MATHEMATICS

Mathematics

Mathematics

Numerical Methods

For this course work we are discussing the solutions to different equations by the help of following three methods which are listed below:

Interval Estimation,

The Newton Raphson Method

And the Re-arrangement Method.

After reviewing these three methods in details and understand the language associated with it. Each method that has been discussed in this paper has its own advantages and disadvantages which are defined in details.

Interval Estimation Method

Consider a classic in applied mathematical statistics parametric estimation problem. Baseline data - sample x 1, x 2, ..., x n, consisting of n real numbers. In the probabilistic model of a simple random sample of its elements x 1, x 2, ..., x n considered to be a set of n realizations independent and identically distributed random variables. We assume that these values ??have a density f (x).In parametric statistical theory, it is assumed that the density f (x) is known up to a finite-dimensional parameter, i.e.,  for some  This, of course, a very strong assumption, which requires justification and validation, but at present the parametric estimation theory is widely used in various application areas.

All observations are determined with some precision, in particular, recorded using a finite number of digits (usually 2 - 5).Consequently, all of the real distribution of the results of observations are discrete. It is generally believed that these discrete distributions well enough continuous approach. Elaborating on this statement, we have to consider a model according to which statistics are available only values

y j = x j +  j, j = 1, 2, ... , N,

where x i - «true" values,  observation errors (including sampling error). In the probabilistic model we assume that n steam

form a simple random sample from a two-dimensional distribution, where x 1, x 2, ..., x n   - Sample from the distribution with density  . Please be aware that  and  - The implementation of dependent random variables (assuming them independent, the distribution of yi will be continuous rather than discrete). Since the bias is usually impossible to exclude [26, p.141], it is necessary to consider the case  There is no reason a priori to accept and normality of errors (according to a summary of the experimental data on the diversity of the distribution of the measurement errors given in [26, p.148] and [27, p.71-77], in most cases, the hypothesis of normal distribution of the errors found unsuitable for measuring various types). Thus, all three common understanding of the properties of the errors are not adequate to reality. The influence of observational errors on the statistical properties of the models to be studied on the basis of different models, namely, models of interval statistics.

Let  be the characteristic size of the error, for example, the mean square error   . In classical mathematical statistics  considered to be negligible (  ) for a fixed sample size n. Overall results are proved in the asymptotic   . Thus, in the classical mathematical statistics is first made the limit   , then the limit   . In the statistics of the interval data assume that the sample size is large enough (  ), But all the measurements correspond to the same specification error   . Useful for the analysis of real data, we obtain limit theorems for   . In statistics, interval data is first made the limit   , then the limit   . So, in the two theories are used the same two limits:  and  But in a different order. The statements of the two theories are fundamentally ...