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Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
Statements about relative errors are sensitive to addition of constants, but not to multiplication by constants. For absolute errors, the opposite is true: are sensitive to multiplication by constants, but not to addition of constants. [5]: 34
Starting from homogeneous initial conditions with only one point-like perturbation, the correct growth process would yield a circle (left). If growth is faster along the coordinate axes, which is caused by the anisotropic effect of central difference discretization, the circle would turn into star-shaped structure, as the errors propagate (right):.
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
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The delta method was derived from propagation of error, and the idea behind was known in the early 20th century. [1] Its statistical application can be traced as far back as 1928 by T. L. Kelley. [2] A formal description of the method was presented by J. L. Doob in 1935. [3] Robert Dorfman also described a version of it in 1938. [4]
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In computing, a roundoff error, [1] also called rounding error, [2] is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. [3]