
Research Article


DataOriented Model of Sine Based on Chebyshev Zeroes 

A. Habibizad Navin,
S.H. Eshagi,
M.N. Fesharaki,
M. Mirnia
and
M. Teshnelab



ABSTRACT

This study presents a new method based on dataoriented
theory for sine modeling. This model of sine made by an array of data
based on Chebyshev zeroes. To compute sine by this model less mathematical
operations are needed comparing to common methods. Hardware and software
implementation of this model provides faster module.







INTRODUCTION
Dataoriented 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 dataoriented
model for sine modeling.
Dataoriented modeling is a useful and applied method which is introduced
and applied as follows:
• 
Discrete structures like probability digraph, probabilistic language,
complete tree walk and ncomplete tree walk have been presented for many
statistical concepts adaptation with computer. In other words requirement
tools, definition and important mathematical theorems for these models have
presented in by HabibizadNavin et al. (2005a) 
• 
The above methods and structures have been utilized for modeling and then
using it for simulating uniform distribution (HabibizadNavin
et al., 2005b) 
• 
Dataoriented models of population and sample named classified image have
been presented and then provided an algorithm to estimate the distribution
of a statistical population based on dataoriented model (HabibizadNavin
et al., 2005c) 
• 
New dataoriented modeling of uniform random variable which is wellmatched
with computing systems has been presented by HabibizadNavin
et al. (2007a) 
• 
A novel method for improving the uniformity of random number generator
named uniformity improving method, or UIM in short, has been introduced
and dataoriented model of uniform random variable named UDPD is simulated
by this approach (HabibizadNavin et al., 2007d) 
• 
Digital probability hyper digraph for modeling random variable as the
hierarchical dataoriented model has been introduced by HabibizadNavin
et al. (2007e) 
• 
The basic mathematical discussions about dataoriented modeling of uniform
random variable have been introduced by HabibizadNavin
et al. (2008) 
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 dataoriented
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 functionoriented methods.
Dataoriented methods require large amount of data with simple algorithms
but functionoriented methods require less data with complex algorithms.
In this study, dataoriented 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 dataoriented 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_{i}x(xx_{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:
• 
Memory mapping unit gets x as the input then provides i as the address
of memory location 
• 
By knowing i as the address of the memory, the interpolation parameters
α_{i} and β_{i} are given 
• 
Calculation unit provides sine(x) by interpolation with using α_{i},
β_{i}, x_{i} and x 
CONCLUSION
Common method computes sine function by using Maclaurin expansion. To
achieve good accuracy more terms of series are needed to compute the sine
by this expansion. Such methods require more mathematical operations to
obtain answer is named function oriented method. Against this method dataoriented
modeling is a new theory that 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 and also can be accessed easily and fast by modern
computing system because of advanced memory technology. In this study
dataoriented model of sine is presented by using an array of data.
To obtain sine in good accuracy of the computation more terms of the
series are required by common methods. As 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
3.
Table 3: 
Obtained results of new method 

MATLAB uses 6 memory locations and 24 operations but common method
uses 1 memory location and 36 operations. While dataoriented model of
sine uses 512 memory locations and computes it by 3 operations. It is
very fast and efficient in modern computing systems with fast memory.
As Table 3 shows, the new model is preferred in computational
point of view.
APPENDIX A
• 
Since sin (x) = sin (x), we need only to consider positive x 
• 
If x<2^{27 }(hx<0x3e400000, 0), return x with inexact
if x! = 0 
• 
sin (x) is approximated by a polynomial of degree 13 on 
• 
Where, sin (x) ≈ x+S1* x^{3}+…+S^{6}* x^{13} 
• 

• 

• 
For better accuracy, let 
• 

• 

S1 
= 
1.66666666666666324348e01,/*0xBFC55555, 0x55555549*/ 
S2 
= 
8.33333333332248946124e03,/*0x3F811111, 0x1110F8A6 / 
S3 
= 
1.98412698298579493134e04,/*0xBF2A01A0, 0x19C161D5*/ 
S4 
= 
2.75573137070700676789e06,/*0x3EC71DE3, 0x57B1FE7D*/ 
S5 
= 
2.50507602534068634195e08,/*0xBE5AE5E6, 0x8A2B9CEB*/ 
S6 
= 
1.58969099521155010221e10;/*0x3DE5D93A, 0x5ACFD57C*/ 

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