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Animation of Gaussian elimination. Red row eliminates the following rows, green rows change their order. In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients.
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations. Classical elimination theory culminated with the work of Francis Macaulay on multivariate resultants, as ...
Lewy's example takes this latter equation and in a sense translates its non-solvability to every point of . The method of proof uses a Baire category argument, so in a certain precise sense almost all equations of this form are unsolvable. Mizohata (1962) later found that the even simpler equation
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]
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: = where the n × n matrix A has a nonzero determinant, and the vector = (, …,) is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns ...
Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form .
This system has the exact solution of x 1 = 10.00 and x 2 = 1.000, but when the elimination algorithm and backwards substitution are performed using four-digit arithmetic, the small value of a 11 causes small round-off errors to be propagated. The algorithm without pivoting yields the approximation of x 1 ≈ 9873.3 and x 2 ≈ 4.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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