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Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one.
One may also use Newton's method to solve systems of k equations, which amounts to finding the (simultaneous) zeroes of k continuously differentiable functions :. This is equivalent to finding the zeroes of a single vector-valued function F : R k → R k . {\displaystyle F:\mathbb {R} ^{k}\to \mathbb {R} ^{k}.}
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
An early iterative method for solving a linear system appeared in a letter of Gauss to a student of his. He proposed solving a 4-by-4 system of equations by repeatedly solving the component in which the residual was the largest [ citation needed ] .
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
The cost of solving a system of linear equations is approximately floating-point operations if the matrix has size . This makes it twice as fast as algorithms based on QR decomposition , which costs about 4 3 n 3 {\textstyle {\frac {4}{3}}n^{3}} floating-point operations when Householder reflections are used.