Numerical Approximation In Real Domain Of Special Function Of Product Of A Variable And Its Double Exponential

TL;DR

This paper introduces a novel numerical method to approximate a transcendental function involving a product of a variable and its double exponential, transforming it into a quadratic equation for efficient solutions.

Contribution

It presents a unique approximation technique for a complex transcendental function using linearization and iterative refinement, with validation through multiple examples.

Findings

The method accurately approximates the inverse of the function.

Iterative process improves approximation precision.

Validated with numerical examples and tabular results.

Abstract

Purpose of writing this paper is to solve a transcendental function containing a product of a variable and its double exponential by a unique method of approximation. If the value of the said product is given, then its inverse function is approximated by use of linear expression in place of natural logarithm of a positive real quantity and, that transforms the function to a quadratic equation. Roots of the equation are, then used for solving the function. For precise approximation, the process is iterated a number of times and more the number of iterations, more precise will be the approximation. To prove truthfulness of the formulae derived, a number of examples are given in tabular form.