Polynomial Interpolation

 

The existence and uniqueness theorem states that for image002ae distinct nodes image006ae, image008ae, with associated function values image004ae, there exists a unique polynomial image010ae, such that

 

image012ae

 

and,

 

image014ae.

 
image010ae is called the interpolating polynomial for the given set of interpolation points. The equations image016ae, image008ae, are called the interpolation conditions.

 

 

If the values image018ae of a function image020ae that belongs to the set of continuous functions in the interval image022ae, image024ae, are known at  the image002ae nodes  image006ae, image008ae, and if  image026ae is the interpolating polynomial for the interpolation points image028ae, then image030ae, wherefrom it is reasonable to assume that  image010ae approximates image020ae in the interval image022ae.

 

Determining a value image032ae, image034ae and image036ae, image008ae, is called interpolation; determining a value image032ae , image039ae, is called extrapolation.

 

 

Sample Polynomial Interpolation and Extrapolation

lagrange

 

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