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In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
Other algorithms for solving linear-programming problems are described in the linear-programming article. Another basis-exchange pivoting algorithm is the criss-cross algorithm . [ 41 ] [ 42 ] There are polynomial-time algorithms for linear programming that use interior point methods: these include Khachiyan 's ellipsoidal algorithm , Karmarkar ...
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
algorithm Gauss–Seidel method is inputs: A, b output: φ Choose an initial guess φ to the solution repeat until convergence for i from 1 until n do σ ← 0 for j from 1 until n do if j ≠ i then σ ← σ + a ij φ j end if end (j-loop) φ i ← (b i − σ) / a ii end (i-loop) check if convergence is reached end (repeat)
Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set. For three variables, each linear equation determines a plane in three-dimensional space , and the solution set is the intersection of these planes.
[1] [2] [3] It was commented on by Liu Hui in the 3rd century. According to Grcar [4] solution of linear equations by elimination was invented independently in several cultures in Eurasia starting from antiquity and in Europe definite examples of procedure were published already by late Renaissance (in 1550's). It is quite possible that already ...
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