Birkhoff interpolation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Birkhoff interpolation is an extension of <a href="/facts/Polynomial_interpolation/DmnuxaKT">polynomial interpolation</a>. It refers to the problem of finding a polynomial 
 
 
 
 P
 (
 x
 )
 
 
 {\displaystyle P(x)}
 
 of degree 
 
 
 
 d
 
 
 {\displaystyle d}
 
 such that only certain <a href="/facts/Derivative/7Ito70Dh">derivatives</a> have specified values at specified points:

P
          
            (
            
              n
              
                i
              
            
            )
          
        
        (
        
          x
          
            i
          
        
        )
        =
        
          y
          
            i
          
        
        
        
          
            for 
          
        
        i
        =
        1
        ,
        …
        ,
        d
        ,
      
    
    {\displaystyle P^{(n_{i})}(x_{i})=y_{i}\qquad {\mbox{for }}i=1,\ldots ,d,}

where the data points 
 
 
 
 (
 
 x
 
 i
 
 
 ,
 
 y
 
 i
 
 
 )
 
 
 {\displaystyle (x_{i},y_{i})}
 
 and the nonnegative integers 
 
 
 
 
 n
 
 i
 
 
 
 
 {\displaystyle n_{i}}
 
 are given. It differs from <a href="/facts/Hermite_interpolation/u36d6yH2">Hermite interpolation</a> in that it is possible to specify derivatives of 
 
 
 
 P
 (
 x
 )
 
 
 {\displaystyle P(x)}
 
 at some points without specifying the lower derivatives or the polynomial itself. The name refers to <a href="/facts/George_David_Birkhoff/3aywz7gs">George David Birkhoff</a>, who first studied the problem in 1906.

Birkhoff interpolation open-in-new

Birkhoff interpolation