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Classical elimination theory culminated with the work of Francis Macaulay on multivariate resultants, as described in the chapter on Elimination theory in the first editions (1930) of Bartel van der Waerden's Moderne Algebra. After that, elimination theory was ignored by most algebraic geometers for almost thirty years, until the introduction ...
Fraction-free algorithm — uses division to keep the intermediate entries smaller, but due to the Sylvester's Identity the transformation is still integer-preserving (the division has zero remainder). For completeness Bareiss also suggests fraction-producing multiplication-free elimination methods. [2]
The notes were widely imitated, which made (what is now called) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century. Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares ...
Another way to understand the operation of the algorithm is as an "elimination method", where the states from 0 to n are successively removed: when state k is removed, the regular expression R k-1 ij, which describes the words that label a path from state i>k to state j>k, is rewritten into R k
LU decomposition on Math-Linux. LU decomposition at Holistic Numerical Methods Institute; LU matrix factorization. MATLAB reference. Computer code. LAPACK is a collection of FORTRAN subroutines for solving dense linear algebra problems; ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, etc. C++ code, Prof. J. Loomis, University ...
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
Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier [ 1 ] who proposed the method in 1826 and Theodore Motzkin who re-discovered it in 1936.
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the ...
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