I. Introduction into GPU programming.
 II. Exception safe dynamic memory handling in Cuda project.
 III. Calculation of partial sums in parallel.
 IV. Manipulation of piecewise polynomial functions in parallel.
 V. Manipulation of localized piecewise polynomial functions in parallel.
 1 Calculus behind the LPoly class.
 2 Crudification operator for LPoly class.
 3 Implementation of LPoly class.

## Crudification operator for LPoly class.

he crudification operator was introduced in the section ( Crudification of piecewise-quadratic representation ). We need to adapt the result of that section to the changes in representation. Most of the definition ( Crudification operator ) stays the same except for reference to the proposition ( Quadratic piecewise interpolation ) that needs to be replaced with an updated proposition. We perform the necessary calculations and present an updated summary.

Proposition

(Quadratic piecewise interpolation 2) A piecewise polynomial of the form satisfies the conditions for given if and only if

Proof

The proposition ( Quadratic piecewise interpolation ) provides expressions for the polynomials such that We convert the result to the localized representation: thus where the expressions for are known from the proposition ( Quadratic piecewise interpolation ). The rest of the calculation is performed by the Mathematica script located in OTSProjects/Cuda/PiecewisePoly/pyd/src/crudification.nb.

Definition

(Crudification operator 2) Let be the class of piecewise quadratic functions with finite support defined on : We define a transformation according to the rule where the piecewise polynomial is constructed according to the following procedure.

Let For every let where and come from the proposition ( Quadratic piecewise interpolation 2 ) with Then

We define for by recursion