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The cost of solving a system of linear equations is approximately floating-point operations if the matrix has size . This makes it twice as fast as algorithms based on QR decomposition , which costs about 4 3 n 3 {\textstyle {\frac {4}{3}}n^{3}} floating-point operations when Householder reflections are used.
The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems () = and = require fewer additions and multiplications to solve, compared with the original system =, though one might require significantly more digits in inexact arithmetic such as floating point.
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
The RQ decomposition transforms a matrix A into the product of an upper triangular matrix R (also known as right-triangular) and an orthogonal matrix Q. The only difference from QR decomposition is the order of these matrices. QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column.
The process is so called because for lower triangular matrices, one first computes , then substitutes that forward into the next equation to solve for , and repeats through to . In an upper triangular matrix, one works backwards, first computing x n {\displaystyle x_{n}} , then substituting that back into the previous equation to solve for x n ...
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
Consider a sparse linear system =. These are often solved by computing the factorization A = L U {\displaystyle A=LU} , with L lower unitriangular and U upper triangular . One then solves L y = b {\displaystyle Ly=b} , U x = y {\displaystyle Ux=y} , which can be done efficiently because the matrices are triangular.
In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual .