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A Note on an Iterative Method for Root Extraction
Abstract
A double iterative method for evaluating y / x 1/ n is derived and it is shown that if −1 ⩽ y n < x < 1 then it can be arranged that all terms occurring in the iteration are also within this range. The rate of convergence is then discussed and some special cases are mentioned.
... with s 0 = x and c 0 = x − 1. This scheme seems to be introduced in [13]. Following [13], it holds that for every n ∈ N, 1 + c n+1 = (1 + c n )(1 − c n /2) 2 , which implies by induction that for every n ∈ N ...
... This scheme seems to be introduced in [13]. Following [13], it holds that for every n ∈ N, 1 + c n+1 = (1 + c n )(1 − c n /2) 2 , which implies by induction that for every n ∈ N ...
In recent work it has been established that deep neural networks are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb{R}^d$ subject to Dirichlet boundary conditions. It is shown that deep neural networks are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.
... Proof. The proof relies on an iterative method for the root extraction originally published in [9], and extended to ReLU neural networks in [10]. Hence, we use some similar idea for the ReQU neural networks. ...
In this work, we explore the approximation capability of deep Rectified Quadratic Unit neural networks for H\"older-regular functions, with respect to the uniform norm. We find that theoretical approximation heavily depends on the selected activation function in the neural network.
... For the determination of cube root, fourth root, etc., of a given number, one may employ an appropriate numerical method, for e.g, Newton's method. Determination of the n th root reduces to solving a non-linear equation of a single variable for which the methods available may be categorized as direct analytical method, graphical method, trial and error method, iterative method, etc. Gower [4] has described an iterative method for the determination of roots. For the iteration methods like bisection method, false position method, Taylor series method, Newton-Raphson method, Muller's method, etc, one may see Burden and Faires [5]. ...
Presently a direct analytical method is available for the digit-by-digit extraction of the square root of a given positive real number. To calculate the nthroot of a given positive real number one may use trial and error method, iterative method, etc. When one desires to determine the nth root, it is found thatsuch methods are inherent with certain weaknesses like the requirement of an initial guess, a large number of arithmetic operations and several iterativesteps for convergence, etc. There has been no direct method for the determination of the nth root of a given positive real number. This paper focusesattention on developing a numerical algorithm to determine the digit-by-digit extraction of the nth root of a given positive real number up to any desiredaccuracy. Examples are provided to illustrate the algorithm.
... For the determination of cube root, fourth root, etc., of a given number, one may employ an appropriate numerical method, for e.g, Newton's method. Determination of the n th root reduces to solving a non-linear equation of a single variable for which the methods available may be categorized as direct analytical method, graphical method, trial and error method, iterative method, etc. Gower [4] has described an iterative method for the determination of roots. For the iteration methods like bisection method, false position method, Taylor series method, Newton-Raphson method, Muller's method, etc, one may see Burden and Faires [5]. ...
Presently a direct analytical method is available for the digit-by-digit extraction of the square root of a given positive real number. To calculate the nth root of a given positive real number one may use trial and error method, iterative method, etc. When one desires to determine the nth root, it is found that such methods are inherent with certain weaknesses like the requirement of an initial guess, a large number of arithmetic operations and several iterative steps for convergence, etc. There has been no direct method for the determination of the nth root of a given positive real number. This paper focuses attention on developing a numerical algorithm to determine the digit-by-digit extraction of the nth root of a given positive real number up to any desired accuracy. Examples are provided to illustrate the algorithm
... This is the process described in [8]. ...
A double iteration process already used to find the nth root of a positive real number is analysed and showed to be equivalent to the Newton's method. These methods are of order two and three. Higher-order methods for finding the nth root are also mentioned.
In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb {R}^d$ subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.
The history of data analysis that is addressed here is underpinned by two themes, -- those of tabular data analysis, and the analysis of collected heterogeneous data. "Exploratory data analysis" is taken as the heuristic approach that begins with data and information and seeks underlying explanation for what is observed or measured. I also cover some of the evolving context of research and applications, including scholarly publishing, technology transfer and the economic relationship of the university to society.
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