Leading-order

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. Deciding whether terms in a model are leading-order (or approximately leading-order), and if not, whether they are small enough to be regarded as negligible, (two different questions), is often a matter of investigation and judgement, and will depend on the context.

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