Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Leading-order term
Summary
The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude. The sizes of the different terms in the equation(s) will change as the variables change, and hence, which terms are leading-order may also change.
A common and powerful way of simplifying and understanding a wide variety of complicated mathematical models is to investigate which terms are the largest (and therefore most important), for particular sizes of the variables and parameters, and analyse the behaviour produced by just these terms (regarding the other smaller terms as negligible). This gives the main behaviour – the true behaviour is only small deviations away from this. This main behaviour may be captured sufficiently well by just the strictly leading-order terms, or it may be decided that slightly smaller terms should also be included. In which case, the phrase leading-order terms might be used informally to mean this whole group of terms. The behaviour produced by just the group of leading-order terms is called the leading-order behaviour of the model.
Consider the equation y = x3 + 5x + 0.1. For five different values of x, the table shows the sizes of the four terms in this equation, and which terms are leading-order. As x increases further, the leading-order terms stay as x3 and y, but as x decreases and then becomes more and more negative, which terms are leading-order again changes.
There is no strict cut-off for when two terms should or should not be regarded as approximately the same order, or magnitude. One possible rule of thumb is that two terms that are within a factor of 10 (one order of magnitude) of each other should be regarded as of about the same order, and two terms that are not within a factor of 100 (two orders of magnitude) of each other should not. However, in between is a grey area, so there are no fixed boundaries where terms are to be regarded as approximately leading-order and where not. Instead the terms fade in and out, as the variables change.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.