1.0 INTRODUCTION
The problem of numerical differentiation does not receive very much attention nowadays. Although the Taylor series plays a key role in much of classical analysis, the poor reputation enjoyed by numerical differentiation has led numerical analysts to construct techniques for most problems which avoid the explicit use of numerical differentiation. One may briefly and roughly define the term numerical differentiation as any process in which numerical values of derivatives are obtained from evaluations of the function at several abscissae near . These numerical differentiation methods are including:
1. Finite Difference Approximation of First and Second Derivative
2. Richardson’s Extrapolation
2.0 OBJECTIVE OF THE REPORT
Objectives for this assignment are:
(a) To reduce the truncation error for approximating the derivative of a function f to coax the results of the higher accuracy out of some numerical formulas.
(b) To show how numerical method very useful in calculation
(c) To expose students about the relationship of numerical method in programming and real condition in engineering field.
(d) To show how numerical method very useful in calculation.
(e) To prepare student for the real life problems.
(f) To build skills among students in order to complete the assignment.
3.0 FINITE DIFFERENCE APPROXIMATION OF FIRST DERIVATIVE
3.1 LITERATURE REVIEW
If a function is specified by means of a table of discrete values, rather than an algebraic expression, then the analytical methods described in earlier chapters cannot be used to determine derivatives. Such situations can occur when experimental measurements are used to obtain data, e.g. distance values at specific times for a moving object, and the function relating the data values is not known. In addition, the calculation of derivative values by means of computers tends to be done numerically, even when the function is known, by operating on discrete values obtained for the function. This topic describes how derivatives can be obtained in such situations.
There are two ways of considering the methods. One is in terms of considering lines joining data values to represent chords between points on a graph of the function. The other, which generates the same equations, involves representing the function by a series. We can represent a function by a series. We can represent a function by a polynomial series, the data values then being points which the series must be represent. In considering, in this topic, various degees of sophistication of numerical methods the equations are developed by both these methods.
3.2 THEORY
3.2.1 Taylor’s series
The basic used in this topic for development of numerical methods based on representing a function by a series, is in terms of the Taylor’s series. The following is a brief discussion of the series.
A function can be represented, provided enough terms are considered, by a polynomial series.
[1]where A, B, C, D, E, etc. are constants. Consider the value of this function at x=a. Then
[2]Equation [1] minus equation [2] enables us to eliminate constant A to give
[3]Differentiating equation [2] with respect to gives
[4]where is used to represent the . Multiplying this equation by and subtracting it from equation [3] eliminates to give
[5]Differentiating equation [4] with respect to gives
[6]where represents . Multiplying this equation by and subtracting it from equation [5] eliminates C and gives
=In general we can write
[7]where we have representing , representing , etc. The equation is known as Taylor’s series or theorem
We can employ Taylor’s series expansions to derive finite-divided-difference approximations of derivatives. We develop forward, backward and centered difference approximations of first and higher derivatives.
3.2.2 Finite difference approximations of first derivative
3.2.2.1 Forward difference approximation of first derivative
The Taylor’s series can be expanded forward by utilizes data i and i+1 to estimate the derivative
So
And we have
Is known as two point forward difference formula.
3.2.2.2 Backward difference approximation of first derivative
The Taylor’s series can be expanded backward to calculate a previous value on the basis of a present value, as in