Linear Approximation Algorithm

A Linear Approximation Algorithm is an approximation algorithm that applies linear functions.

• See: Linear Regression Algorithm, Affine Function, Jacknife Algorithm, Jacobian Matrix, Non-Linear Approximation Algorithm.

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References

2011

• http://en.wikipedia.org/wiki/Linear_approximation
  • In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. Given a twice continuously differentiable function [math]\displaystyle{ f }[/math] of one real variable, Taylor's theorem for the case n = 1 states that [math]\displaystyle{ f(x) = f(a) + f'(a)(x - a) + R_2\ }[/math] where [math]\displaystyle{ R_2 }[/math] is the remainder term. The linear approximation is obtained by dropping the remainder [math]\displaystyle{ f(x) \approx f(a) + f'(a)(x - a). }[/math] This is a good approximation for x when it is close enough to a since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of [math]\displaystyle{ f }[/math] at (a,f(a)). For this reason, this process is also called the tangent line approximation. Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix

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