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Elimination theory culminated with the work of Leopold Kronecker, and finally Macaulay, who introduced multivariate resultants and U-resultants, providing complete elimination methods for systems of polynomial equations, which are described in the chapter on Elimination theory in the first editions (1930) of van der Waerden's Moderne Algebra.
Gaussian elimination can be performed over any field, not just the real numbers. Buchberger's algorithm is a generalization of Gaussian elimination to systems of polynomial equations. This generalization depends heavily on the notion of a monomial order. The choice of an ordering on the variables is already implicit in Gaussian elimination ...
In other situations, the system of equations may be block tridiagonal (see block matrix), with smaller submatrices arranged as the individual elements in the above matrix system (e.g., the 2D Poisson problem). Simplified forms of Gaussian elimination have been developed for these situations. [6]
Drug elimination, clearance of a drug or other foreign agent from the body; Elimination, the destruction of an infectious disease in one region of the world as opposed to its eradication from the entire world; Hazard elimination, the most effective type of hazard control; Elimination (pharmacology), processes by which a drug is eliminated from ...
When solving systems of equations, b is usually treated as a vector with a length equal to the height of matrix A. In matrix inversion however, instead of vector b , we have matrix B , where B is an n -by- p matrix, so that we are trying to find a matrix X (also a n -by- p matrix):
In the field of computer science, the method is called generate and test (brute force). In elementary algebra, when solving equations, it is called guess and check. [citation needed] This approach can be seen as one of the two basic approaches to problem-solving, contrasted with an approach using insight and theory.
How to Solve It suggests the following steps when solving a mathematical problem: . First, you have to understand the problem. [2]After understanding, make a plan. [3]Carry out the plan.
The quality of implemented systems has benefited from the existence of a large library of standard benchmark examples—the Thousands of Problems for Theorem Provers (TPTP) Problem Library [25] —as well as from the CADE ATP System Competition (CASC), a yearly competition of first-order systems for many important classes of first-order problems.