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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.
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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 MATH

Proposition

(Conversion of representations) If MATH then MATH

Proof

We consider a single interval MATH for some $k$ .

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

Proposition

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

Proof

Let MATH We seek MATH to satisfy MATH Also, MATH Hence, if MATH then $b=0$ and $2^{a}x_{a}=x_{b}$ .

If MATH and MATH then $b=$ floor MATH .

Let MATH , thus MATH We arrive to the set of rules MATH


Proposition

(Upscaling) Suppose a piecewise polynomial $P\left( x\right) $ is represented on two different scales: MATH

then MATH where MATH

Proof

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

Proposition

(Localized integral) Suppose a piecewise polynomial $P\left( x\right) $ is given by MATH Then MATH

Proof

We calculate MATH

Proposition

(Localized transport) Suppose a piecewise polynomial $P\left( x\right) $ is given by MATH Then MATH

Proof

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

Proposition

(Localized scaling) Suppose a piecewise polynomial $P\left( x\right) $ is given by MATH Then MATH

Proof

We calculate MATH





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