How to Understand Numerical Analysis

The field of numerical analysis is traditionally used where solutions to various equations must be found. Numerical analysis often arrives at practical approximations, not exact answers. Even the best algorithm may require infinite arithmetic precision and infinite iterations to arrive at an exact solution. Questions subject to numerical analysis may be as complicated as differential equations modeling the weather, or as simple as determining an adequate approximation to square root of two. A famous example of numerical analysis is Newton's Method, used to find the root of a function. Numerical analysis is also used to account for computation error and algorithm stability.

Things You'll Need

• Calculator and spreadsheet software
• Numerical methods text with explanations and exercises

Instructions

Definition and Applications

1. By definition, numerical analysis is use of algebraic and numeric manipulation to answer problems in continuous mathematics. Understanding numerical analysis is tied to understanding algorithmic iterations---steps done repeatedly in order to converge on a solution.

2. 

   A famous use of numerical analysis is Newton's Method, used to converge on the root of a function. Recall that Newton's Method only gives the exact solution in the limit of infinite steps. Realistically, Newton's Method is used until it is clear that a root precise enough for an application is achieved (see first Reference below).

3. Similarly to functions, numerical analysis is used to solve otherwise impossible differential equations. Differential equations are a topic within calculus that helps to describe a vast array of physical processes. For example, the arrows indicating pressure or wind gradients on a weather map are dictated by differential equations (see second Reference below).

4. In computers, even mathematically correct operations may give clearly wrong answers. This occurs because numbers can only be so precise when in computer memory. Round-off and other types of error have to be accounted and incorporated into statements on solution reliability. For example, if operation (x -- y) contains an error due to limited precision, then (x -- y -- z) would have even greater error for the same reason. The same idea applies to multiplication and division, exponential and trigonometric operations (see third Reference below).

5. Stability refers to accurate solution approximations. Stability can be an issue if a denominator with a variable occurs. If an algorithm gives nonsensical solutions, it is unstable. If an algorithm converges on a solution smoothly, then the algorithm is stable. The topic is explored in depth by Tuncer Cebeci in "Stability and Transition: Theory and Application: Efficient Numerical Methods with Computer Applications."

Tips & Warnings

• Practice problems done by hand and on computers are immensely helpful in aiding comprehension given the computationally focused nature of the field.

References

• Knowledge Representation Laboratory. "Numerical Analysis Overview"
• University of Karlschule, Institute for Hydromechanics. "Numerical Methods"
• California State University, Department of Mathematics. "Numerical Analysis"

Resources

• University of South Florida. "Transforming Numerical Methods Education"