Numerical Differential & Integration

Introduction• If a function f(x) is defined as an expression, its

derivative or integral is determined using analytical techniques.

• If f(x) is complicated or when it is given in a tabular form, numerical methods are used to determine the derivative or integral.

• The accuracy of these methods would depend on the given function & the order of the polynomial used. If the fitted polynomial is exact the error is likely to be zero.

• The numerical methods would be avoided if an alternative exists.

Numerical Differentiation

• Find a suitable interpolating polynomial to represent the function.

12

This provides the value of dy/dx at any which is not in the table

3

Putting x = a and u = 0 in 3

3

4

4

Maxima & Minima of Tabulated Functions

Maxima/Minima of a y=fx) can be found by equating dy / dx = 0

Differentiating it w. r. t u we get

For simplicity truncate terms after the third term and by solving this quadratic equation we get two values of u.

Reading Assignment

• Read on errors in numerical differntiation and write a comprehensive report.

Numerical Integration

Newton-Cote’s Quadrature Formula

This is a Newton-Cote’s Formula. Trapezoidal Rule, Simpson’s one-third and three-eigth rules, and Weddle’s rule all can be deduce from this by putting n=1,2,3 & 6.

Assignment No. 4