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.

## Calculus behind the LPoly class.

he code behind the LPoly class is based on calculations of the sections ( Calculus behind the poly module ) and ( Calculus behind the PiecewisePoly project ). There are, however, several differences covered below.

We use the notation

Proposition

(Conversion of representations) If then

Proof

We consider a single interval for some .

According to the proposition ( Taylor decomposition in Peano form ), and Similarly, Thus

Proposition

(Adjustment of scale) Let be an pair , . We would like to find a pair with the properties and either or We accomplish it via the following calculation

Proof

Let We seek to satisfy Also, Hence, if then and .

If and then floor .

Let , thus We arrive to the set of rules

Proposition

(Upscaling) Suppose a piecewise polynomial is represented on two different scales:

then where

Proof

We pick a and consider the interval We make a change thus We repeat calculations of the proposition ( Conversion of representations ) and obtain and Thus, We conclude

Proposition

(Localized integral) Suppose a piecewise polynomial is given by Then

Proof

We calculate

Proposition

(Localized transport) Suppose a piecewise polynomial is given by Then

Proof

We calculate We make a change and arrive to the desired claim.

Proposition

(Localized scaling) Suppose a piecewise polynomial is given by Then

Proof

We calculate