Numerical Methods are critical to almost every “real” application of mathematics. Despite being incredibly important, most students just don’t learn or aren’t taught them directly. Implicit in any number of operations, we use numerical methods to put space shuttle on the ground and price exotic options on the stock market, and the uses are growing daily.
What are Numerical Methods?
A numerical method is exactly what it sounds like. It is a method of solving a problem without the complications of symbolic operations and algebra. Ideally, it has a few steps, typically looped, that get you closer and closer to an answer, and at the end, you are close enough to claim to know the solution.
A simple example is done by young students in the form of “Guess and Check.” While not incredibly systematic, imagine the problem of “Bob has $13 and buys baseball cards for $0.35 each. How many cards can he buy?” For an algebra student, this is the equation “0.35x = 13″ but for a student of arithmetic, this might be a “guess and check” problem. They might guess 100 cards and test it, only to find that 100 cards cost $35. Then they might guess 30 cards, seeing as they know to go down since they had too much. That gives them $10.50, which is closer. Guessing 35 cards takes them even closer, and 37 cards is perfect. The operation of guessing and checking is repeated, with each answer getting closer to correct. That is the essence of Numerical Methods.
What are some Numerical Methods
Early numerical methods that students learn, outside of guess and check, come up in calculus. In calculus, students learn about using Riemann Sums to approximate integrals, Newton’s method for finding roots, Euler’s Method for graphing, and a few other algorithmic methods for approximating answers. They have the advantage of being less difficult to implement for the mathematician than finding perfect, symbolic answers.
By design, numerical methods are great for computers. Computers are built to follow instructions quickly and accurately, and it is far simpler to make a computer do a numerical method than it is to get a computer to do algebra. Many computational answers are calculated using these methods, and in some cases (especially integrals) no simpler method exists.
What Are Some Disadvantages of Numerical Methods
Numerical methods are not without their flaws. While simple to implement, they lose accuracy and computational power. For instance, Euler’s Method for graphing gets worse and worse the further you go with it, eventually creating a terrible estimate of the initial line. Newton’s method can fail to find an answer at all, even when the answer is obvious to the mathematician. Guess and check, the old standby, could take all day if you are a bad guesser.
Ideally, use of a numerical method is based on knowing what the problems with the method are. Often, you can mitigate the problems or at least put an upper bound on the error of the answer. This is important for real life engineering, where a misplaced number could mean life or death in the final product.
Wolfram-Mathworld: Newton’s Method
Wolfram-Mathworld: Euler Forward Method
Wolfram-Mathworld: Riemann Sum
Scholarpedia.org: Numerical Analysis