Schröder-Based Inverse Function Approximation

Abstract

Schröder approximations of the first kind, modified for the inverse function approximation case, are utilized to establish general analytical approximation forms for an inverse function. Such general forms are used to establish arbitrarily accurate analytical approximations, with a set relative error bound, for an inverse function when an initial approximation, typically with low accuracy, is known. Approximations for arcsine, the inverse of x − sin(x), the inverse Langevin function and the Lambert W function are used to illustrate this approach. Several applications are detailed. For the root approximation of a function, Schröder approximations of the first kind, based on the inverse of a function, have an advantage over the corresponding generalization of the standard Newton–Raphson method, as explicit analytical expressions for all orders of approximation can be obtained.

Relationship between Inverse Langevin Function and r0-r1-Lambert W Function

“…Consistent with (21) and Theorem 1, the roots of h r 0 (y),r 1 (y) (w) = r 0 (y) + r 1 (y)w + we w , 0 < y < 1 are −2 1+y and −2 1+y − 2L −1 (y). A desirable extension to these results would be a specification of the roots of h α,β (w) = α + βw + we w (66)…”

“…The variation of h r 0 (y),r 1 (y) (w), with w, is illustrated in Figure 4 along with the first negative root of w = −2/(1 + y) as defined by (21). The required root is to the left of the local maximum of h r 0 (y),r 1 (y) (w) that is located left of the first negative root.…”

“…The following general approximations for an inverse function arise from the Schröder root approximation of the first kind, e.g., [25] (Equation ( 21)), [19] (Equation ( 17)), [20] (Section 3) and [21]: Theorem 2. Schröder Based Approximations for an Inverse Function Consider a real function f that is differentiable up to at least order n and is also strictly monotonic in an interval around a point x 0 which includes the root x o of g(x) = f (x) − y o that is close to x 0 .…”

“…This result arises from a Taylor series approximation for g −1 at the point (y 0 , x 0 ) and the inverse function theorem. See, for example, [21]. □ 5.1.…”

“…The derived relationship leads to new approximations for the inverse Langevin function with potentially lower relative error bounds than comparable approximations. Improved approximations, based on Schröder's root approximations of the first kind, e.g., [19][20][21], are detailed. Several applications of the results are also detailed.…”