Applying Computer Algebra Systems in Approximating the Trigonometric Functions

Quan, Le Phuong; Nhan, Thái Anh

doi:10.3390/mca23030037

Cited by 4 publications

(14 citation statements)

References 8 publications

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“…Here, we choose Algorithm 2 in [2] to be implemented with MAPLE (or any of Computer Algebra Systems (CAS), if possible), because it has the great advantage: only using arithmetic calculations on finite rational numbers and comparisons. The choice of MAPLE is due to its powerfulness of symbolic computation and its ability to display exact number of significant digits for the obtained numerical results.…”

Section: Introductionmentioning

confidence: 99%

“…The choice of MAPLE is due to its powerfulness of symbolic computation and its ability to display exact number of significant digits for the obtained numerical results. The description of Algorithm 2 is clear, but examples on numerical integration and graphical plots in [2] seem to be as of the first kind of implementation as mentioned above. Therefore, we come back to the article [2] with aspiration to provide the beginners or occasional users with a complete MAPLE procedure, the output of which being easy and convenient to exploit when implementing Algorithm 2.…”

Section: Introductionmentioning

confidence: 99%

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A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

Quan

2018

MCA

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A complete MAPLE procedure is designed to effectively implement an algorithm for approximating trigonometric functions. The algorithm gives a piecewise polynomial approximation on an arbitrary interval, presenting a special partition that we can get its parts, subintervals with ending points of finite rational numbers, together with corresponding approximate polynomials. The procedure takes a sequence of pairs of interval–polynomial as its output that we can easily exploit in some useful ways. Examples on calculating approximate values of the sine function with arbitrary accuracy for both rational and irrational arguments as well as drawing the graph of the piecewise approximate functions are presented. Moreover, from the approximate integration on [ a , b ] with integrands of the form x m sin x , another MAPLE procedure is proposed to find the desired polynomial estimates in norm for the best L 2 -approximation of the sine function in the vector space P ℓ of polynomials of degree at most ℓ, a subspace of L 2 ( a , b ) .

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

Quan

2018

MCA

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“…, i, are the nodes mentioned above. This block gives a so-called speading technique that takes nodes together with the polynomials A j (x), by using Block 2 successively, as clearly described in [6,Sec. 5].…”

Section: Implementation By Maple Proceduresmentioning

confidence: 99%

“…The choice of MAPLE is due to its powerfulness of symbolic computation and its ability to display exact number of significant digits for the obtained numerical results. The description of Algorithm 2 is clear, but examples on numerical integration and graphical plots in [6, seem to be as of the first kind of implementation mentioned above. Therefore we come back to the article [6] with aspiration to provide the beginners or occasional users with a complete MAPLE procedure, easy and comfortable to explore its output when implementing Algorithm 2.…”

Section: Introductionmentioning

confidence: 99%

“…Algorithm 2 in [6] that is chosen here to be implemented with MAPLE (or any of Computer Algebra Systems (CAS), if possible) has the great advantage: only using arithmetic calculations on finite rational numbers and comparisons. The choice of MAPLE is due to its powerfulness of symbolic computation and its ability to display exact number of significant digits for the obtained numerical results.…”

Section: Introductionmentioning

confidence: 99%

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A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

Quan¹

2018

Preprint

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Abstract:A complete MAPLE procedure is designed to implement effectively an algorithm for approximating the trigonometric functions. The algorithm gives a piecewise polynomial approximation on an arbitrary interval, presenting a special partition that we can get its parts, subintervals with ending points of finite rational numbers, together with corresponding approximate polynomials. The procedure takes a sequence of pairs of interval-polynomial as its output that we can easily explore in some useful ways. Examples on calculating approximate values of the sine function with arbitrary accuracy for both of rational and irrational arguments as well as drawing the graph of the piecewise approximate functions will be presented. Moreover, from the approximate integration of integrands of the form x m sin x on [a, b], another MAPLE procedure is proposed to find the desired polynomial estimates in norm for the best L 2 -approximation of the sine function in the vector space P of polynomials of degree at most , a subspace of L 2 (a, b).

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Symbolic Regression Based Genetic Approximations of the Colebrook Equation for Flow Friction

Praks¹,

Brkić²

2018

Preprint

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Widely used in hydraulics, the Colebrook equation for flow friction relates implicitly to the input parameters; the Reynolds number, Re and the relative roughness of inner pipe surface, ε/D with the output unknown parameter; the flow friction factor, λ; λ=f(λ, Re, ε/D). In this paper, a few explicit approximations to the Colebrook equation; λ≈f(Re, ε/D), are generated using the ability of artificial intelligence to make inner patterns to connect input and output parameters in explicit way not knowing their nature or the physical law that connects them, but only knowing raw numbers, {Re, ε/D}→{λ}. The fact that the used genetic programming tool does not know the structure of the Colebrook equation which is based on computationally expensive logarithmic law, is used to obtain better structure of the approximations which is less demanding for calculation but also enough accurate. All generated approximations are with low computational cost because they contain a limited number of logarithmic forms used although for normalization of input parameters or for acceleration, but they are also sufficiently accurate. The relative error regarding the friction factor λ, in best case is up to 0.13% with only two logarithmic forms used. As the second logarithm can be accurately approximated by the Padé approximation, practically the same error is obtained also using only one logarithm.

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Applying Computer Algebra Systems in Approximating the Trigonometric Functions

Cited by 4 publications

References 8 publications

A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

Symbolic Regression Based Genetic Approximations of the Colebrook Equation for Flow Friction

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