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The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floating point arithmetic. This is accomplished by using a polynomial of high degree , and/or narrowing the domain over which the polynomial has to approximate the function.
Several progressively more accurate approximations of the step function. An asymmetrical Gaussian function fit to a noisy curve using regression.. In general, a function approximation problem asks us to select a function among a well-defined class [citation needed] [clarification needed] that closely matches ("approximates") a target function [citation needed] in a task-specific way.
For example, for Newton's method as applied to a function f to oscillate between 0 and 1, it is only necessary that the tangent line to f at 0 intersects the x-axis at 1 and that the tangent line to f at 1 intersects the x-axis at 0. [19] This is the case, for example, if f(x) = x 3 − 2x + 2.
Linear approximations in this case are further improved when the second derivative of a, ″ (), is sufficiently small (close to zero) (i.e., at or near an inflection point). If f {\displaystyle f} is concave down in the interval between x {\displaystyle x} and a {\displaystyle a} , the approximation will be an overestimate (since the ...
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions .
These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. [ 1 ] [ 2 ] One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision.
Bresenham's line algorithm is a line drawing algorithm that determines the points of an n-dimensional raster that should be selected in order to form a close approximation to a straight line between two points.
For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10: 2 10 = 1024 ≈ 1000 = 10 3 . {\displaystyle 2^{10}=1024\approx 1000=10^{3}.} Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.