UNIT-2

Finite Differences

and

Interpolation

with

Equal Intervals

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Differences:

1. Forward Difference: The forward difference is

denoted as ∆f.

If for given arguments are

and its functional values are

The forward difference is respectively define as:

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Calculation of Higher order Difference: (Second order)

In this way one can calculate----------

In this way one can obtain-------

Exm. Construct difference table for following data.

(2) Back ward Difference: The Back ward difference is denoted as f.

If for given arguments are

and its functional values are

The Backward difference is respectively define as:

Calculation of Higher order Difference: (Second order)

In this way one can obtain-------

Exm. Construct difference table for following data.

In this way one can obtain-------

Exm. Construct difference table for following data.

The Central difference

Define as:

The Shift Operator (E)

Define as:

The Mean or Average Operator

Define as:

In usual notation prove:

Exa. Find the missing term of the following data.

f(1)=8, f (2)=12, f(3)=19, f(4)=29

Sol.: First of all we construct difference table for the given data.

We have to find out f(6):

First of all we note the following formula

From the given data the difference value of x means

h=1, initial value x0=1 we have to given four values,

So we can obtain expansion up to ∆3----

In calculation of f(6)=f(x5)

In usual notation prove that

Let y=f(x) be a continuous function and the value of x are equally space .

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Expression of Polynomial

Let f(x) be a given polynomial of degree n which is to be expressed in terms of factorial notation.

Let

Expression of Polynomial

Let f(x) be a given polynomial of degree n which is to be expressed in terms of factorial notation.

Let

Substituting x=0, in all the above equations, we have

Sub. Ai’s we get

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Exa. Express in factorial notation and hence find its differences in factorial notation, the interval of differencing being unity.

Sol. Here given polynomial of degree four is given.

So, we need in our difference table up to fourth difference.

We have

Find the polynomial f(x), which satisfy the following

data and hence find the value of f(1.5).

Sol. We have to construct a difference table for

Given data.

As the third difference is constant.

So, we have a polynomial of degree three.

We know that

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Self-Practice example: Write down the polynomial which satisfy the following data.

Interpolation:

The method of finding the value of f(x) for a given

Value of x is known as interpolation.

NEWTON’S FORWARD DIFFERENCE INTERPOLATION FORMULA:

Let y=f(x) be a function of x. Given the set of n+1

value

Where x is independent and y dependent variable.

The value of x is at equally spaced.

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So, y=f(x) is a polynomial of x degree n.

Such that the value of y agree at the given point

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can be written as

(1)

Where are constant to be determine.

Put in equation (1)

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Again, put in equ.(1)

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Put in equation (1)

Similarly putting

Now

Similarly, we can find remaining terms, using all the above values in equation (1), we get

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Exa.: The population of town were as under

When Finding is starting of the table than we can apply Newton’s forward formula

Estimate the population for the year 1895.

Solution: We have to find x=1895 which is starting of the table, so we can apply Newton’s forward

interpolation formula.

NOW, NEWTON’S FORWARD DIFFERENCE INTERPOLATION FORMULA:

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Difference table:

For given data:

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Sub. All these values in N.F..I.F.

Self-Practice Example:

Find f(1.02) having given:

Stirling ‘s Interpolation Formula

Let y=f(x) is a function of x.

Given set of n+1 value

Of x & y. The value of x is equally spaced.

To find a polynomial of a x of degree n we consider, the following diff. table in which central co-ordinate as , corresponding to

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By Gauss’s forward formula, we have

By Gauss’s Backward formula, we have

Adding equation (1) and (2) gives the formula

To find a polynomial of a x of degree n we consider, the following diff. table in which central co-ordinate as , corresponding to

Bessel’s Interpolation Formula

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Difference table:

The Bessel’s formula is of the form

Where are constant to be determine.

We know that

Sub. these values of (a), (b),(c) and (d) in equation (1)

Comparing the coefficient of from

Equ. (2) and (3)

Using all these values in equation (1), we get

Bessel’s Formula

To find a polynomial of a x of degree n we consider, the following diff. table in which central co-ordinate as , corresponding to

Everett’s Interpolation Formula

Let y=f(x) is a function of x. Given a set of n+1

values of x and y. The value of x are at equally spaced.

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Difference table:

For this formula we can use even order term from

The table.

The Everett’s formula is of the form

Where are constant to be determine.

We know that

Sub. these values of (a), (b),(c) and in equation (1)

Comparing the coefficient of equ. (2) and (3)

Consider

Using the values of In equation (1)

Everett’s Formula