相关论文: Efficient computation of pi by the Newton - Raphso…
In this paper we propose a new method for determination of the two-term Machin-like formula for pi with arbitrarily small arguments of the arctangent function. This approach excludes irrational numbers in computation and leads to a…
In this paper we present a two-term Machin-like formula for pi \[\frac{\pi}{4} = 2^{k - 1}\arctan\left(\frac{1}{u_1}\right) + \arctan\left(\frac{1}{u_2}\right)\] with small Lehmer's measure $e \approx 0.245319$ and describe iteration…
In our earlier publication we have shown how to compute by iteration a rational number ${u_{2,k}}$ in the two-term Machin-like formula for pi of kind…
Lehmer defined a measure $$ \mu=\sum\limits_{j=1}^J\frac{1}{\log_{10}\left(\left|\beta_j\right|\right)}, $$ where the $\beta_j$ may be either integers or rational numbers in a Machin-like formula for $\pi$. When the $\beta_j$ are integers,…
We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \frac{\pi}{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction…
In this work, we develop a new iterative method for computing the digits of $\pi$ by argument reduction of the tangent function. This method combines a modified version of the iterative formula for $\pi$ with squared convergence that we…
In this work, we obtain an iterative formula that can be used for computing digits of $\pi$ and nested radicals of kind $c_n/\sqrt{2 - c_{n - 1}}$, where $c_0 = 0$ and $c_n = \sqrt{2 + c_{n - 1}}$. We also show how with the help of this…
We present a simple recurrent formula to generate the Machin-like expression for calculating $\pi/4$. The method works for any denominator in the starting term and always provides a finite decomposition. We show that the terms in the…
In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $\pi$ by using its rational approximation. In this approximation, both terms are constructed by using a…
We present a new form of the Machin-like formula for $\pi$ that can be generated by using iteration. This form of the Machin-like formula may be promising for computation of the constant $\pi$ due to rapidly increasing integers at each step…
We study three questions related to Machin's type formulas. The first one gives all two terms Machin formulas where both arctangent functions are evaluated $2$-integers, that is values of the form $b/2^a$ for some integers $a$ and~$b$.…
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 \[…
Previously we have proposed a new method of transforming quotients into integer reciprocals in the Machin-like formulas for $\pi$. As a further development, here we show how to generate a multi-term Machin-like formula for $\pi$ with a…
This paper is concerned with the numerical solution of nonlinear ill-posed operator equations involving convex constraints. We study a Newton-type method which consists in applying linear Tikhonov regularization with convex constraints to…
In this paper, we develop efficient and accurate algorithms for evaluating $\varphi(A)$ and $\varphi(A)b$, where $A$ is an $N\times N$ matrix, $b$ is an $N$ dimensional vector and $\varphi$ is the function defined by…
In this paper, we use a biorthogonal approach (Appell system) to construct and characterize the spaces of test and generalized functions associated to the fractional Poisson measure $\pi_{\lambda,\beta}$, that is, a probability measure in…
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…
A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.
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 $\pi = 4 \arctan \left( 1 \right)$. Our…
We exhibit a way of "forcing a functional to be an effective operation" for arbitrary partial combinatory algebras (pcas). This gives a method of defining new pcas from old ones for some fixed functional, where the new partial functions can…