Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$
locally as a function of $x$. In this article, we express divided differences
of $y$ in terms of bivariate divided differences of $g$, generalizing a recent
result on divided differences of inverse functions

Floater, Michael S.

Muntingh, Georg

English

arXiv

This thesis studies two aspects of polynomial interpolation theory. The first part sets forth explicit formulas for the coefficients of polynomial interpolants to implicit functions. A formula for the higher-order derivatives of implicit functions appears as a limiting case of these formulas. The second part delves into certain configurations of points in space — generalized principal lattices — that are well suited as interpolation points. Applying the theory of real algebraic curves then allows the construction of many new examples of such configurations

NORA - Norwegian Open Research Archives

Topics in Polynomial Interpolation Theory

Under general conditions, the equation 

  
    
      g
      (
      x
      ,
      y
      )
      =
      0
    
    g(x,y) = 0
  

 implicitly defines 

  
    y
    y
  

 locally as a function of 

  
    x
    x
  

. In this article, we express divided differences of 

  
    y
    y
  

 in terms of bivariate divided differences of 

  
    g
    g
  

, generalizing a recent result on divided differences of inverse functions.</p

Georg Muntingh

Michael Floater

Crossref

Mathematics of Computation

Divided differences of implicit functions

Under general conditions, the equation g(x, y)  = 0 implicitly defines y locally as a function of x. In this article, we express divided differences of y in terms of bivariate divided differences of g, generalizing a recent result on divided differences of inverse functions

Divided Differences of Implicit Functions

Abstract

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