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INTERPOLATION, STIRLING FORMULA, DEFINITION, DISADVANTAGES, ADVANTAGES,…
INTERPOLATION
purpose of interpolation
approximating unknown values
filling in missing data
creating smooth curves
For Equally Spaced Intervals
Central Difference Method
DEFINITION : This formula is a technique used to approximate the value of a function at a specific point by considering the function values at nearby points. It utilizes finite differences and provides higher accuracy compared to simpler interpolation methods.
TYPES
GAUSS FORMULA
Gauss backward formula
The gauss backward interpolation formula is a method used to approximate the value of a function at a given point within a given range . It is a variant of Newtons divided difference interpolation formula and provides an estimate based on a set of equidistant data points.
Limited applicability , Sensitivity to data spacing , susceptible to cumulative errors . Limited flexibility , Higher degree polynomials, decrease accuracy
Simplicity , good for Equidistant data points , computationally efficienr
APPLICATIONS
1) Interpolating x near centre of the table
DISADVANTGES
1)Decreses more rapidly
2)Gives better accuracy
ADVANTAGES
This method is only applicable if there is a uniform difference between two consecutive x
Gauss forward formula
ADVANTAGES
Simple , good for equidistant data points , computationally more efficient as compared to langrange formula.
disadvantages
Limited applicability , Sensitivity to data spacing , susceptible to cumulative errors , limited flexibility , higher degree polynomials decrease accuracy
DEFINITION : Forward interpolation formula is a method used to approximate the value of a function at a given point within a given range. It is a variant of Newtons divided difference interpolation formula and provides an estimate based on a set of equidistant data points.
Newton Forward Difference Method
DISADVANTAGES
It is only applicable for the equidistant values of a function.
Sometimes they tend to overfit the data.
Newton's forward difference formula is a finite difference identity giving an interpolated value between tabulated points in terms of the first value and the powers of the forward difference .
ADVANTAGES :
These are simpler to evaluate than the approximations of non-polynomials.
The data sets can fit exactly for higher-order polynomials.
Newton backward difference Method
The differences y1 – y0, y2 – y1, ……, yn – yn–1 when denoted by dy1, dy2, ……, dyn, respectively, are called first backward difference.
In order to reduce the numerical computations associated to the repeated application of the existing interpolation formula in computing a large number of interpolated values, a formula has been derived from Newton's backward interpolation formula for representing the numerical data
Applications
This method is used to interpolate the values of y which is near the end of the difference table.
Disadvantages
Only applicable for the equidistant values of a function.
Sometimes they tend to overfit the data.
Advantages
Simpler to evaluate than the approximations of non-polynomials.
For higher-order polynomials.
data sets fit exactly
For unequally spaced intervals
Newton divided difference formula
NEWTON FORWARD DIVIDED DIFFERENCE FORMULA
ADVANTAGES
More data points can be added, for improved accuracy
The data sets can fit exactly for higher-order polynomials
DISADVANTAGES
It is less efficient than Lagrange's interpolation
NEWTON BACKWARD DIVIDED DIFFERENCE FORMULA
ADVANTAGES
It has almost the same advantages as newton forward divided difference formula
DISADVANTAGES
Disadvantages are also similar to Forward divided difference formula
Lagrange Interpolation
To find a polynomial which takes on certain values at arbitrary points.
ADVANTAGES
Used to determine the function's value even when the parameters are not evenly spaced
Answers for higher order polynomials are more efficient
DISADVANTAGES
For higher order polynomials it becomes very difficult task to do as number of points increase
STIRLING FORMULA
DEFINITION
DISADVANTAGES
ADVANTAGES
formula
formula
