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.

1 2 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