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S. M. Abrarov

Publications and source records attributed to S. M. Abrarov.

At least 19 recordsLinked to original sources
arXiv2019

A rational approximation of the sinc function based on sampling and the Fourier transforms

In our previous publications we have introduced the cosine product-to-sum identity [17] $$ \prod\limits_{m = 1}^M {\cos \left( {\frac{t}{2^m}} \right)} = \frac{1}{2^{M - 1}}\sum\limits_{m = 1}^{2^{M - 1}} {\cos \left( {\frac{2m - 1}{2^M}t} \right)} $$ and applied it for sampling [1, 2] as an incomplete cosine expansion of the sinc function in order to obtain a rational approximation of the Voigt/complex error function that with only $16$ summation terms can provide accuracy ${\sim 10^{ - 14}}$. In this work we generalize this approach and show as an example how a rational approximation of the sinc function can be derived. A MATLAB code validating these results is presented.

math.NA↗
arXiv2019

Carbon dioxide retrieval of Argus 1000 space data by using GENSPECT line-by-line radiative transfer model

The micro-spectrometer Argus 1000 being in space continuously monitors the sources and sinks of the trace gases. It is commonly believed that among other gases $\text{CO}_\text{2}$ is the major contributor causing the greenhouse effect. Argus 1000 along its orbit gathers the valuable spectral data that can be analyzed and retrieved. In this paper we present the retrieval of $\text{CO}_\text{2}$ gas in the near infrared window $1580$ to $1620$ nm by using line-by-line code GENSPECT. The retrieved Argus 1000 space data taken over British Columbia on May 31, 2010 indicates an enhancement of $\text{CO}_\text{2}$ by about $30\%$.

physics.ao-ph↗
arXiv2019

A single-domain implementation of the Voigt/complex error function by vectorized interpolation

In this work we show how to perform a rapid computation of the Voigt/complex error over a single domain by vectorized interpolation. This approach enables us to cover the entire set of the parameters $x,y \in \mathbb{R}$ required for the HITRAN-based spectroscopic applications. The computational test reveals that within domains $x\in\left[0,15\right]\cap y\in\left[10^{-8},15\right]$ and $x\in\left[0,50000\right]\cap y\geq 10^{-8}$ our algorithmic implementation is faster in computation by factors of about $8$ and $3$, respectively, as compared to the fastest known C/C++ code for the Voigt/complex error function. A rapid MATLAB code is presented.

math.GM↗
arXiv2018

A formula for pi involving nested radicals

We present a new formula for pi involving nested radicals with rapid convergence. This formula is based on the arctangent function identity with argument $x=\sqrt{2-{{a}_{k-1}}}/{{a}_{k}}$, where \[ {{a}_{k}}=\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{k\,\,\text{square}\,\,\text{roots}} \] is a nested radical consisting of $k$ square roots. The computational test we performed reveals that the proposed formula for pi provides a significant improvement in accuracy as the integer $k$ increases.

math.GM↗
arXiv2018

A sampling-based approximation of the complex error function and its implementation without poles

Recently we developed a new sampling methodology based on incomplete cosine expansion of the sinc function and applied it in numerical integration in order to obtain a rational approximation for the complex error function $w\left(z \right) = e^{- {z^2}}\left(1 + \frac{2i}{\sqrt π}\int_0^z e^{t^2}dt\right),$ where $z = x + iy$. As a further development, in this work we show how this sampling-based rational approximation can be transformed into alternative form for efficient computation of the complex error function $w\left(z \right)$ at smaller values of the imaginary argument $y=\operatorname{Im}\left[z \right]$. Such an approach enables us to avoid poles in implementation and to cover the entire complex plain with high accuracy in a rapid algorithm. An optimized Matlab code utilizing only three rapid approximations is presented.

math.NA↗
arXiv2018

Efficient computation of pi by the Newton - Raphson iteration and a two-term Machin-like formula

In our recent publication we have proposed a new methodology for determination of the two-term Machin-like formula for pi with small arguments of the arctangent function of kind $$ \frac{π}{4} = {2^{k - 1}}\arctan \left( {\frac{1}{β_1}} \right) + \arctan \left( {\frac{1}{β_2}} \right), $$ where $k$ and ${β_1}$ are some integers and ${β_2}$ is a rational number, dependent upon ${β_1}$ and $k$. Although ${1/\left|β_2\right|}$ may be significantly smaller than ${1/β_1}$, the large numbers in the numerator and denominator of $β_2$ decelerate the computation. In this work we show how this problem can be effectively resolved by the Newton--Raphson iteration method.

math.GM↗
arXiv2017

A rational approximation of the Dawson's integral for efficient computation of the complex error function

In this work we show a rational approximation of the Dawson's integral that can be implemented for high-accuracy computation of the complex error function in a rapid algorithm. Specifically, this approach provides accuracy exceeding $\sim {10^{ - 14}}$ in the domain of practical importance $0 \le y < 0.1 \cap \left| {x + iy} \right| \le 8$. A Matlab code for computation of the complex error function with entire coverage of the complex plane is presented.

math.NA↗
arXiv2017

An iteration procedure for a two-term Machin-like formula for pi with small Lehmer's measure

In this paper we present a two-term Machin-like formula for pi \[\fracπ{4} = 2^{k - 1}\arctan\left(\frac{1}{u_1}\right) + \arctan\left(\frac{1}{u_2}\right)\] with small Lehmer's measure $e \approx 0.245319$ and describe iteration procedure for simplified determination of the required rational number $u_2$ at $k = 27$ and $u_1 = 85445659$. With these results we obtained a formula that has no irrational numbers involved in computation and provides $16$ digits of pi at each increment by one of the summation terms. This is the smallest Lehmer's measure ever reported for the Machin-like formulas for pi.

math.GM↗
arXiv2017

A set of the Viète-like recurrence relations for the unity constant

Using a simple Viète-like formula for $π$ based on the nested radicals $a_k = \sqrt{2 + a_{k-1}}$ and $a_1 = \sqrt{2}$, we derive a set of the recurrence relations for the constant $1$. Computational test shows that application of this set of the Viète-like recurrence relations results in a rapid convergence to unity.

math.GM↗
arXiv2016

The Fourier expansion approximation for high-accuracy computation of the Voigt/complex error function at small imaginary argument

It is known that the computation of the Voigt/complex error function is problematic for highly accurate and rapid computation at small imaginary argument $y << 1$, where $y = \operatorname{Im} \left[ z \right]$. In this paper we consider an approximation based on the Fourier expansion that can be used to resolve effectively such a problem when $y \to 0$.

math.NA↗
arXiv2016

A simple identity for derivatives of the arctangent function

We present an identity for the derivatives of the arctangent function as an alternative to the Adegoke - Layeni - Lampret formula. We show that algorithmic implementation of the proposed identity can significantly accelerate the computation since this approach requires no symbolic programming in determination of the derivatives for the arctangent function.

math.GM↗
arXiv2016

Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi

We describe a method of integration to obtain identities of the arctangent function and show how this method can be applied to the high-accuracy computation of the constant pi using the equation $π= 4 \arctan \left( 1 \right)$. Our approach combines the midpoint method with the Taylor expansion series to enhance accuracy in the subintervals. The accuracy of this method of integration is determined by number of subintervals $L$ and by order of the Taylor expansion $M$. This approach provides significant flexibility in computation since the required convergence in resulting equations can be optimized through appropriate choices for the integers $L$ and $M$. Sample computations are presented to illustrate that even with relatively small values of the integers $L$ and $M$ the constant $π$ can be computed with high accuracy.

math.GM↗
arXiv2016

A rational approximation of the arctangent function and a new approach in computing pi

We have shown recently that integration of the error function ${\rm{erf}}\left( x \right)$ represented in form of a sum of the Gaussian functions provides an asymptotic expansion series for the constant pi. In this work we derive a rational approximation of the arctangent function $\arctan \left( x \right)$ that can be readily generalized it to its counterpart $ - {\rm{sgn}}\left( x \right)π/2 + \arctan \left( x \right)$, where ${\rm{sgn}}\left( x \right)$ is the signum function. The application of the expansion series for these two functions leads to a new asymptotic formula for $π$.

math.GM↗
arXiv2016

A new asymptotic expansion series for the constant pi

In our recent publications we have introduced the incomplete cosine expansion of the sinc function for efficient application in sampling [Abrarov & Quine, Appl. Math. Comput., 258 (2015) 425-435; Abrarov & Quine, J. Math. Research, 7 (2) (2015) 163-174]. Here we show that it can also be utilized as a flexible and efficient tool in mathematical analysis. In particular, an application of the incomplete cosine expansion of the sinc function leads to expansion series of the error function in form of a sum of the Gaussian functions. This approach in integration provides a new asymptotic formula for the constant $π$.

math.GM↗