Dimensionality of parameters and variables is a fundamental issue in physics but mostly ignored from a mathematical point of view. Diffculties arising from dimensional inconsistence are overcome by scaling analysis and, often, both concepts, dimensionality and scaling, are confused. In the particular case of iterative methods for solving non-linear equations, dimensionality and scaling affects their robutness: while some classical methods, such as Newton, are adimensional and scale independent, some other iterations as Steffensen's are not; their convergence depends on the scaling, and their evaluation needs a dimensional congruence. In this paper we introduce the concept of adimensional form of a function in order to study the behavior of iterative methods, thus correcting, if possible, some pathological features. From this adimensional form we will devise an adimensional and scale invariant method based on Steffensen's which we will call ASIS method.

中文翻译：

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一维尺度不变斯特芬森 (ASIS) 方法

参数和变量的维数是物理学中的一个基本问题，但从数学的角度来看大多被忽略。维度不一致引起的困难可以通过缩放分析来克服，而且维度和缩放这两个概念常常被混淆。在求解非线性方程的迭代方法的特殊情况下，维数和标度会影响它们的稳健性：虽然一些经典方法，如牛顿，是无量纲和标度无关的，但 Steffensen 的一些其他迭代不是；它们的收敛性取决于缩放，它们的评估需要维度一致。在本文中，我们引入了函数的无量纲形式的概念，以研究迭代方法的行为，从而在可能的情况下纠正一些病理特征。