Data-Oriented Model of Sine Based on Chebyshev Zeroes

INTRODUCTION

Data-oriented theory presents methods by which any concepts can be modeled in terms of data structures. Now a days large amount of data can be managed cheaply, accessed easily and fast by modern computing system because of advanced memory technology. Any concept can be modeled in this way with large amount of data to recognize and identify easily by modern computing systems. The main contribution of this study is to introduce data-oriented model for sine modeling.

Data-oriented modeling is a useful and applied method which is introduced and applied as follows:

The questions can be answered by data processing or by fewer amounts of mathematical operations by using these models. The methods to answer the questions by using large amount of data easily are called data-oriented methods. Because of less memory storage locations available and slowness of computing in the past, the problems were solved with fewer amounts of data using complex algorithms which are called function-oriented methods. Data-oriented methods require large amount of data with simple algorithms but function-oriented methods require less data with complex algorithms. In this study, data-oriented model of sine is presented.

COMMON METHOD FOR COMPUTING SINE FUNCTION

The common method for computing sine function is reviewed here. This method computes sine function by using Maclaurin expansion as follows (Thomas and Finy, 1996):

To obtain an approximation with a good accuracy more terms of the series are required to be considered. For example using 4 terms of this series to approximate sin(x) as:

results an approximation with an error about

(Thomas et al., 1996).

To obtain more accurate result more terms of this series, are required to be taken. Computing sine, by this series more operations including subtractions, which are due to cancellation error, are required.

Such methods are named function oriented method because they need more operations. The operation counts, errors and required memory locations in computing sine function using MATLAB software and common method with 7 terms of this series are shown in Table 1. The algorithm used to compute sine by MATLAB is presented in appendix A. MATLAB uses 6 memory location and 24 operations but common method uses 1 memory location and 36 operations.

By using the rule that by exceeding time complexity (need operation) memory complexity is decreased and vice versa the data-oriented theory has been introduced which models the concepts by data structures (large amount of data) for decreasing the time complexity of algorithms. Such a model for sine is mentioned further.

Table 1:	Comparison of common method and MATLAB software method

NEW MODELING METHOD

Here, a new model of sine is presented by using an array of data. To make a model for sine in [a,b], n equidistant points, x_i, a≤x_i≤b, i, = 0, 1,….,n, are chosen and then compute:

Next let

which is a linear interpolation approximation for sine x in [x_i, x_i+1]. The interpolation error in [x_i, x_i+1] is

where, α is some point in [x_i, x_i+1] (Thomas et al., 1996). This is easy to understand that its maximum absolute value is obtained for x = (x_i, x_i+1)/2. To model sine in

Then the interpolations corresponding to Chebyshev zeroes perform at the points:

And

Where:

Now α_i and β_i are computed as

and stored in fast memory. Then to find sin(x) for given x, first an interval [x_i, x_i+1] containing x is defined from which sin(x) = α_ix(x-x_i)β_i,x_i,≤x≤x_i+1 is computed with probably less errors.

Table 2:	Obtained result of new method

Fig. 1:	Architecture of the model

Table 2 shows the simulation results of presented methods.

New model of sine uses 512 memory locations involve 3 operations. It is very fast and efficient in modern computing systems with fast memory. Figure 1 shows the architecture of such system that operates as follows: