This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Presentationon
Interpolation and forward ,backward ,
central method
In partial fulfillment of the subject
CVNM
Submitted by:
Mitesh Patel (130120119155) / Mechanical / 4C1
Mitul Patel (130120119156) / Mechanical / 4C1
Neel Patel (130120119157) / Mechanical / 4C1
(2140001)
G ANDH INAG AR INSTITUTE O F TECH NO LO G Y
INTERPOLATION AND EXTRAPOLATION
The process of finding the values inside the interval ๐ฅ0<๐ฅ< ๐ฅ๐ is known as interpolation.
The process of finding the values outside the interval ๐ฅ0<๐ฅ< ๐ฅ๐ is known as extrapolation.
Interpolation
Forward interpolation
Backward interpolation
POLYNOMIAL INTERPOLATION
For a two point data a first order(linear)polynomial connecting two
points is used.
For a three point data a second order (quadratic) polynomial connecting
three point used
For four point data a third order (cubic) polynomial connecting three
point.
FINITE DIFFERENCES
FINITE DIFFRENCES
FORWARD DIFFERENCES
CENTRAL DIFFERENCES
BACKWARD DIFFERENCES
Finite differences are of three types :-
RULES OF INTERPOLATIONInterpolation formulas can be used only when the values of the argument ๐ฅ are equidistant.
The point ๐ฅ0 should be selected very close to the point at which interpolation is required.
Usually in the forward interpolation the very first value of ๐ฅ is taken equal to ๐ฅ0.
Backward interpolation is suitable for interpolation near the end of tabulated values in the backward interpolation.
In backward interpolation the last value of ๐ฅ is taken equal to ๐ฅ๐.
FIRST FORWARD DIFFERENCES
The ๐ฆ1- ๐ฆ0 , ๐ฆ2 - ๐ฆ1 , ๐ฆ๐ - ๐ฆ๐โ1.differences are called the first forward differences of the function.
y = f (x) and we denote these difference by
โ๐ฆ0 , โ๐ฆ1 , โ๐ฆ2โฆโฆโฆ.., โ๐ฆ๐respectively, where ฮ is called the descending or forward difference operator.
In general, the first forward differences is defined by
ฮ๐ฆ๐ฅ= ๐ฆ๐ฅ+1โ ๐ฆ๐ฅ.
where ฮ is called first forward difference operator.
SECOND FORWARD DIFFRENCE OPERATOR
The differences of first forward differences are called second forward differences.