There are two methods for interpolation. One is Graphical method and the other one is algebraic method.

**Methods of interpolation**

There are two methods for interpolation. One is Graphical method
and the other one is algebraic method.

We are given the â€˜*n*â€™ values of *x* and the
corresponding values of *y* for given function *y *=* f *(*x*)* *. we
plot these* n *observed points* *(* x _{i} *,

**Example 5.12**

Using graphic method, find the value of *y* when *x* =
38 from the following data:

*Solution:*

From the graph in Fig. 5.1 we find that for x = 38, the value of y
is equal to 35

Take
a suitable scale for the values of x and y, and plot the various points on the
graph paper for given values of x and y.

Draw a suitable curve passing through the plotted points.

Find
the point corresponding to the value x = 38 on the curve and then read the
corresponding value of y on the y- axis, which will be the required
interpolated valu

Newtonâ€™s Gregory forward interpolation formula (or) Newtonâ€™s
forward interpolation formula (for equal intervals).

Let *y* =
*f* ( *x*) denote a polynomial of
degree *n* which takes (*n* +
1) values. Let them be *y* _{0} , *y*_{1} , *y* _{2}
,... *y _{n}* corresponding to the values

The
values of (x_{0}, x_{1}, x_{2},â€¦.. x_{n}) are
at equidistant.

(i.e.)
x_{1}=x_{0}+h, x_{2}=x_{0}+2h, x_{3}=x_{0}+3h,â€¦â€¦.
X_{n}=x_{0}+nh,

Then
the value of *f*(x) at x = x_{0}+nh
is given by

**Note**

Newtonâ€™s forward interpolation formula is used when the value of
*y* is required near the beginning of the table.

Newtonâ€™s forward interpolation formula cannot be used when the
value of *y* is required near the end of the table. For this we use
another formula, called Newtonâ€™s Gregory backward interpolation formula.

**Note**

Newtonâ€™s backward interpolation formula is used when the value
of *y* is required near the end of the table.

Tags : Numerical Methods , 12th Business Maths and Statistics : Chapter 5 : Numerical Methods

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12th Business Maths and Statistics : Chapter 5 : Numerical Methods : Methods of interpolation | Numerical Methods

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