Control Engineering

Charlie Masi

What do people mean when they say “To first order?”
September 24, 2007

The phrase “To first order” usually pops up when scientists, engineers, and applied mathematicians are using approximation theory, or its close cousin perturbation theory. The concept derives from series expansions.

The best known series expansion Taylor’s series. [mathworld.wolfram.com/TaylorSeries.html] Also commonly referred to as a “power series,” you can approximate any sufficiently well behaved scalar function to arbitrary accuracy as the sum of terms of the form:

where n is an integer identifying the “order” of the term, C_n is a constant giving its magnitude, and x₀ is a starting point at which the value of the function is presumably well known.

A function is sufficiently well behaved if it is single valued and has a finite first derivative everywhere in the problem’s domain. In theory, any such function is characterized by the set of constants needed to form a Taylor expansion whose value is within some specified error limit e of the function’s true value at that point. The fly in the ointment, however, is that you may need a whole lot of non-zero terms.

The series starts with n = 0, then proceeds through the counting numbers (1, 2, 3, …). The zero-order term is a constant equal to the function’s starting value. The first order term (n = 1) provides a straight line that intersects the function at the starting point. The second-order (quadratic) term gives a parabola, and so forth. An expansion is of order n when all coefficients higher than the nth are zero.

The expansion’s domain encompasses those x values for which the expansion is required to be sufficiently close to the actual function. Of course, that gives you three adjustable parameters (domain, order, and error limit) that you can use to assure the expansion’s success.

Can’t reach your error goals over the whole domain? You can either narrow the domain or use more non-zero terms.

Tired of evaluating more and more constants? Relax the error limit.

So, when someone says “to first order,” they are saying their approximation works with only two terms in the power series (0 and 1). It uses only the zero-order and first-order terms.

In a realistic situation, suppose you want to write a function characterizing the rate of fluid flow from a barrel filled with water. The barrel is standing on one of its flat ends and has a fixed-diameter hole in the side near its base. This was a real problem faced by the Roman architect and engineer Vitruvius as part of his design for a water clock.

If you just take a snapshot at any one time, the flow rate might be F₀. To zero order, you could quote the flow rate when the barrel was half empty and say it was pretty close to some “average” flow rate.

Generally, however, the flow rate will go down as the barrel empties. That is, the flow will be highest when the barrel is almost full because there’s more pressure to push water out through the same–size hole. Your first order correction, then, is to assume the flow will drop at a constant rate.

Because of the barrel’s shape, however, the rate of change of the flow will itself change with time. When the level is near the top or bottom, the flow rate will change slightly more rapidly than average because the barrel is narrow there. When the level is near the middle, where the barrel is fattest, the flow will change less quickly. This is the second order correction.

If you realize that this second order correction is much smaller than you care about for any physically meaningful fill level, then you’d characterize the flow rate as a linear function and specify that it was accurate “to first order.”

This basic scalar expansion can be extended to an arbitrary number of dimensions, and the idea can be implemented for a wide range of functions, such as Bessel functions or spherical harmonics.

Posted by Charlie Masi on September 24, 2007 | Comments (0)