Abstract: This article proposes a new numerical method to calculate the integral of a function, using the trapezoid rule, but spacing the intervals fractally, with the purpose of discovering other complementary observations in relation to the degree of fluctuation of the function studied, a parameter that can be included in conventional numerical integration methods. The results are compared with the conventional method of generalized Trapezoid Rule, in which the intervals are equally spaced.
This is then a new contribution that extends universal knowledge on the subject.
AMS subject classifications: 65D30, 65Yxx, 65M12.
Key words : Numerical Integration; Computer aspects of numerical algorithms, Fractal dimension.
1. Introduction
A semi-geometric element is “fractal” when it contains a repetitive structure in different scales with self-similarity characteristics. The fractal concept was developed by the mathematician Benoit Mandelbrot in the 1970s [1]. In fact, nature provides examples of fractals such as snowflakes, ferns, peacock feathers, and romanesco broccoli. Fractal theory has been used to analyze data and obtain relevant information in highly complex problems such as biology [2,3], health sciences [4–8], securities markets [9], network communications [10–12], and others. Other interesting applications are related with the electrical power. In [13], authors presented a new methodology to analyze the dynamics of the daily power demand by using the curve formed by attractors that move in the complex plane over the Mandelbrot set according to the law dictated by the load curve.
The fractal dimension is concept that indicate the degree of fluctuation of any function. In [14] authors explored fractal analysis in order to discriminate between pathological and healthy retinal texture.
Other example is [15], where bentonite is studied when it is affected by a corrosion process in alkaline solution. The fractal dimensions of bentonite specimens at different stages of the corrosion process were obtained from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {N}_{2}} adsorption and swelling deformation tests, and the results of both methods suggested that the fractal dimension tends to increase with duration of the corrosion process.
[16] used fractal dimension to describe the behavior of auriferous veins across the boreholes.
[17] used fractal dimension as an indicator of compactness. The fractal dimension of demolition waste was calculated by fitting the particle size demolition curves into the fractal model.
The exhaust stream of internal combustion engines contains soot agglomerates with fractal geometries. In [18] the fractal dimension of an agglomerate is estimated from the combination of its volume and a characteristic radius.
In [19] the fractal dimensions of aggregates and cement are discussed. A comprehensive understanding of the fractal properties of concrete meso-structure is important because it is associated with the complex and random mechanical behaviors of the concrete.
In [20], authors present a novel method to estimate pedestrian flow in uncontrolled environments by using the fractal dimension measured through the box-counting algorithm, which does not neither require the use of image pre-processing and nor intelligent algorithms.
In [21] fractal dimension is applied to study the growth of urban people of Beijing city, in China.
Finally, [22] present the spatial morphology of the soft agglomerate, such as maltodextrin, quantified by fractal dimension.
The previous Articles are the basis for the new contribution, considering the following hypothesis: The degree of fluctuation of a function is calculated using fractal dimension method. The fractal segments obtained also may be used to calculate the area under the curve, integrating the function using standard trapezoid rule, with fractally spaced data.
The most outstanding reference is [23], where a method to do numerical integration is studied when the original ‘‘signal’’ shows experimentally some kind of self-similarity.
Finally, this document is organized as follows. Section 2 provides a brief explanation of fractal dimension and the methods to do the numerical integration using trapezoid rule. Section 3 presents the method applied in this research and the algorithm used. Section 4 reports and discusses the most relevant results. Finally, the main conclusions of this research work are summarized in Sec. 5
The concept of fractal dimension is introduced with respect to the fact that most objects of nature do not have an entire dimension, but they are in a fractional dimension and this dimension must be greater than its topological dimension. The fractal dimension is a dimensionless numerical measure that indicates how the length, area, or volume varies, depending on the size of the measurement element used. It is calculated with various box sizes, making a linear fit to a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Log(N\left( r\right) )} graph over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Log(r)} . As the size of the boxes reduces their area, the points in the logarithmic graph align more and more and the fractal dimension can be calculated as the slope of the line that joins them, by the expression:
|
(1) |
Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
is the size of the measure element used, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N\left( r\right)}
is the length, area or volume related.
In conclusion, D allows to determine the complexity of the set space under study. [24]
A detailed study on the numerical methods of integration is beyond the scope of this unit. Full information is provided in (Mathews, J, Fink, K, 2000). A recount on the matter, are presented below:
- '2.2.1' Newton-Cótes methods.
It is sought to calculate the definite integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \int_{a}^{b}f\left( x\right) dx} , using the Newton-Cótes methods, which integrates an interpolation polynomial that approximates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right)}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]} . Therefore, these are methods, according to the interpolation polynomial considered. In the case of linear and quadratic interpolations, these methods are called Trapeze Method and Simpson Method, respectively.
- '2.2.2' Trapezoid method
The Trapezoid Method is a Newton-Cotes Method based on linear interpolation. With the purpose of integrating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right)}
from the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( a,f\left( a\right) \right)} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( b,f\left( b\right) \right)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right)} is approximated by its linear interpolation polynomial in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right) \approx {P}_{1}\left( x\right) =\frac{x-b}{a-b}f\left( a\right) +} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{x-a}{b-a}f(b) , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \forall x\in \left[ a,b\right]}
And so
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): I=\int_{a}^{b}f\left( x\right) dx\approx \int_{a}^{b}{P}_{1}\left( x\right) dx=\frac{b-a}{2}\left( f\left( a\right) +f(b)\right) |
The value of the integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle I}
approximates the area of the trapezoid determined by the lines Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a,\, x=b
, the axis of abscissa and the line that joins the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( a,f\left( a\right) \right)}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( b,f\left( b\right) \right)} . Another important aspect is also the knowledge of the degree of precision of the numerical solution.
The expression of the linear interpolation error, assuming that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right)}
is continuous and derivable twice in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]} :
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f\left( x\right) ={P}_{1}\left( x\right) +\, \epsilon \left( x\right) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon \left( x\right) =\frac{{f}^{''}\left( \delta \right) }{2}(x-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a)(x-b) , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a\leq \delta \leq b}
We will have then
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): I=\int_{a}^{b}f\left( x\right) dx=\frac{b-a}{2}\left( f\left( a\right) +f(b)\right) +E |
Where the error of numerical integration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E}
is:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): E=\int_{a}^{b}\epsilon \left( x\right) dx=\frac{{f}^{''}(\delta )}{2}\int_{a}^{b}\left( x-a\right) \left( x-b\right) dx |
Integrating this last expression and naming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): b-a
, it is concluded that:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): E=-\frac{{h}^{3}}{12}{f}^{''}\left( \delta \right) \Rightarrow \left| E\right| \leq \left| \frac{{h}^{3}}{12}{M}_{2}\right| |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_{2}}
is the maximum value reached by the second derivative of the function in the given interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]}
. The demonstration of this general result can be found in advanced texts on numerical integration.
- 2.2.3 Composite Trapezoid method.
If the interval in which the integral is carried out is large, the Simple Trapeze Method is usually very imprecise. To improve accuracy, it is possible to subdivide the interval into smaller ones and apply the Simple Method to each one.
Thus, the compound or generalized Trapeze Method consists in taking a partition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ {x}_{0},{x}_{1},\ldots ,{x}_{n}\right\}
from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( {x}_{0}=\right. }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. a,\, \, {x}_{n}=b\right)
, equiespaced, that is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {x}_{i+1}-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {x}_{i}=h , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \forall i=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1,\ldots ,n. We will have so:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h=\frac{b-a}{n} |
Taking into account the basic properties of the definite integral:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int_{a}^{b}f\left( x\right) dx=\int_{{x}_{0}}^{{x}_{1}}f\left( x\right) dx+\int_{{x}_{1}}^{{x}_{2}}f\left( x\right) dx+\ldots +
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int_{{x}_{n-1}}^{{x}_{n}}f\left( x\right) dx |
And applying to each integral the simple method:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int_{a}^{b}f\left( x\right) dx\approx \frac{h}{2}\left( f\left( {x}_{0}\right) +f\left( {x}_{1}\right) \right) +\frac{h}{2}\left( f\left( {x}_{1}\right) +f\left( {x}_{2}\right) \right) +\ldots +\frac{h}{2}\left( f\left( {x}_{n-1}\right) +f\left( {x}_{n}\right) \right) |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): =\frac{h}{2}\left( f\left( {x}_{0}\right) +2\left( f\left( {x}_{1}\right) +f\left( {x}_{2}\right) +\ldots +\right. \right.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. \left. f\left( {x}_{n-1}\right) \right) +f\left( {x}_{n}\right) \right) |
Therefore, we have the final expression for the Generalized Trapeze Method:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int_{a}^{b}f\left( x\right) dx\cong \frac{h}{2}\left( f\left( a\right) +2\sum _{i=1}^{n-1}f\left( {x}_{i}\right) +f\left( b\right) \right) |
With regard to the integration error, it will be equal to the sum of the errors of each of the applications of the simple method:
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): E={E}_{1}+{E}_{2}+\ldots +{E}_{n}=-\frac{{h}^{3}}{12}{f}^{''}\left( {\epsilon }_{1}\right) -
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{h}^{3}}{12}{f}^{''}\left( {\epsilon }_{2}\right) -\ldots -\frac{{h}^{3}}{12}{f}^{''}\left( {\epsilon }_{n}\right) |
If we call Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {M}_{2}}
the maximum of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {f}^{''}\left( x\right)}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]} , we will have:
|
(2) |
This means that the error is of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle O\left( {h}^{2}\right)}
.
- '2.2.4' Numerical integration for unequally spaced data.
In [25] is indicated that all numerical integration formulas have been based on equally spaced data. However, in practice there are many situations in which this assumption is not met and there are segments of unequal sizes. For example, experimentally obtained data, or fractally spaced data, are often of this type. In such cases, one method is to apply the trapezoid rule to each segment and add the results, as indicated in equation (2), where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {h}_{i}=} the width of segment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} .
Assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f\left( x\right)
be continuous, and possess continuous derivatives in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ {x}_{0},{x}_{n}\right] .}
Expanding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f\left( x\right)
in Taylor polynomials [26] around Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {x}_{0} , the result obtained is
|
(3) |
Now, using (3)
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int_{{x}_{0}}^{{x}_{1}}ydx=\int_{{x}_{0}}^{{x}_{1}}\left[ {y}_{0}+\frac{\left( x-{x}_{0}\right) }{1!}{y}_{0}^{'}+\frac{{\left( x-{x}_{0}\right) }^{2}}{2!}{y}_{0}^{''}+\ldots \right] dx |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ={\left[ {y}_{0}x+\frac{{\left( x-{x}_{0}\right) }^{2}}{2}{y}_{0}^{'}+\frac{{\left( x-{x}_{0}\right) }^{3}}{6}{y}_{0}^{''}+\ldots \right] }_{{x}_{0}}^{{x}_{1}} |
|
(4) |
But
|
(5) |
Putting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {x=x}_{1}}
in (3)
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y\left( {x}_{1}\right) \approx {y}_{1}={y}_{0}+\frac{{\left( {x}_{1}-{x}_{0}\right) }^{}}{1!}{y}_{0}^{'}+
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\left( {x}_{1}-{x}_{0}\right) }^{2}}{2!}{y}_{0}^{''}+\ldots |
Using the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {y}_{1}} in (5), the result obtained is
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int_{{x}_{0}}^{{x}_{1}}ydx=\frac{\left( {x}_{1}-{x}_{0}\right) }{2}\left( {y}_{0}+{y}_{1}\right) =\frac{\left( {x}_{1}-{x}_{0}\right) }{2}\left[ {y}_{0}+{y}_{0}+\frac{{\left( {x}_{1}-{x}_{0}\right) }^{}}{1!}{y}_{0}^{'}+\frac{{\left( {x}_{1}-{x}_{0}\right) }^{2}}{2!}{y}_{0}^{''}+\ldots \right] |
|
(6) |
Now, for the error term subtracting (6) from (4)
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int_{{x}_{0}}^{{x}_{1}}ydx-\frac{{\left( {x}_{1}-{x}_{0}\right) }^{}}{2}\left( {y}_{0}+\right.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {y}_{1}\right) =\frac{{\left( {x}_{1}-{x}_{0}\right) }^{3}}{6}{y}_{0}^{''}- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{{\left( {x}_{1}-{x}_{0}\right) }^{3}}{4}{y}_{0}^{''}+\ldots |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): =\left( \frac{1}{6}-\frac{1}{4}\right) {\left( {x}_{1}-{x}_{0}\right) }^{3}{y}_{0}^{''}+\ldots =
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -\frac{1}{12}{\left( {x}_{1}-{x}_{0}\right) }^{3}{y}_{0}^{''}+\ldots |
This is the error for the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ {x}_{0},{x}_{1}\right]}
In the same way, the error for remaining intervals and the total error are
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): E=-\frac{1}{12}\left[ {\left( {x}_{1}-{x}_{0}\right) }^{3}{y}_{0}^{''}+{\left( {x}_{2}-{x}_{1}\right) }^{3}{y}_{1}^{''}+\ldots +\right.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\left( {x}_{n}-{x}_{n-1}\right) }^{3}{y}_{n-1}^{''}\right] |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): =\sum _{i=1}^{n}{\left( {x}_{i}-{x}_{i-1}\right) }^{3}{y}_{i-1}^{''}=O\left( {h}_{i}^{3}\right) \left[ {h}_{i}=\right] |
This is the error of Trapezoid rule for unequal spacing
The literature presents multiple articles on adaptations of the trapezoid rule to perform approximations to the integral of a function. However, there are few articles when the fractal dimension of the area is related. For example, in [28] were estimate the error in replacing an integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \int_{}^{}f.d\mu \,}
with respect to a fractal measure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu} with a discrete sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sum _{x\in E}^{}w\left( x\right) f(x)\,} over a given sample set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E}
with weights Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w} .The model is the classical Koksma-Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a discrepancy that depends only on the geometry of the sample set and weights, and variance that depends only on the smoothness of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f} .
In order to obtain universally valid results with respect to calculate the fractal dimension and the numerical integration, of a function divided with unequal spaced data, the procedure described below was followed. The numerical integration calculated with unequal segments will be compared with trapezoid rule equispaced and with the exact result. The algorithm proposed is:
Defining Input Variables
Assume a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right)}
defined in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]} , represented by the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\left\{ {S}_{j}\left( x,y\right) \right\} }_{j=1}^{NTer}}
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NTer} is the number of terms, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\left\{ {P}_{j}\left( x,y\right) \right\} }_{j=1}^{NI}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NI} is the number of intervals of the succession. It is interesting to calculate its fractal dimension, and area under the curve.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i:} Counter variable related with the iterations.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r} : Radius between two points.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]}
: Interval.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle inc} : Increment step to evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right)}
from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle b} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NTer:} Variable related with the Term Numbers.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle j:} Counter variable related with the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NTer} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NI:} Variable related with the Interval Numbers.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k:} Counter variable related with the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NI} .
Step 1. Read Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right) ,\, a,\, b,\, inc} .
Step 2. Calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\left\{ {S}_{j}.y=f({S}_{j}.x)\right\} }_{j=1}^{NTer}} , according to the following algorithm:
j = 1 //Counter variable
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Value=a |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): WHILE\left( Value\leq b\right) \, Do |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S}_{j}.x=Value |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S}_{j}.y=f({S}_{j}.x) |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Value=Value+inc |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): j=j+1 |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): EndWHILE |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): NTer=j-1 |
Step 3. Making Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NI=1,\, i=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
as the number of subintervals in the first iteration. Counter variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): NI
. The first fractal partition of the curve is calculated with the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {P}_{(1)}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S}_{(1)}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {P}_{(2)}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S}_{(NTer)}
. They join together and calculate the distance, radius, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
between them.
|
(7) |
Step 4. A first approach to the area under the curve formed by the trapezoid is calculated
|
(8) |
Step 5. Make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i=i+}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
. A new radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
and equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r/2
is calculated.
Step 6. Calculating the number of Intervals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NI} with the new radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r} . Searching intercepts from the starting point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {P}_{(1)}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {S}_{(1)} , and using (7), a sequential search is made of the new points, which with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r} intercepts the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\left( x\right)}
. The algorithm is the following
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): NI=1 |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L\_Index=0}
//Left_Index
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R\_Index=1} //Right_Index
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): WHILE\left( R\_Index\leq NTer\right) \, Do |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Hypotenuse=\, \sqrt{{\left( {S}_{(R\_Index)}.x-{S}_{(L\_Index)}.x\right) }^{2}+{\left( {S}_{(R\_Index)}.y-{S}_{(L\_Index)}.y\right) }^{2}} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): IF\left( Hypotenuse\geq \, r\right) Then |
// The point of intersection was found
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {P}_{(NI)}={S}_{(R\_Index)}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NI=NI+1}
// Increase variable about Interval Numbers.
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): L\_Index=R\_Index |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): EndIF |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R\_Index=R\_Index+1 |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): EndWHILE |
This procedure replaces the standard “compass dimension” method [29] that manually determining the new point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {P}_{(k+1)}}
, with a compass centered on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {P}_{(k)}}
and radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
.
Step 7. The areas of each of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NI} trapezoids made up of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {P}_{(N)}} unequally spaced points are calculated, using equation (8). The total area is calculated as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle A\_Total=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sum _{k=1}^{NI}{A}_{(k)}
Step 8. Save iteration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
, radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
and number of subintervals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NI.}
Step 9. Return to the step 5 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m}
times, repeating the process recursively, reducing the size of the radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{{r}_{i}}{2}\,}
in each new iteration.
Step 10. Once the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m} iterations are finished, for each one the points are calculated as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( Log\left( 1/{r}_{(i)}\right) ,\, Log\left( {Longitude}_{(i)}\right) \right)}
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {Longitude}_{(i)}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {r}_{(i)}.{NTer}_{(i)} .
Step 11. The regression line is calculated by the least squares method. The slope of the line obtained is an estimate of the fractal dimension of the curve studied.
4. NUMERICAL RESULTS AND DISCUSSION.
For the following function, due to its degree of fluctuation or variation, it is interesting to calculate its integral and fractal dimension in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ \mathit{\boldsymbol{0,\, \, 4.2}}\right]}
.
|
(9) |
In order not to saturate the tables of hundreds of results, the calculations will be shown to up to the fourth fractal partition.
Integrating the function analytically, the area under the curve is
==
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int_{\mathit{\boldsymbol{0}}}^{\mathit{\boldsymbol{4.2}}}\mathit{\boldsymbol{f}}\left( \mathit{\boldsymbol{x}}\right) \mathit{\boldsymbol{dx=}}{\left[ \frac{\mathit{\boldsymbol{2}}}{\mathit{\boldsymbol{6}}}{\mathit{\boldsymbol{x}}}^{\mathit{\boldsymbol{6}}}\mathit{\boldsymbol{-}}\frac{\mathit{\boldsymbol{25}}}{\mathit{\boldsymbol{5}}}{\mathit{\boldsymbol{x}}}^{\mathit{\boldsymbol{5}}}\mathit{\boldsymbol{+}}\frac{\mathit{\boldsymbol{112}}}{\mathit{\boldsymbol{4}}}{\mathit{\boldsymbol{x}}}^{\mathit{\boldsymbol{4}}}\mathit{\boldsymbol{-}}\frac{\mathit{\boldsymbol{212}}}{\mathit{\boldsymbol{3}}}{\mathit{\boldsymbol{x}}}^{\mathit{\boldsymbol{3}}}\mathit{\boldsymbol{+}}\frac{\mathit{\boldsymbol{144}}}{\mathit{\boldsymbol{2}}}{\mathit{\boldsymbol{x}}}^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{+3}}\mathit{\boldsymbol{x}}\right] }_{\mathit{\boldsymbol{0}}}^{\mathit{\boldsymbol{4.2}}}\mathit{\boldsymbol{=53.212}} |
The function is numerically integrated in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ 0,\, 4.2\right]}
, with the partitions indicated in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NT}
subintervals, all with the same upper base length. The results are shown in figure 1, and are presented in table 1.
In Figure 1(a), a grid with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle NT=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1 fractal segment with an upper base Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4.21 is superimposed on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ 0,\, 4.2\right]}
. The resulting area of 7.464 is a first approximation to the area of the region (table 1, case a,). In order to improve it, in Figure 1 (b) a variation is made, partitioning the curve with 3 fractal segments with an upper base Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2.105 , for a calculated total area of 38,195, with which a new coating of the R region (table 1, case b,) is formed. In Figure 1(c), a variation is presented, partitioning the curve with 9 fractal segments with an upper base Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1.052 for a calculated total area of 51.221 (table 1, case c). In Figure 1 (d) a last variation is made by partitioning the curve with 21 fractal segments with an upper base Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.526 . The integral of the curve is calculated by adding the area of the subintervals and equal to 54.306 (table 1, case d).
(1) Iter. 1, NT=1, r= 4.21 |
(2) Iter. 2. NT=3, r= 2.105 |
(3) Iter. 3, NT=9, r= 1.052 |
(4) Iter. 4, NT=21, r= 0.526 |
Figure 1. Area calculated by modified trapezoid rule.
| Case a: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{r}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{4}.\boldsymbol{211} | ||||||
Area |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{h}}}_{\mathit{\boldsymbol{1}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{h}}}_{\mathit{\boldsymbol{0}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{0}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{h}}\mathit{\boldsymbol{=}}\mathit{\boldsymbol{h}}}_{\mathit{\boldsymbol{1}}}-
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{h}}}_{\mathit{\boldsymbol{0}}} |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{A=}}\frac{\mathit{\boldsymbol{h}}\left( {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{0}}}\mathit{\boldsymbol{+}}{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}\right) }{\mathit{\boldsymbol{2}}} |
| A1 | 4.2 | 0 | 0.553 | 3 | 4.2 | 7.464 |
Case b: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{r}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{2}.\boldsymbol{105} | ||||||
| A1 | 0.151 | 0 | 20.282 | 3 | 0,151 | 1.757 |
| A2 | 1.557 | 0.151 | 7.373 | 20.282 | 1.406 | 19.453 |
| A3 | 3.656 | 1.557 | 8.806 | 7.373 | 2.098 | 16.981 |
| Total | 38.195 | |||||
| Case c: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{r}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{1},\boldsymbol{052} | ||||||
| A1 | 0.067 | 0 | 11.729 | 3 | 0.067 | 0.493 |
| A2 | 0.152 | 0.067 | 20.370 | 11.729 | 0.085 | 1.364 |
| A3 | 0.28 | 0.152 | 29.007 | 20.370 | 0.128 | 3.160 |
| A4 | 1.032 | 0.280 | 22.906 | 29.007 | 0.751 | 19.519 |
| A5 | 1.284 | 1.032 | 14.468 | 22.906 | 0.252 | 4.727 |
| A6 | 1.617 | 1.284 | 6.2416 | 14.468 | 0.332 | 3.448 |
| A7 | 2.636 | 1.617 | 8.4341 | 6.2416 | 1.018 | 7.477 |
| A8 | 3.689 | 2.636 | 8.2926 | 8.4341 | 1.052 | 8.806 |
| A9 | 4.186 | 3.689 | 0.6556 | 8.2926 | 0.497 | 2.223 |
| Total | 51.221 | |||||
| Case d: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{r}}=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol{0}.\boldsymbol{526} | ||||||
| A1 | 0.032 | 0 | 7.394 | 3 | 0.032 | 0.166 |
| A2 | 0.067 | 0.032 | 11.729 | 7.394 | 0.035 | 0.334 |
| A3 | 0.107 | 0.067 | 16.114 | 11.729 | 0.040 | 0.556 |
| A4 | 0.153 | 0.107 | 20.456 | 16.114 | 0.046 | 0.841 |
| A5 | 0.209 | 0.153 | 24.811 | 20.456 | 0.056 | 1.267 |
| A6 | 0.282 | 0.209 | 29.106 | 24.811 | 0.073 | 1.967 |
| A7 | 0.405 | 0.282 | 33.336 | 29.106 | 0.123 | 3.840 |
| A8 | 0.795 | 0.405 | 30.414 | 33.336 | 0.390 | 12.431 |
| A9 | 0.934 | 0.795 | 26.208 | 30.414 | 0.139 | 3.935 |
| A10 | 1.058 | 0.934 | 21.979 | 26.208 | 0.124 | 3.011 |
| A11 | 1.182 | 1.058 | 17.756 | 21.979 | 0.123 | 2.463 |
| A12 | 1.314 | 1.182 | 13.553 | 17.756 | 0.131 | 2.066 |
| A13 | 1.466 | 1.314 | 9.403 | 13.553 | 0.151 | 1.744 |
| A14 | 1.669 | 1.466 | 5.398 | 9.403 | 0.202 | 1.502 |
| A15 | 2.132 | 1.669 | 3.333 | 5.398 | 0.462 | 2.021 |
| A16 | 2.484 | 2.132 | 6.568 | 3.333 | 0.351 | 1.742 |
| A17 | 2.783 | 2.484 | 10.141 | 6.568 | 0.298 | 2.498 |
| A18 | 3.219 | 2.783 | 12.569 | 10.141 | 0.435 | 4.951 |
| A19 | 3.602 | 3.219 | 9.593 | 12.569 | 0.382 | 4.244 |
| A20 | 3.840 | 3.602 | 5.728 | 9.593 | 0.237 | 1.823 |
| A21 | 4.076 | 3.840 | 1.854 | 5.728 | 0.236 | 0.894 |
| Total | 54.306 | |||||
4.1 Calculating fractal dimension
Next, the fractal dimension of the function (9), obtained with the trapezoid rule modified method, is calculated by dividing the curve with fractal segments, results recorded in Table 2, and represented in Figure 2.
In table 2, the slope Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m} of the regression line that makes up the points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left( Log\left( 1/r\right) ,\, Log\left( Longitude\right) \right)}
is calculated. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m}
is an estimate of the fractal dimension of the area under study.
| Iter. | Fig | r | NI | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{Longitude=r.NI}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{LOG(1/r)}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{LOG(Longitude)}} |
| 1 | 1(a) | 4.211 | 1 | 4.211 | -0,624 | 0,624 |
| 2 | 1(b) | 2.105 | 3 | 6.317 | -0,323 | 0,800 |
| 3 | 1(c) | 1.058 | 9 | 9.475 | -0,022 | 0,976 |
| 4 | 1(d) | 0.526 | 21 | 11.055 | 0,278 | 1,043 |
| 5 | - | 0.2632 | 43 | 11.318 | 0,579 | 1,053 |
| 6 | - | 0.1312 | 86 | 11.318 | 0,880 | 1,053 |
| 7 | - | 0.0658 | 171 | 11.252 | 1,181 | 1,051 |
Table 2. Fractal dimension partitioning the curve using fractal segments.
Note that the results are shown with three additional iterations.
The fractal dimension is a parameter that characterizes the sinuous or irregular of an object. For the case under study. The fractal dimension determines the variability of the results in the different iterations. This means that when there is no variability, an error is entered into the calculations.
Note that the critical point of the Figure 2 was reached in iteration 4, from which no significant differences can be seen. For that reason, the fractal dimension calculated, using the least squares method, was delimited between 1 to 4 iterations, to eliminate the error due to invariability of the results in the interactions of 5 to 7. The result was rounded to 3 decimals, considering that the error obtained by the adjustment with least squares is in this range.
The slope Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m} of the regression line is an estimate of the fractal dimension of the curve under study. With respect to the topological dimension Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {(D}_{top})} of a line, the number of dimensions needed to fix a point on the line is 1. And because the fractional dimension Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {(D}_{frac})} must be larger than its dimension topological, this is calculated as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {D}_{frac}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {D}_{top}+m=1+m. The calculation shows a measure of the fractal dimension of the curve with a value of 1.476, which means that it is a curve with a medium degree of fluctuation.
The previous function is integrated in the same interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ a,b\right]}
, with the partitions indicated in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n}
subintervals, all of the same length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (b-a)/n , results reported in table 3 and represented in figure 3.
In Figure 3(a), a grid with 1 trapezium in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ 0,\, 4.2\right]}
is superimposed on the region. The resulting area of 7.477, is a first approximation to the area of the region. In order to improve it, in Figure 3(b), a variation is made with 3 trapezoids in the intervals for a total calculated area of 32.502, with which forms a new superimposed of the R region. In Figure 3(c), a variation with 9 trapezoids is presented for a total calculated area of 52.281. In Figure 3(d), a last variation is made with 21 trapezoids. The approximate area under the curve is calculated by adding the area of the trapezoids and equal to 54.490 (table 4).
(1) n=1, h=4.2 |
(2) n=3, h=1.4 |
(3) n=9, h=0.466, |
(4) n=21, h=0.2 |
Figure 3. Area calculating by trapezoid rule equiespaced.
| Case a: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{r}}\boldsymbol{=0.200}}
| ||||||
Area |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{h}}}_{\mathit{\boldsymbol{1}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{h}}}_{\mathit{\boldsymbol{0}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{0}}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{h}}\mathit{\boldsymbol{=}}\mathit{\boldsymbol{h}}}_{\mathit{\boldsymbol{1}}}-
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\mathit{\boldsymbol{h}}}_{\mathit{\boldsymbol{0}}} |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{A=}}\frac{\mathit{\boldsymbol{h}}\left( {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{0}}}\mathit{\boldsymbol{+}}{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}\right) }{\mathit{\boldsymbol{2}}} |
| A1 | 0.200 | 0.00 | 24.176 | 3.000 | 0.2 | 2.717 |
| A2 | 0.400 | 0.200 | 33.228 | 24.176 | 0.2 | 5.740 |
| A3 | 0.600 | 0.400 | 34.187 | 33.228 | 0.2 | 6.741 |
| A4 | 0.800 | 0.600 | 30.279 | 34.187 | 0.2 | 6.446 |
| A5 | 1.000 | 0.800 | 24.000 | 30.279 | 0.2 | 5.427 |
| A6 | 1.200 | 1.00 | 17.192 | 24.000 | 0.2 | 4.119 |
| A7 | 1.400 | 1.200 | 11.124 | 17.192 | 0.2 | 2.831 |
| A8 | 1.600 | 1.400 | 6.563 | 11.124 | 0.2 | 1.768 |
| A9 | 1.800 | 1.600 | 3.855 | 6.563 | 0.2 | 1.041 |
| A10 | 2.000 | 1.800 | 3.000 | 3.855 | 0.2 | 0.685 |
| A11 | 2.200 | 2.000 | 3.728 | 3.000 | 0.2 | 0.672 |
| A12 | 2.400 | 2.200 | 5.580 | 3.728 | 0.2 | 0.930 |
| A13 | 2.600 | 2.400 | 7.979 | 5.580 | 0.2 | 1.356 |
| A14 | 2.800 | 2.600 | 10.311 | 7.979 | 0.2 | 1.829 |
| A15 | 3.000 | 2.800 | 12.000 | 10.311 | 0.2 | 2.231 |
| A16 | 3.200 | 3.000 | 12.584 | 12.000 | 0.2 | 2.458 |
| A17 | 3.400 | 3.200 | 11.796 | 12.584 | 0.2 | 2.438 |
| A18 | 3.600 | 3.400 | 9.635 | 11.796 | 0.2 | 2.143 |
| A19 | 3.800 | 3.600 | 6.447 | 9.635 | 0.2 | 1.608 |
| A20 | 4.000 | 3.800 | 2.999 | 6.447 | 0.2 | 0.944 |
| A21 | 4.200 | 4.00 | 0.560 | 2.999 | 0.2 | 0.356 |
| Total | 54.490 | |||||
The area of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle R} region bounded by the function (9) is compared by the standard simple trapezoid rule, partitioning the curve with fractal segments.
| Method | Result | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{\boldsymbol{E}}\mathit{\boldsymbol{RP=}}\frac{\left| {\mathit{\boldsymbol{A}}}^{\mathit{\boldsymbol{\ast }}}\mathit{\boldsymbol{-A}}\right| }{\mathit{\boldsymbol{A}}}}
*100 |
| Exact Area | 53.212 | |
| Our trapezoid rule method for fractally (unequally) spaced data | 54.306 | 2.014 |
| Trapezoid rule for equallly spacing data | 54.490 | 2.345 |
The value of the function integral (9) used as a reference is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathit{\boldsymbol{A}}}^{\mathit{\boldsymbol{\ast }}}=} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 53.212 . Table 4 presents the results obtained, comparing the values of the integral of the curve studied, the relative percentage error (ERP). It is concluded that both methods are equally comparable
Next, the validation of the proposed method is presented, with the following three polynomial functions, all with different degrees of irregularity or fluctuation.
| Polynomial functions | |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y={x}^{4}-13{x}^{3}+49{x}^{2}-49x+45 | (10) |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y={\left( x-3\right) }^{5}-8{\left( x-3\right) }^{3}+10\left( x-3\right) +60 | (11) |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y=-3{x}^{4}+20{x}^{3}-41{x}^{2}+22x+88 | (12) |
The polynomial functions (10), (11) and (12) will be integrated in the intervals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ 0,\, 7\right]}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ 0,\, 5.9\right]} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left[ 0,\, 4\right]} . The exact areas are 274.869, 349.230 and 318.976. Dividing the functions using fractal procedure, in the iteration number 5, there are 41, 24 and 19 segments respectively.
Figure 4 shows the partitions of the areas of the curves under study, using equally spaced data in the first case, and unequally spaced in the second.
| Trapezoid rule equally spaced | Trapezoid rule fractally spaced |
(1) Function (10), NI = 41 equispaced. |
(2) Function (10), NI=41 unequally spaced |
(3) Function (11), NI=24 equispaced |
(4) Function (11), NI= 24 unequally spaced |
(5) Function (12), NI=19 equispaced |
(6) Function (12), NI=19 unequally spaced |
Figure 4. Intervals equispaced and not equispaced for other functions.
Tables 6, 7 and 8 show the calculation of the integral and fractal dimension of the polynomial functions under study (10), (11) and (12) respectively.
| Iter | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{r}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{NI}} | Area | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{Longitude=r.NI}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{LOG(1/r)}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{LOG(Longitude)}} |
| 1 | 7.000 | 1 | 315.000 | 7.000 | -0.845 | 0.845 |
| 2 | 3.500 | 4 | 302.463 | 14.000 | -0.544 | 1.146 |
| 3 | 1.750 | 8 | 287.793 | 14.000 | -0.243 | 1.146 |
| 4 | 0.875 | 20 | 277.548 | 17.500 | 0.057 | 1.243 |
| 5 | 0.437 | 41 | 274.791 | 17.937 | 0.359 | 1.253 |
| 6 | 0.218 | 83 | 274.832 | 18.156 | 0.660 | 1.259 |
| 7 | 0.109 | 165 | 274.699 | 18.046 | 0.961 | 1.256 |
Table 6. Fractal dimension, polynomial function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {x}^{4}-13{x}^{3}+49{x}^{2}-49x+45
.
| Iter | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{r}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{NT}} | Area | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{Longitude=r.NT}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{LOG(1/r)}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{LOG(Longitude)}} |
| 1 | 8.218 | 1 | 300.898 | 8.218 | -0.914 | 0.914 |
| 2 | 4.109 | 2 | 270.113 | 8.218 | -0.613 | 0.914 |
| 3 | 2.054 | 5 | 339.050 | 10.273 | -0.312 | 1.011 |
| 4 | 1.027 | 11 | 346.767 | 11.300 | -0.011 | 1.053 |
| 5 | 0.513 | 24 | 348.170 | 12.327 | 0.289 | 1.090 |
| 6 | 0.256 | 48 | 347.850 | 12.327 | 0.590 | 1.090 |
| 7 | 0.128 | 96 | 348.443 | 12.327 | 0.891 | 1.090 |
Table 7. Fractal dimension, polynomial function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\left( x-3\right) }^{5}-8{\left( x-3\right) }^{3}+10\left( x-3\right) +60
| Iter | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{r}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{NT}} | Area | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{Longitude=r.NT}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{LOG(1/r)}} | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{LOG(Longitude)}} |
| 1 | 4.693 | 1 | 239.831 | 4.693 | -0.671 | 0.671 |
| 2 | 2.346 | 2 | 294.116 | 4.693 | -0.370 | 0.671 |
| 3 | 1.173 | 4 | 306.375 | 4.693 | -0.069 | 0.671 |
| 4 | 0.586 | 9 | 315.353 | 5.279 | 0.231 | 0.722 |
| 5 | 0.293 | 19 | 318.351 | 5.573 | 0.532 | 0.746 |
| 6 | 0.146 | 38 | 318.650 | 5.573 | 0.833 | 0.746 |
| 7 | 0.073 | 76 | 318.942 | 5.573 | 1.134 | 0.746 |
Table 8. Fractal dimension, polynomial function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y=}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -3{x}^{4}+20{x}^{3}-41{x}^{2}+22x+88
Next, the fractal dimension of the functions (10, 11 and 12), are calculated by dividing the curve with fractal segments, results are represented in Figures 5, 6 and 7.
Figure 5. Regression line, function (10). |
Figure 6. Regression line, function (11). |
Note that in figures 5, 6 and 7 the critical points were reached in iterations 4, 5 and 5 respectively, from which no significant differences can be seen. For that reason, the fractal dimension calculated, using the least squares method, was delimited between 1 to 4, 1 to 5, 3 to 5 iterations respectively, to eliminate the error due to invariability of the results in the interactions of 5 to 7 (Figure 5), 6 to 7 (Figure 6), 1 to 2 and 6 to 7 (Figure 7), respectively.
Table 9 presents the results obtained, comparing the values of the integral of the curve studied, the relative percentage error (ERP), using the methods Trapezoid rule standard and Trapezoid rule modified, partitioning the curve with fractal segments.
As evidenced in Table 9, the method proposed in this article, dividing the trapezoids from the fractal fragmentation of its upper side, presents results comparable with those obtained with the traditional composite trapezoid method, in which the lower sides of the sets are equally spaced.
| Function | Exact Area | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{NI}}
|
Area equally spaced | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{ERP}} | Area fractally spaced | Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathit{\boldsymbol{ERP}} | Fractal Dim. |
| (10) | 274.869 | 41 | 275.340 | 0.171 | 274.791 | 0.028 | 1.396 |
| (11) | 349.230 | 24 | 349.040 | 0.054 | 348.170 | 0.304 | 1.162 |
| (12) | 318.976 | 19 | 318.430 | 0.171 | 318.351 | 0.196 | 1.124 |
Table 9. Area comparison.
With respect to the fractal dimension of the curves studied, their values are 1.396, 1.162 and 1.124 respectively, an index that effectively reveals their degree of fluctuation. The polynomial function (10) has a maximum point over two deep basins at different slopes. With respect to (10), the direction of the function (11) changes more smoothly between its two points of maximum and minimum (all of them almost at the same level), respectively. Finally (12) it is the function with smoother changes, with negative bias to the left of the study interval.
5. Conclusions
In the present paper the working hypothesis was validated: The degree of fluctuation of a function is calculated using fractal dimension method. The fractal segments obtained also may be used to calculate the area under the curve, integrating the function using standard trapezoid rule, with fractally spaced data, from which other relevant observations may be obtained from its analysis, such as:
(1) The great advantage of the proposed method is that it unifies the calculation of the numerical integration of a function, together with its corresponding fractal dimension, a task not performed with conventional numerical integration methods.
(2) To calculate the fractal dimension, the standard method uses a compass that with a specific radius, an arc is drawn to find the point that intercepts the curve. In this article a compass was not used, because an algorithm was developed to find the point that intercepts the curve studied with a specific radius.
(3) The fractal dimension is a quantitative estimator of spatial complexity.
(4) It is confirmed that the fractal dimension satisfactorily describes the degree of irregularity of a function, an impossible measure to make using Euclidean (classical) geometry that is limited to regular shapes.
(5) The calculation of the fractal dimension of the studied surface cannot be obtained exactly but estimated, being very sensitive to numerical or experimental noise, and particularly to the limitations in the amount of data
This work was supported by the Agencia de Educación Superior de Medellín (SAPIENCIA) under the specific agreement celebrated with the Institución Universitaria Pascual Bravo. The project is part of the Energy System Doctorate Program of the Universidad Nacional de Colombia, Sede Medellín, Facultad de Minas.
Conflicts of Interest: The authors declare no conflict of interest.
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