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
  Linear approximations (red) to green curve. Note that successive approximations are progressively closer to exact solution.
  
  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."