Peano kernel theorem

In numerical analysis, the Peano kernel theorem is a general result on error bounds for a wide class of numerical approximations (such as numerical quadratures), defined in terms of linear functionals. It is attributed to Giuseppe Peano.[1]

StatementEdit

Let   be the space of all differentiable functions   defined for   that are of bounded variation on  , and let   be a linear functional on  . Assume that   is   times continuously differentiable and that   annihilates all polynomials of degree  , i.e.

 
Suppose further that for any bivariate function   with  , the following is valid:
 
and define the Peano kernel of   as
 
introducing notation
 
The Peano kernel theorem then states that
 
provided  .[1][2]

BoundsEdit

Several bounds on the value of   follow from this result:

 

where  ,   and  are the taxicab, Euclidean and maximum norms respectively.[2]

ApplicationEdit

In practice, the main application of the Peano kernel theorem is to bound the error of an approximation that is exact for all  . The theorem above follows from the Taylor polynomial for   with integral remainder:

 

defining   as the error of the approximation, using the linearity of   together with exactness for   to annihilate all but the final term on the right-hand side, and using the   notation to remove the  -dependence from the integral limits.[3]

See alsoEdit

ReferencesEdit

  1. ^ a b Ridgway Scott, L. (2011). Numerical analysis. Princeton, N.J.: Princeton University Press. pp. 209. ISBN 9780691146867. OCLC 679940621.
  2. ^ a b Iserles, Arieh (2009). A first course in the numerical analysis of differential equations (2nd ed.). Cambridge: Cambridge University Press. pp. 443–444. ISBN 9780521734905. OCLC 277275036.
  3. ^ Iserles, Arieh (1997). "Numerical Analysis" (PDF). Retrieved 2018-08-09.