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.
 Downloads. Index. Contents.

## Manipulation of localized piecewise polynomial functions in parallel.

resented in the previous chapter library runs into numerical problems when performing wavelet calculations.

Consider a linear polynomial for large and small . Assume that the value of on is small but the slope is significant. Because the value is large, the coefficient has to be a large number and the terms and must have alternating sign. The same consideration extends to polynomials of higher order. Calculations with such functions lead to quick accumulation of floating point errors. To avoid such difficulty, it is better to operate in terms of where . To have symmetry, we make a choice

Another numerical difficulty comes from definition ( Scale and transport operators 2 ). We are interested in increasing precision of wavelet decomposition. Thus, the parameter tends to get large and, consequently, function supports get small. On the other hand, we need to keep wavelet bases - normalized, see the formula ( Definition of Galerkin basis 1 ) and proposition ( Galerkin approximation of stationary problem ). Hence, we are forced to deal with rapidly oscillating functions. Such difficulty needs another modification.

We enhance the representation ( Piecewise polynomial representation ) as follows. A piecewise polynomial is given by where are some numbers and the parameter is an integer defined uniquely by satisfying both of the following rules: This way a large scale does not lead to numerical problems because it is represented by an integer parameter.

Implementation of such representation strategy is given by the class LPoly within the same project PiecewisePoly discussed in the previous chapter.

 1 Calculus behind the LPoly class.
 2 Crudification operator for LPoly class.
 3 Implementation of LPoly class.
 Downloads. Index. Contents.
 Copyright 2007