Post-grad. student Golikov A. N.

Anton Chekhov State Pedagogical Institute, Russia

A piecewise polynomial computer scheme for approximation

of functions and double integrals on a circle

 

Introduction. By computer simulation of electronic devices based on two dimensional quantum refinement nano-structures it is needed to calculate real bivariate functions and double integrals, for example overlap integrals for scattering rates [1] or coefficients in the Galerkin [2, 3] method by electronic structure simulating.

Many physical model of quantum electronics considers devises, which are based on nano-elements with a circle cross-section. In this case a domain of a function and an integration domain is circle also. Approximation accuracy questions play a leading role, because the calculated double integrals are coefficients of an eigen value problem or a system of linear algebraic equations, which stability depend significantly of a input data perturbation.

The numerical stability of the mostly used Monte-Carlo method depends of a weight functions selection [1]. This selection is not possible at all times, because we are forced to use approximations of integrated functions instead functions themselves.

In this paper it is report the computer numerical piecewise polynomial scheme for approximation of bivarate functions and double integrals on a circle, at that the scheme provided an a priori given accuracy for function approximation. The presented scheme needs not a weight functions selection in principle and ties to an approximated function conditions of a general kind, which is sufficient for interpolation polynomial constructing.

Problem formulation. Let be considered the real function of a kind

,                                               (1)

on the circle domain

,   (2)

where center coordinate ,  and radius  are preset a priori.

The domain (2) is continuous mapped in the rectangle

,                            (3)

where  and  are the standard polar coordinates, for which it is valid

                                         (4)

as well

                          (5)

On the domain (2) it is defined two family of lines ,  and , , by which the domain (2) is subdivided into the disjoint rectangular sub-domains , at that  is calculated as

                                         (6)

On each sub-domain  using the Romm’s matrix scheme [4] it is constructed the Newton’s interpolation polynomial and transformed to a canonical form

.                                   (7)

Thereafter the following condition is checked:

,                                      (8)

where  is pre-set a priori. The condition (8) is checked at the uniformly spaced along the both coordinate axis points, which not consisted the interpolation nodes.

The numbers  and , which define a number of the sub-domains , as well the polynomial degrees  and  are chosen algorithmically in the following way. The construction of the polynomial (7) is began by  and . If at that the condition is valid on all of the sub-domains, then polynomial (7) is implied as constructed, else the construction is done by   and . And so on, while the condition will be valid, or while an a priori give limit  will be reached. In the last case the degrees are taken the minimal values , as well as  and  are increased by one. And so on.

If the coefficients  are calculated in the presented way, then they are entered into a computer memory for a future use. Thereafter the polynomial (7) is used for approximation of the function (1), also double integrals of the function (2) over the domain (2) are approximated by respective double integrals of the polynomial (7). At that in the both cases the polynomial (7) is calculated by the Horner’s rule.

Hereinafter the piecewise polynomial schemes for approximation of the function (1) on the domain (2) is constructed using the presented methodology.

1. The piecewise polynomial scheme for approximation of bivariate functions on a base of the Newton’s interpolation polynomials. The transformation of the interpolation polynomial is invariant respective to the index-number  of the sub-domain , thereby it is taken  is chosen in the way written hereinbefore.

On  it is defined the set of the interpolation points

(9)

at which it is constructed the polynomial

.      (10)

Next the notations are introduced

 and ,                   (11)

as well

                                          (12)

thereafter the polynomial (10) have the form

.                       (13)

For calculation of the polynomial coefficients from   and  the Romm’s [4] matrix scheme is used. After calculation of the coefficients we will have

,

where . In the end the polynomial is written as

,                                     (14)

where  .

After the coefficients from (14) is computed, the condition is checked, and if it is valid at all of the check points, then polynomial (14) is declared to be desired. For computing the polynomial (14) it can use the Horner’s rule with preliminary calculation of the index-number for the sub-domain by (6) and calculation of the polar coordinates by (5).

The proposed computer scheme is oriented to a routine library creation target to a computer simulation of nano-elements with the two-dimensional quantum refinement.

2. Using the piecewise polynomial scheme for calculation of double integrals. Let be constructed the polynomial (14) on the presented base. In this case a double integral of the function (1) over the domain (2) is approximated by a respective double integral of the polynomial (14). At that, taking into account the additivity of double integrals over an integration domain, the double integral of the polynomial is calculated over each sub-domain , thereafter all of the gotten values are added, and finally resulting sum approximate the double integral of the function. So, it have a place

.              (15)

By analogy with (14) for calculation of the right-hand site of (15) it can use the Horner’s rule.

3. Numerical experiment. For the numerical experiment it was used the PC on a base of Intel Core Duo 2 Quad Q6600 2.4 GHz processor with 8 Gb DDR2 RAM. For the numerical test it was used the well known Franke’s function [5]. It is taken  and  from (2).

In the Table 1 it is shown the numbers of the sub-domains, polynomial degrees and factual maximal approximation error () for the different  from (8).

Table 1

The numerical experiment for function approximation

;

Number

of the sub-domains

4; 4

4096

5; 5

16384

7; 7

262144

 

The double integrals over the domain (2) of the functions (1), which are sufficiently smooth and have not a singularity, are approximated with the maximal absolute error lesser than .

 

References

 

1.       Borzdov A.V., Pozdnyakov D.V., Galenchik V.O., Borzdov V.M., Komarov F.F. Self-consistent calculations of phonon scattering rates in GaAs transistor structure with one-dimensional electron gas // Phys. Stat. Sol. (b). – 2005. – Vol. 242. – ¹ 15. – pp. 134 – 136.

2.       Golikov A. N. Self-consistent calculation of electron-phonon scattering in nano-wires on a base of piecewise polynomial schemes / TSPI. – Taganrog, 2011. – 123 p. Dep. in the ARISTI ¹ 488-Â2011.

3.       Wu Z. and Ruden P.P. Self-consistent calculation of the electronic structure of semiconductor quantum wires: Semiclassical and quantum mechanical approaches // J. Appl. Phys. – 1993. – Vol. 74. – ¹ 10. – pp. 6234 – 6241.

4.       Ya. E. Romm, Sorting-based localization and stable computation of zeros of a polynomial. I. Cybernetic and System Analysis. – 2007. – Vol. 47. – pp. 139-154.

5.       R. Franke, A Critical Comparison of Some Methods for Interpolation of Scattered Data, Naval Postgraduate School Technical Report, NPS-53-79-003 (1979).