Numerical Methods


Numerical Method

Numerical methods almost never fail to provide an answer to any particular situation, but they can never furnish a general solution of any problem. The mathematical details outlined here include both analytic and numerical techniques useful in obtaining solutions to problems. Our discussion to this point has been confined to those areas in which the governing laws are well known. However, in many areas, information on the governing laws is lacking. Interest in the application of statistical methods to all types of problems has grown rapidly since World War II. Broadly speaking, statistical methods may be of

use whenever conclusions are to be drawn or decisions made on the basis of experimental evidence. Since statistics could be defined as the technology of the scientific method, it is primarily concerned with the first two aspects of the method, namely, the performance of experiments and the drawing of conclusions from experiments.

Traditionally the field is divided into two areas:
  1. Design of experiments. When conclusions are to be drawn or decisions made on the basis of experimental evidence, statistical techniques are most useful when experimental data are subject to errors. The design of experiments may then often be carried out in such a fashion as to avoid some of the sources of experimental error and make the necessary allowances for that portion which is unavoidable.
  2. Second, the results can be presented in terms of probability statements which express the reliability of the results. Third, a statistical approach frequently forces a more thorough evaluation of the experimental aims and leads to a more definitive experiment than would otherwise have been performed.
  3. Statistical inference. The broad problem of statistical inferenceis to provide measures of the uncertainty of conclusions drawnfrom experimental data. This area uses the theory of probability,enabling scientists to assess the reliability of their conclusions in termsof probability statements.
Both of these areas, the mathematical and the statistical, are intimately intertwined when applied to any given situation. The methods of one are often combined with the other. And both in order to be successfully used must result in the numerical answer to a problem—that is, they constitute the means to an end. Increasingly the numerical answer is being obtained from the mathematics with the aid of computers.

MISCELLANEOUS MATHEMATICAL CONSTANTS

Numerical values of the constants that follow are approximate to the number of significant digits given.

p = 3.1415926536 Pi
e = 2.7182818285 Napierian (natural) logarithm base
g = 0.5772156649 Euler's constant
ln p = 1.1447298858 Napierian (natural) logarithm of pi, base e
log p = 0.4971498727 Briggsian (common logarithm of pi, base 10
Radian = 57.2957795131°
Degree = 0.0174532925 rad
Minute = 0.0002908882 rad
Second = 0.0000048481 rad

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