Numerical Analysis Lecture 27

Chapter 7 Numerical Differentiation and Integration

INTRODUCTION DIFFERENTIATION USING DIFFERENCE OPERATORS DIFFERENTIATION USING INTERPOLATION RICHARDSON’S EXTRAPOLATION METHOD NUMERICAL INTEGRATION

NEWTON-COTES INTEGRATION FORMULAE THE TRAPEZOIDAL RULE (COMPOSITE FORM ) SIMPSON’S RULES (COMPOSITE FORM ) ROMBERG’S INTEGRATION DOUBLE INTEGRATION

DIFFERENTIATION USING DIFFERENCE OPERATORS

Introduction Consider a function of single variable y = f (x). If the function is known and simple, we can easily obtain its derivative (s) or can evaluate its definite integral

However, if we do not know the function as such or the function is complicated and is given in a tabular form at a set of points x0,x1,…,xn, we use only numerical methods for differentiation or integration of the given function.

We shall discuss numerical approximation to derivatives of functions of two or more variables in the lectures to follow when we shall talk about partial differential equations.

In what follows, we shall derive and illustrate various formulae for numerical differentiation of a function of a single variable based on finite difference operators and interpolation.

Subsequently, we shall develop Newton-Cotes formulae and related trapezoidal rule and Simpson’s rule for numerical integration of a function.

DIFFERENTIATION USING DIFFERENCE OPERATORS

We assume that the function y = f (x) is given for the values of the independent variable x = x0 + ph, for p = 0, 1, 2, … and so on. To find the derivatives of such a tabular function, we proceed as follows

Case I: Using forward difference operator ∆ and combining equations and

Remember the Differential operator, D is known to represents the property

This would mean that in terms of ∆ :

Therefore,

In other words,

Also, we can easily verify

Case II: Using backward difference operator , we have On expansion, we have

we can also verify that

Hence,

and

The formulae for Dy0and D2y0 are useful to calculate the first and second derivatives at the beginning of the table of values in terms of forward differences; while formulae for y’n and y’’n

are used to compute the first and second derivatives near the end points of the table, in terms of backward differences. Similar formulae can also be derived for computing higher order derivatives.

Case III: Using central difference operator δ and following the definitions of differential operator D, central difference operator δ and the shift operator E, we have

Therefore, we have But,

Using the last two equations we get That is,

Therefore, Also

Hence

For calculating the second derivative at an interior tabular point, we use the equation

while for computing the first derivative at an interior tabular point, we in general use another convenient form for D, which is derived as follows. Multiply the right hand side of

which is unity and noting the Binomial expansion by which is unity and noting the Binomial expansion

We get

On simplification, we obtain Therefore the equation can also be written in another useful form as

The last two equations for y” and y’ respectively are known as Stirling’s formulae for computing the derivatives of a tabular function. Similar formulae can be derived for computing higher order derivatives of a tabular function.

The equation for y’ can also be written as

In order to illustrate the use of formulae derived so far, for computing the derivatives of a tabulated function, we shall consider some examples in the next lecture.

Numerical Analysis Lecture 27