Gregory M. Slugocki
Warsaw Technical University, Poland E-mail: gms@poland.com
ABSTRACT
Summary: The present paper is devoted to the orthogonal polynomials and their application to numerical local approximation and higher order differentiation algorithms used to the data, which can be either equally or randomly sampled. The Gram polynomials norms spectra will be compared to the respective spectra associated with the Chebyshev polynomials in view to the Runge phenomenon. The applications of such algorithms in flight dynamics and biomechanics will be shown in the separate paper.
INTRODUCTION
The numerical methods of local approximation and higher order differentiation, which are based on the use of orthogonal Gram polynomials [1,2,6,10-14], still attest their performances facing to other methods, polynomial [3,4,7,8,16,19,21] or other, e.g. spline method [5,15,17]. Nevertheless, they are limited to the case of strictly equidistant samples and then it is necessary to generalize the set of orthogonal polynomials to the fall of randomly but orderly distributed nodes in the standard interval.
Contrary to the Gram polynomials, which once developed for the desired odd number of local nodes are valid for all positions of the local grid on the global set of nodes, the generalized orthogonal polynomials must be designed on each local set of nodes. Such procedure was adapted for the wavelet networks (called wavenets) [20] by Espiau et al. [18].
The author has developed the generalized orthogonal polynomial method and designed the associated software as the programs THOR and TOR (1999) computing the derivatives up to the 2nd, and the program TORFL (2000) which can compute all allowed arbitrary order derivatives.
GENERALIZED LOCAL ORTHOGONAL POLYNOMIALS Let consider the GN randomly distributed nodes being a dense set in the closed interval
such that:
and the N (odd) nodes local subset of the global nodes dense in the standard interval:
such that N≤GN and:
Then it can exist M=GN-N+1 possible local subsets of nodes denoted by the index ls satisfying the inequalities 0≤ls ≤M-1.
The generalized orthogonal polynomials set with the discrete type of orthogonality is defined as follows:
where their scalar products and norms in the Hilbert space l2[-1,1] are commonly defined below:
where δk,l denotes the Kronecker symbol.
To develop the sets of generalized orthogonal polynomials for each ls, we use the recurrent formula:
where:
and the symbol Ar,r denotes the leading coefficient of the polynomial φr of r-th order.
The length of each subinterval of the interval [a,b] is determined (for given ls) as below:
then the scaling of the given subinterval is defined by the following formula:
which will be further useful.
Let now consider the function f(z) for a≤z≤b and sampled in the knots (nodes) defined earlier. Then the approximating polynomial y(z) can be defined for each subinterval ls comprising a respective set of local nodes as follows:
where r≤N-1 and:
The sets:
are the spectra of norms defined for all local subintervals (containing N knots) of the global interval, and when the knots are equidistant then for the all subintervals then one spectrum of norms exists. The Chebyshev polynomials [1,2,6] have the interesting properties concerning their spectra of norms L2 and l2 respetively:
In the case of discrete type norm the nodes are zeros of the TN(x) which are a dense set in the [-1,1]
standard interval, also:
1
The central point of the each local set has the label ls+0.5(N-1) and generally does not lie at the midpoint of the respective subinterval i.e zls+0.5(N-1) ≠0.5(zls+zls+N-1), this means that referring to the standard interval x0.5(N-1)≠0. One can prove that at the midpoint of the local set of knots the derivatives of the function f(z) are the most precise and least sensitive to data disturbances, when the grid is equidistant such proof is mathematically easy to do [12-14].
The differentiating polynomials for all subintervals assume the following form:
where the inequality n≤r denotes the capacity of the differentiation order, and the polynomial order r
can not exceed r≤2(N-1)0.5 when the nodes are equally distributed, i.e. for Gram polynomials.
RESULTS
Due to the large amount of data files we can not include all them in the paper even as plots. However, some examples will be given.
The list of files concerning the norms spectra is following: GRAMNO5, GRAMNO7, GRAMNO9, GRAMNO11, GRAMNO13, GRAMNO15, GRAMNO17, GRAMNO19, GRAMNO21, GRAMNO23, GRAMNO25, GRAMNO27, GRAMNO29, GRAMNO31, GRAMNO33, GRAMNO35, GRAMNO37, GRAMNO39, GRAMNO41, GRAMNO43, GRAMNO45, GRAMNO47, GRAMNO49, GRAMNO51, GRAMNO53, GRAMNO55. Here, the numbers are the amounts of nodes N of the local grids.
The list of files concerning the norms spectra of the modified Chebyshev polynomials is following:
TJEBNO5, TJEBNO7, TJEBNO9, TJEBNO11, TJEBNO13, TJEBNO15, TJEBNO17, TJEBNO19, TJEBNO21, TJEBNO23, TJEBNO25, TJEBNO27, TJEBNO29, TJEBNO31, TJEBNO33, TJEBNO35, TJEBNO37, TJEBNO39, TJEBNO41, TJEBNO43, TJEBNO45, TJEBNO47, TJEBNO49, TJEBNO51, TJEBNO53, TJEBNO55.
The modified set of Chebyshev polynomials is: T0(x), 2T1(x), 2T2(x), ..., 2TN-1(x), and they have the identical leading coefficients as the orthogonal Gram polynomials described above.
The example of comparison of the norm spectra either for Gram and Chebyshev polynomials (modified) for N=11 is shown in the Tab.1.
J ║pj║2 ║mTj║2
5 26.17245695999999 22
6 22.40215896436364 22
7 16.21227011821029 22
8 9.277233159408746 22
9 3.722382097397441 22
10 .783659388925784 22
Tab.1. Spectra of norms for Gram polynomials (file GRAMNO11) and modified Chebyshev polynomials (file TJEBNO11) where: m=1 for j=0, m=2 for j≠0. The Runge phenomenon limit is 2(N-1)0.5=6.324555320336759.
CONCLUSIONS
1. The numerical development of the orthogonal polynomials set must be done using the double precision numbers regardless the data numbers type.
2. Such procedure is valid for N≤39 if the interpolation is required and N≤55 is nowadays the absolute limit.
( ) ∑
3. The use of randomly distributed nodes makes more realistic the computations of data derivatives when the sampling sway occurs.
4. The limit r≤2(N-1)0.5 can not be overtaken for practical computations REFERENCES:
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