Lei et al. [29] proposed a protocol for outsourcing matrix inversion operations that uses a similar transformation technique. Using the Monte Carlo algorithm, random vectors are used in the verification step to assess the correctness of the result. Unlike matrix multiplication and inversion, matrix factorization is a term that encompasses a number of different computational tasks such as non-negative matrix factorization, singular and nonsingular value decomposition, eigenvalue decomposition, etc. Matrix decomposition into the product of a lower and upper triangular matrix. All these methods are used to divide a matrix into two same dimensional matrices. There are many LU decomposition algorithms like Modified form of gaussian elimination [30], doolittle factorization [31], crout factorization [32], cholesky factorization [33], QR factorization [34] etc. From LU decomposition a lower triangular matrix L and upper triangular matrix L are generated which are basically used for matrix privacy if the matrix is encrypted in the cloud.