Unit-3

Numerical Integration and Differentiation

3.1 Introduction: Numerical differentiation of Newton’s forward and backward formula

Newton’s forward Difference formula:

This method is useful for interpolation near the beginning of a set of tabular values.

    Where
 

Differentiating both side with respect to p, we get

h

This formula is applicable to compute the value of for non tabular values of x.

For tabular values of x , we can get formula by putting 

Therefore

In similar manner we can get the formula for higher order by differentiating the previous order formulas

   Again differentiating with respect to p, we get

Hence

Also

And so on.

Example1: Given that

Find at .

Here the first derivative is to be calculated at the beginning of the table, therefore forward difference formula will be used

Forward difference table is given below:

By Newton’s forward differentiation formula for differentiation

Here 

Example2: Find the first and second derivatives of the function given below at the  point :

Here the point of the calculation is  at the beginning of the  table,

Forward difference table is given  by:

By Newton’s forward differentiation formula for differentiation

Here  , 0.

Again 

At

Example3: From the following table of values of x and y find   for 

 Here the value of the derivative is to be calculated at the beginning of the table.

Forward difference table is given by

From Newton’s forward difference formula for differentiation we get

Here

     =0.48763

Newton Backward Difference Method:

This method is useful for interpolation near the ending of a set of tabular values.

Where 

Differentiating both side with respect to p, we get

This formula is applicable to compute the value of for non tabular values of x.

For tabular values of x , we can get formula by putting 

Therefore

In similar manner we can get the formula for higher order by differentiating the previous order formulas

Differentiating both side with respect to p, we get

Also

Example1: Given that

Find ?

Backward difference table:

Newton’s Backward formula for differentiation

Here

Example2: Given that

Find the derivative of y at ?

The  difference table is  given below:

Since the point  is at the beginning of the table therefore

From Newton’s forward difference formula for differentiation, we get

Here

Since the point is at the end of the table therefore

Backward difference table is:

Newton’s Backward formula for differentiation

Key takeaways-

3.2 Stirling’s, Bessel’s, Everett’s formula

Stirling’s formula-

Stirling’s formula is defined as-

Note-

The formula gives the best estimates when

Example: By using stirling formula to find , given-

Sol.

Suppose the origin is at 30 and h = 5

a + hu = 28

Where h = 5 and a = 35 then-

u = -0.4

The difference table will be as follows-

By using stirling formula, we get-

= 47691.8256

So that, we have

Example: By using stirling’s formula, compute from the table given below

Sol.

Here let the origin is at , h = 1,

Then using stirling’s formula-

Bessel’s formula-

The formula given below is called Bessel’s formula-

Note- this formula is useful when  u =1/2 and gives best estimates when ¼<u<3/4

Example: By using Bessel’s formula to find . Given

Sol.

The central difference table-

By using Bessel’s formula-

We get-

Everette’s formula-

The Everette’s formula is defined as-

Here w = 1 – u,

When u > ½, it gives best estimate.

Example: By using Everette’s formula, Evaluate f(30) if

F(20) = 2854,             f(28) = 3162

F(36) = 7088,             f(44) = 7984

Sol.

Lets origin is 28,

A = 28, h = 8

A +hu = 30

28 + 8u = 30

U = 0.25

And w = 1-u = 1-0.25 = 0.75

The difference table is-

By using Everette’s formula-

So that the value of f(30) = 4064

Key takeaways-

2.     Bessel’s formula-

3.     Everette’s formula-

 3.3 Lagrange’s Interpolation and Newton Divided difference formula

Finite differences-

Divided Difference:

In the case of interpolation, when the value of the arguments are not equi-spaced (unequal intervals) we use the class of differences called divided differences.

Definition:   The   difference which are defined by taking into consideration   the change in the value of the argument are known as  divided differences.

Let be a function defined as

Where are unequal i.e. it is case of unequal interval.

The first order divided differences are:

And so on.

The second order divided difference is:

And so on.

Similarly the nth order divided difference is:

With the help of these we construct the divided difference table:

Newton’s Divided difference Formula:

Let be a function defined as

Where are unequal i.e. it is case of unequal interval.

.

Example1: By means of Newton’s divided difference formula, find the values of from the following table:

We construct the divided difference table is given by:

By Newton’s Divided difference formula

.

Now, putting in above we get

.

Example2: The following values of the function f(x) for values of x are given:

Find the value of and also the value of x for which f(x) is maximum or minimum.

We construct the divide difference table:

By Newton’s divided difference formula

.

Putting  in above we get

For maximum and minimum of , we have

Or 

Example3: Find a polynomial satisfied by

 , by the use of Newton’s interpolation formula with divided difference.

Here

We will construct the divided difference table:

By Newton’s divided difference formula

.

This is the required polynomial.

Lagrange interpolation

Given a set of values of x and y, the process of finding the value of x for a certain value of y is called inverse interpolation.

Lagrange’s Inverse interpolation:

Let , be defined function we get

Where the interval is not necessarily equal. We assume f(x) is a polynomial of degree n.  Then Lagrange’s inverse interpolation formula is given by

Example1: Use the inverse interpolation to find value of x at for the following data:

Here , we have the data

The Lagrange’s inverse interpolation formula is given by

.

Example2: Use the inverse   Lagrange’s method to find the root of the equation , give data

Here , we have the data

Also.

The Lagrange’s inverse interpolation formula is given by

Thus, the approximate   root of the given equation is .

Example3: Find the value of x at for the following data:

Here , we have the data

Also.

The Lagrange’s inverse interpolation formula is given by

Thus, the value.

Key takeaways-

Lagrange’s inverse interpolation formula is given by

3.4 Numerical Integration: Newton cotes formula, Trapezoidal rule, Simpson’s 1/3 and 3/8 rules, Boole’s rule, Waddle’s rule.

Numerical Integration

Numerical integration is a process of evaluating or obtaining a definite integral from a set of numerical values of the integrand f(x).In case of function of single variable, the process is called quadrature

Newton cotes formula-

Suppose where y takes the values

And let the integration interval (a, b) is divided into n equal sub-intervals, each of width h = b – a /n, so that,

The above formula is known as Newton’s cotes formula.

This is also known as general quadrature formula.

Trapezoidal Method:

Let the interval be divided into n equal intervals such that <<…. <=b.

Here.

To find the value of.

Setting n=1, we get

Or I=

The above is known as Trapezoidal method.

Note: In this method second and higher difference are neglected and so f(x) is a polynomial of degree 1.

Geometrical Significance: The curve y=f(x), is replaced by n straight lines with the points ();() and ();…….;() and ().

The area bounded by the curve y=f(x), the ordinates, and the x axis is approximately equivalent to the sum of the area of the n trapeziums obtained.

Example1: State the trapezoidal rule for finding an approximate area under the given curve. A curve is given by the points (x, y) given below:

Estimate the area bounded by the curve, the x axis and the extreme ordinates.

We construct the data table:

Here length of interval h =0.5, initial value a = 0 and final value b = 4

By Trapezoidal method

Area of curve bounded on x axis =

Example2: Compute the value of   ?

Using the trapezoidal rule with h=0.5, 0.25 and 0.125.

Here

For h=0.5, we construct the data table:

By Trapezoidal rule

For h=0.25, we construct the data table:

By Trapezoidal rule

For h = 0.125, we construct the data table:

By Trapezoidal rule

[(1+0.5) +2(0.98461+0.94117+0.87671+0.8+0.71910+0.64+0.56637)]

Example3: Evaluate, using trapezoidal rule with five ordinates

Here

We construct the data table:

Simpson’s Rule:

Let the interval be divided into n equal intervals such that <<…. <=b.

Here.

To find the value of.

Setting n = 2,

Which is known as Simpson’s 1/3- rule or Simpson’s rule.

Note: In this rule third and higher differences are neglected a so f(x) is a polynomial of degree 2.

Example1: Estimate the value of the integral

 By Simpson’s rule with 4 strips and 8 strips respectively.

For n=4, we have

Construct the data table:

By Simpson’s Rule

For n = 8, we have

By Simpson’s Rule

Example2: Evaluate

Using Simpson’s 1/3 rule with .

For , we construct the data table:

By Simpson’s Rule

Example3: Using Simpson’s 1/3 rule with h = 1, evaluate

For h = 1, we construct the data table:

By Simpson’s Rule

= 177.3853

Simpson’s 3/8 rule

Let the interval be divided into n equal intervals such that <<…. <=b.

Here.

To find the value of .

Setting n=3, we get

Is known as Simpson’s 3/8 rule which is not as accurate as Simpson’s rule.

Note: In this rule the fourth and higher differences are neglected and so f(x) is a polynomial of degree 3.

Example1: Evaluate

By Simpson’s 3/8 rule.

Let us divide the range of the interval [4, 5.2] into six equal parts.

For h=0.2, we construct the data table:

By Simpson’s 3/8 rule

= 1.8278475

Example2: Evaluate

Let us divide the range of the interval [0,6] into six equal parts.

For h=1, we construct the data table:

By Simpson’s 3/8 rule

+3(0.0385) +0.027]

=1.3571

Error in Integration

 Error in Trapezoidal method

The total error in trapezoidal method is given by

Let is the largest value of the n quantities on the right-hand side of the above equation then

 Error in Simpson’s Rule

The error in the Simpson’s rule is given by

Where is the largest value of the fourth derivative of y(x)

Error in Simpson’s 3/8 Rule

The error in this rule is given by

Where is the largest value of the derivative of y(x)

Boole’s rule & Waddle’s rule-

The formula given below is known as Boole’s rule-

And the waddle’s rule is defined as-

Example: A river is 80 m wide. The depth ‘b’ of the river at a distance ‘a’ from one bank is given by the table-

Find the approximate area of cross-section of the river using Boole’s rule.

Sol.

The area of the cross-section of the river will be-

The number of sub-intervals here is 8.

Then by Boole’s rule-

So that the area of the cross-section of the river is 708 square meters.

Example: Evaluate by using Boole’s.

Sol.

Take h = , so that there four sub intervals-

Using Boole’s method-

Key takeaways-

2.     Simpson’s Rule:
 

3.     Simpson’s 3/8 rule

4.     Boole’s rule & Waddle’s rule-

The formula given below is known as Boole’s rule-

And the waddle’s rule is defined as-

References