Study: Polynomials and Their Applications in Engineering
I’m interested in creating wealth through Engineering. I see Engineering as re-configuring reality by leveraging the laws of Physics and the layers of abstraction built on top of Physics.
But the real world is complicated. There can be many moving pieces to any single problem domain, with too much distracting resolution to meaningfully understand or operate on that reality.
The second problem is that reality is rarely how I perceive it to be. How things look and feel are rarely how empirical evidence (science) tells us they are.
I think this is what brought me to Rebooting Robotics; I suspect a healthy dose of Mathematics and Physics will remedy my problems. Mathematics can help with the first problem, modelling reality to those few(er) elements that drive outcomes, and both Math and Physics help with the second problem, helping us remain rooted in real cause-effect relationships and avoid fooling ourselves.
This is the first in hopefully a series of posts that refresh my memory of Mathematical and Physical concepts that have fallen out of memory and use. Let’s turn to the subject of this post: polynomials.
Polynomials are a class of equations that quantify the effect of scaling and summing various degrees of a quantity of interest. To simplify discussion let’s refer to this quantity of interest, which could be the distance a car travels or a feature in a machine learning dataset, as “x”. The degree of “x” is the largest number of times x is multiplied by itself. The degree of a polynomial is the largest degree of x in the polynomial.
Here’s the general from of a polynomial, represented by a function f(x).
Here, the degree of the polynomial is n. The a’s are called coefficients; they scale the contribution of the corresponding degree expression to the overall output.
Many real-world phenomena can be represented as polynomials:
A batsman hits a cricket ball and their fate—whether it lands on the other side of the boundary or into the hands of a fielder—rests on the height of the ball’s trajectory. The balls’ height is governed by a polynomial of degree 2 (also known as a quadratic).
The efficiency of a jet engine in a space shuttle depends on the heat capacity of the gas used in that engine, and the heat capacity of a gas follows a 4th degree polynomial relationship with respect to its temperature.
Polynomials capture trends in a machine learning model more accurately (i.e. with lower regression error) where a linear model doesn’t suffice.
So if I’m designing this mythical (hopefully not for long) Engineering solution that will bring me wealth and fortune, and I encounter a Polynomial, better to know how to solve it.
Solving a polynomial means finding the values of x where f(x) vanishes (becomes zero), and such values are known as the zeroes or the roots of the polynomial. The roots are useful because the points where f(x) vanishes often represent useful system states, transition points or boundary conditions in real-world applications.
For example:
Projectile motion: the roots of the following height equation of a ball’s trajectory give the time when the ball hits the ground.
Financial Analysis: A company’s profit model can be represented as a polynomial and the roots indicate points where profit is zero (breakeven points).
Machine Learning optimization: ML Optimization functions are often polynomials trying to minimize cost and zeros indicate critical points (i.e. points where derivative/slope of the cost function is zero).
Solving Polynomials
The approach to solve a polynomial depends on its degree. There’s also a theorem that states that a nth-degree polynomial will have n roots.
For degree 1, only one root exists and we can get to the answer directly by setting f(x) to zero. The root is x=-0.5.
For degree 2, two roots exist and are given by the following formula, where the alphas are the roots.
For higher degrees it’s gets more complicated to present a straightforward formula to plug analytically. While closed forms exist for degrees 3 and 4, they’re somewhat complicated to use. For degrees higher than 4, closed form analytical solutions don’t exist (yet?) and we resort to numerical methods.
For higher degree polynomials, it’s often useful and sufficient to understand the nature of the roots rather than finding exact values for the roots. It suffices to know the nature of the roots.
Useful characteristics to know about roots include:
how many roots are there? how many of them are repeated (duplicated) and how many are unique (simple)?
are the roots real (as in a real number) or are they complex (as in a complex number), which I think just means there’s some abstract notion attached to the answer, like rotation. I’ll expand on this as I understand more.
are the roots positive or negative?
We can understand things about physical systems by knowing the nature of the roots, without knowing exact values for the roots. For example, modern car cruise control systems can be modelled with nth degree polynomials, and real negative roots predict a stable system (speed will remain constant at set point), real positive roots predict an unstable system (cruise control will speed up or brake uncontrollably), complex roots imply that the system will oscillate (the car will speed up to a maximum then brake and start again in a loop) and repeated roots predict poor damping and that there’s a risk of instability.
So for higher degree polynomials it’s often enough to understand the nature of the roots.
Investigating roots without calculating them
A cubic equation (degree 3 polynomial) can be analyzed to investigate whether it has one root or three by looking at the first derivative function.
For locations of x where the first derivative vanishes, the slope of the cubic is parallel to the x-axis. These arguments x are known as critical points, where f(x) could take on a minimum point, a maximum point, or a point of inflection. Combining this with knowledge of whether the function changes signs (goes from +ve to -ve, or vice versa) from one critical point to the next tells us whether the critical points lie on either side of the x-axis, and consequently, whether there’s a point between them where the value of f(x) vanishes (which is a root of f(x).
From here, we can study how f(x) behaves for very large positive values of x (+∞) and very large negative values of x (-∞). This tells us what’s happening on either side of the critical points. For example, if f(x) goes to positive infinity for large positive values of x greater after a minimum point, we can say that f(x) will again cross the x-axis (and therefore has a root to the right of the minimum point). Similarly, if f(x) goes to negative infinity for large negative values of x to the left of a maximum point, we know f(x) will again have to cross the x-axis at some point.
For higher-degree polynomials, we would need to take successive derivatives, which themselves become cubic or higher-degree polynomials, and try to find information about the roots of those derivatives.
For a much deeper treatment of this topic, I refer the reader to Chapter 1.1 of Mathematical Methods for Physics and Engineering.
Conclusion
This article was meant to look at polynomials, their importance, and some analytical methods to study them, such that I can solve for equations of this nature that show up in the quest for developing useful Engineering systems.