Numerical Analysis
Unlike most fields of mathematics, numerical analysis is not about proving theorems or finding generalized solutions to problems. Instead, numerical analysis is the field that focuses on finding a usable solution to a specific question about a known function. Some of the most common problems in which numerical analysis is used are integrating specific regions of a curve and finding the zeroes of a function. Instead of finding the desired answer exactly, numerical analysis gives a way to find the answer within a specified margin of error.
There are many popular techniques in numerical analysis. All involve the knowledge of the function itself, which yields the value at all points. This value can be used to approximate distance from the axis or other useful approximation quantities in the process of calculating the desired quantity. For some results there will be numerous techniques for numerical solution, all of which converge upon the same result, but at different speeds and with different degrees of certainty. For example, for numerical integration, one may have occasion to use the trapezoid rule, Simpson's rule, Boole's rule, or Gauss-Legendre quadrature. The trapezoid (or trapezoidal) rule divides the area of interest into a selected number of trapezoids and calculates the area of each trapezoid, then sums the areas to get the total integration area. (This could also be expressed by rectangles with heights at the midpoint of each interval.) Simpson's rule takes the same type of subdivided regions and approximates the area of each region using a quadratic polynomial fit instead of a straight line across the top forming a trapezoid. Boole's rule uses a recursive relationship of the type of quadratic fits used above in Simpson's rule. Gauss-Legendre quadrature uses uneven choices for values of the function and tables of abscissa and weights to find the average values of the function across the interval, from which the area is calculated. Each of these rules has an error rate and several methods of improving the accuracy of the procedure. This is only one example of the many problems for which numerical analysis can give multiple paths to an approximately correct solution. Of course factors such as interval size and placement will come to bear on this particular example, as will the errors inherent in the techniques themselves, so that even the poorest technique can be made fairly feasible with technical improvements.
Other useful applications of numerical analysis are to find polynomial interpolations and extrapolations of functions, solutions to matrices or linear systems of equations, curve fitting (including least-squares fit), numerical differentiation, functional optimization, solution of eigenvalue and eigenvector equations, and solutions of ordinary and partial differential equations.
While numerical analysis techniques have been known for hundreds of years, they have become much more widely used since the advent of computers and electronic calculators. First, many of the simple computations performed by these devices are approximations within a certain range of error. The number of decimal places available in some of the computations will provide the source of that error. Second, the availability of computer programs to do more complex calculations allows many more iterations on an approximation, bringing it ever closer to an exact answer and thus rendering it more useful. The number of available calculations per minute means that a problem that would take hours to approximate closely by hand would take less than a minute to approximate using a computer program. Because of this capability, numerical analysis is more widely used today than it ever has been before. With the knowledge of a structured programming language, numerical analysis can be a very useful tool in scientific or engineering applications where exact solutions would prove to be impossible or much more difficult.
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