Interpolation,Extrapolation,Interpolation Formulae

Graphical representation of Interpolation and Extrapolation

In this post we define some definitions of numerical analysis and their formulas which are very important related to the topic.There are seven types of it.

Definition:(Interpolation)

            The process of finding the value of function inside the given range of arguments is known as interpolation

Definition:(Extrapolation)

             The process of finding the value of function outside the given range of arguments is known as interpolation.

Interpolation Formulae:

           There are following Interpolation Formulae that are used in interpolation.

  1. Geogory-Newton forward Interpolation Formula
  2. Geogory-Newton backward Interpolation Formula
  3. Gauss Forward Interpolation Formula
  4. Gauss Backward Interpolation Formula
  5. Stirling Formula
  6. Bessel’s Formula
  7. Laplace-Everett’s Formula

1. Geogory-Newton forward Interpolation Formula

               The formula of the form given below is called Geogory-Newton forward Interpolation Formula.

\fn_cm \small y_p=y_o+p\Delta y_o+\frac{p(p-1)}{2!}\Delta^2 y_o+...+\frac{p(p-1)(p-2)...[p-(n-1)]}{n!}\Delta^n y_o

2. Geogory-Newton Backward Interpolation Formula

               Geogory-Newton Backward Interpolation Formula is given as

\fn_cm \small y_p=y_n+p\bigtriangledown y_n+\frac{p(p-1)}{2!}\bigtriangledown^2 y_n+...+\frac{p(p-1)(p-2)...[p+n-1)]}{n!}\bigtriangledown^n y_n

3.Gauss Forward Interpolation Formula

                Gauss Forward Interpolation Formula given as

\fn_cm \small y_p=y_o+p\Delta y_o+\frac{p(p-1)}{2!}\Delta^2 y_{-1}+...+\frac{(p+1)(p-1)}{3!}\Delta^3 y_{-1}+\frac{(p+1)p(p-1)(p-2)}{4!}\Delta^4 y_{-2}+...

is called Gauss Forward Interpolation Formula

4. Gauss Backward Interpolation Formula

               The formula of the form given below is called Gauss Backward Interpolation Formula.

\fn_cm \small y_p=y_o+p\Delta y_{-1}+\frac{p(p+1)}{2!}\Delta^2 y_{-1}+\frac{(p+1)p(p-1)}{3!}\Delta^3 y_{-2}+\frac{(p+1)p(p-1)(p+2)}{4!}\Delta^4 y_{-2}+...

5.Stirling Formula

           The formula  given below is called Stirling Formula if

 \fn_cm \small y_p=y_o+pu\delta y_o+\frac{p^2}{2!}\delta^2 y_o+\frac{p(p^2-1)}{3!}\delta^3 y_o+\frac{p^2(p^2-1)}{4!}\delta^4 y_o+...

6.Bessel’s Formula

               The formula of the form given below is called Bessel’s Formula

\fn_cm \small y_p=\frac{1}{2}(y_o+y_1)+(p-\frac{1}{2})\Delta y_o+\frac{(p-1)}{2!}[\frac{\Delta^2y_{-1}+\Delta^2y_o}{2}]+...

7.Laplace-Everett’s Formula

The formula of the form given below is called Laplace-Everett’s Formula

 \fn_cm \small y_p=((1-p)y_o+py_1)+\frac{p(p-1)(p-2)}{2!}\Delta^2y_{-1}+\frac{p(p-1)(p+1)}{3!}\Delta^2y_{o}+...

You can also attempt mcqs about numerical analysis test 01 and  so on.
pakmath team will provide these results in the next posts and also with their proofs. But these proofs will be provided in the separated posts.

 

One Reply to “Interpolation,Extrapolation,Interpolation Formulae”

  1. Oh my goodness! an amazing article dude. Thanks However I’m experiencing situation with ur rss . Don’t know why Unable to subscribe to it. Is there anybody getting equivalent rss drawback? Anybody who is aware of kindly respond. Thnkx

Leave a Reply

Your email address will not be published. Required fields are marked *