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It can therefore be important that considerations of computation efficiency for such problems extend to all of the auxiliary quantities required for such analyses, and are not restricted to the formal solution of the linear least squares problem. Matrix calculations, like any other, are affected by rounding errors. An early summary of these ...
For most linear programs solved via the revised simplex algorithm, at each step, most columns (variables) are not in the basis. In such a scheme, a master problem containing at least the currently active columns (the basis) uses a subproblem or subproblems to generate columns for entry into the basis such that their inclusion improves the ...
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
The minimum of f is 0 at z if and only if z solves the linear complementarity problem. If M is positive definite, any algorithm for solving (strictly) convex QPs can solve the LCP. Specially designed basis-exchange pivoting algorithms, such as Lemke's algorithm and a variant of the simplex algorithm of Dantzig have been used for decades ...
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
To solve the underdetermined (<) linear problem = where the matrix has dimensions and rank , first find the QR factorization of the transpose of : =, where Q is an orthogonal matrix (i.e. =), and R has a special form: = [].
The conjugate gradient method with a trivial modification is extendable to solving, given complex-valued matrix A and vector b, the system of linear equations = for the complex-valued vector x, where A is Hermitian (i.e., A' = A) and positive-definite matrix, and the symbol ' denotes the conjugate transpose.
Cyclic reduction is a numerical method for solving large linear systems by repeatedly splitting the problem. Each step eliminates even or odd rows and columns of a matrix and remains in a similar form. The elimination step is relatively expensive but splitting the problem allows parallel computation.