Isaac Newton
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Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform
World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:13, No:8, 2019 Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform Lagrange and Newton Curves Taweechai Nuntawisuttiwong, Natasha Dejdumrong Abstract—Newton-Lagrange Interpolations are widely used in quadratic computation time, which is slower than the others. numerical analysis. However, it requires a quadratic computational However, there are some curves in CAGD, Wang-Ball, DP time for their constructions. In computer aided geometric design and Dejdumrong curves with linear complexity. In order to (CAGD), there are some polynomial curves: Wang-Ball, DP and Dejdumrong curves, which have linear time complexity algorithms. reduce their computational time, Bezier´ curve is converted Thus, the computational time for Newton-Lagrange Interpolations into any of Wang-Ball, DP and Dejdumrong curves. Hence, can be reduced by applying the algorithms of Wang-Ball, DP and the computational time for Newton-Lagrange interpolations Dejdumrong curves. In order to use Wang-Ball, DP and Dejdumrong can be reduced by converting them into Wang-Ball, DP and algorithms, first, it is necessary to convert Newton-Lagrange Dejdumrong algorithms. Thus, it is necessary to investigate polynomials into Wang-Ball, DP or Dejdumrong polynomials. In this work, the algorithms for converting from both uniform and the conversion from Newton-Lagrange interpolation into non-uniform Newton-Lagrange polynomials into Wang-Ball, DP and the curves with linear complexity, Wang-Ball, DP and Dejdumrong polynomials are investigated. Thus, the computational Dejdumrong curves. An application of this work is to modify time for representing Newton-Lagrange polynomials can be reduced sketched image in CAD application with the computational into linear complexity. -
Practical Calculations for Designing a Newtonian Telescope
Practical Calculations for Designing a Newtonian Telescope Jeff Beish (Rev. 07 February 2019) INTRODUCTION A Newtonian reflecting telescope can be designed to perform more efficiently than any other type of optical system, if one is careful to follow the laws of nature. One must have optics made from precision Pyrex or a similar material and figured to a high quality. The mounting hardware must be well planned out and properly constructed using highest quality materials. The Newtonian can be used for visual or photographic work, for "deep sky" or "planetary" observing, or a combination of all these. They are easy to layout and to construct using simple household tools. The design mathematics is simple and can be easily accomplished by hand calculator. The Newtonian reflector can be easily modified for other types of observing such a photometry, photography, CCD imaging, micrometer work, and more. Since a reflecting telescope does not suffer for chromatic aberration we don't have to worry about focusing each color while observing or photographing with filters as we would in a single or double lens refractor. This is a problem especially associated with photography. Since the introduction of the relatively low cost Apochromatic refractors (APO) in the past few years these problems no longer hamper the astrophotographer as much. However, the cost of large APO's for those requiring large apertures is prohibitive to most of us who require instruments above 12 or so inches. Remember, even with the highest quality optics a Newtonian can be rendered nearly useless by tube currents, misaligned components, mirror stain, and a secondary mirror too large for the application of the instrument. -
Telescopes and Binoculars
Continuing Education Course Approved by the American Board of Opticianry Telescopes and Binoculars National Academy of Opticianry 8401 Corporate Drive #605 Landover, MD 20785 800-229-4828 phone 301-577-3880 fax www.nao.org Copyright© 2015 by the National Academy of Opticianry. All rights reserved. No part of this text may be reproduced without permission in writing from the publisher. 2 National Academy of Opticianry PREFACE: This continuing education course was prepared under the auspices of the National Academy of Opticianry and is designed to be convenient, cost effective and practical for the Optician. The skills and knowledge required to practice the profession of Opticianry will continue to change in the future as advances in technology are applied to the eye care specialty. Higher rates of obsolescence will result in an increased tempo of change as well as knowledge to meet these changes. The National Academy of Opticianry recognizes the need to provide a Continuing Education Program for all Opticians. This course has been developed as a part of the overall program to enable Opticians to develop and improve their technical knowledge and skills in their chosen profession. The National Academy of Opticianry INSTRUCTIONS: Read and study the material. After you feel that you understand the material thoroughly take the test following the instructions given at the beginning of the test. Upon completion of the test, mail the answer sheet to the National Academy of Opticianry, 8401 Corporate Drive, Suite 605, Landover, Maryland 20785 or fax it to 301-577-3880. Be sure you complete the evaluation form on the answer sheet. -
Mials P
Euclid's Algorithm 171 Euclid's Algorithm Horner’s method is a special case of Euclid's Algorithm which constructs, for given polyno- mials p and h =6 0, (unique) polynomials q and r with deg r<deg h so that p = hq + r: For variety, here is a nonstandard discussion of this algorithm, in terms of elimination. Assume that d h(t)=a0 + a1t + ···+ adt ;ad =06 ; and n p(t)=b0 + b1t + ···+ bnt : Then we seek a polynomial n−d q(t)=c0 + c1t + ···+ cn−dt for which r := p − hq has degree <d. This amounts to the square upper triangular linear system adc0 + ad−1c1 + ···+ a0cd = bd adc1 + ad−1c2 + ···+ a0cd+1 = bd+1 . adcn−d−1 + ad−1cn−d = bn−1 adcn−d = bn for the unknown coefficients c0;:::;cn−d which can be uniquely solved by back substitution since its diagonal entries all equal ad =0.6 19aug02 c 2002 Carl de Boor 172 18. Index Rough index for these notes 1-1:-5,2,8,40 cartesian product: 2 1-norm: 79 Cauchy(-Bunyakovski-Schwarz) 2-norm: 79 Inequality: 69 A-invariance: 125 Cauchy-Binet formula: -9, 166 A-invariant: 113 Cayley-Hamilton Theorem: 133 absolute value: 167 CBS Inequality: 69 absolutely homogeneous: 70, 79 Chaikin algorithm: 139 additive: 20 chain rule: 153 adjugate: 164 change of basis: -6 affine: 151 characteristic function: 7 affine combination: 148, 150 characteristic polynomial: -8, 130, 132, 134 affine hull: 150 circulant: 140 affine map: 149 codimension: 50, 53 affine polynomial: 152 coefficient vector: 21 affine space: 149 cofactor: 163 affinely independent: 151 column map: -6, 23 agrees with y at Λt:59 column space: 29 algebraic dual: 95 column -
Lecture 9: Newton Method 9.1 Motivation 9.2 History
10-725: Convex Optimization Fall 2013 Lecture 9: Newton Method Lecturer: Barnabas Poczos/Ryan Tibshirani Scribes: Wen-Sheng Chu, Shu-Hao Yu Note: LaTeX template courtesy of UC Berkeley EECS dept. 9.1 Motivation Newton method is originally developed for finding a root of a function. It is also known as Newton- Raphson method. The problem can be formulated as, given a function f : R ! R, finding the point x? such that f(x?) = 0. Figure 9.1 illustrates another motivation of Newton method. Given a function f, we want to approximate it at point x with a quadratic function fb(x). We move our the next step determined by the optimum of fb. Figure 9.1: Motivation for quadratic approximation of a function. 9.2 History Babylonian people first applied the Newton method to find the square root of a positive number S 2 R+. The problem can be posed as solving the equation f(x) = x2 − S = 0. They realized the solution can be achieved by applying the iterative update rule: 2 1 S f(xk) xk − S xn+1 = (xk + ) = xk − 0 = xk − : (9.1) 2 xk f (xk) 2xk This update rule converges to the square root of S, which turns out to be a special case of Newton method. This could be the first application of Newton method. The starting point affects the convergence of the Babylonian's method for finding the square root. Figure 9.2 shows an example of solving the square root for S = 100. x- and y-axes represent the number of iterations and the variable, respectively. -
CS321-001 Introduction to Numerical Methods
CS321-001 Introduction to Numerical Methods Lecture 3 Interpolation and Numerical Differentiation Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 Polynomial Interpolation Given a set of discrete values, how can we estimate other values between these data The method that we will use is called polynomial interpolation. We assume the data we had are from the evaluation of a smooth function. We may be able to use a polynomial p(x) to approximate this function, at least locally. A condition: the polynomial p(x) takes the given values at the given points (nodes), i.e., p(xi) = yi with 0 ≤ i ≤ n. The polynomial is said to interpolate the table, since we do not know the function. 2 Polynomial Interpolation Note that all the points are passed through by the curve 3 Polynomial Interpolation We do not know the original function, the interpolation may not be accurate 4 Order of Interpolating Polynomial A polynomial of degree 0, a constant function, interpolates one set of data If we have two sets of data, we can have an interpolating polynomial of degree 1, a linear function x x1 x x0 p(x) y0 y1 x0 x1 x1 x0 y1 y0 y0 (x x0 ) x1 x0 Review carefully if the interpolation condition is satisfied Interpolating polynomials can be written in several forms, the most well known ones are the Lagrange form and Newton form. Each has some advantages 5 Lagrange Form For a set of fixed nodes x0, x1, …, xn, the cardinal functions, l0, l1,…, ln, are defined as 0 if i j li (x j ) ij -
Psychology of Aesthetics, Creativity, and the Arts
Psychology of Aesthetics, Creativity, and the Arts Foresight, Insight, Oversight, and Hindsight in Scientific Discovery: How Sighted Were Galileo's Telescopic Sightings? Dean Keith Simonton Online First Publication, January 30, 2012. doi: 10.1037/a0027058 CITATION Simonton, D. K. (2012, January 30). Foresight, Insight, Oversight, and Hindsight in Scientific Discovery: How Sighted Were Galileo's Telescopic Sightings?. Psychology of Aesthetics, Creativity, and the Arts. Advance online publication. doi: 10.1037/a0027058 Psychology of Aesthetics, Creativity, and the Arts © 2012 American Psychological Association 2012, Vol. ●●, No. ●, 000–000 1931-3896/12/$12.00 DOI: 10.1037/a0027058 Foresight, Insight, Oversight, and Hindsight in Scientific Discovery: How Sighted Were Galileo’s Telescopic Sightings? Dean Keith Simonton University of California, Davis Galileo Galilei’s celebrated contributions to astronomy are used as case studies in the psychology of scientific discovery. Particular attention was devoted to the involvement of foresight, insight, oversight, and hindsight. These four mental acts concern, in divergent ways, the relative degree of “sightedness” in Galileo’s discovery process and accordingly have implications for evaluating the blind-variation and selective-retention (BVSR) theory of creativity and discovery. Scrutiny of the biographical and historical details indicates that Galileo’s mental processes were far less sighted than often depicted in retrospective accounts. Hindsight biases clearly tend to underline his insights and foresights while ignoring his very frequent and substantial oversights. Of special importance was how Galileo was able to create a domain-specific expertise where no such expertise previously existed—in part by exploiting his extensive knowledge and skill in the visual arts. Galileo’s success as an astronomer was founded partly and “blindly” on his artistic avocations. -
Lagrange & Newton Interpolation
Lagrange & Newton interpolation In this section, we shall study the polynomial interpolation in the form of Lagrange and Newton. Given a se- quence of (n +1) data points and a function f, the aim is to determine an n-th degree polynomial which interpol- ates f at these points. We shall resort to the notion of divided differences. Interpolation Given (n+1) points {(x0, y0), (x1, y1), …, (xn, yn)}, the points defined by (xi)0≤i≤n are called points of interpolation. The points defined by (yi)0≤i≤n are the values of interpolation. To interpolate a function f, the values of interpolation are defined as follows: yi = f(xi), i = 0, …, n. Lagrange interpolation polynomial The purpose here is to determine the unique polynomial of degree n, Pn which verifies Pn(xi) = f(xi), i = 0, …, n. The polynomial which meets this equality is Lagrange interpolation polynomial n Pnx=∑ l j x f x j j=0 where the lj ’s are polynomials of degree n forming a basis of Pn n x−xi x−x0 x−x j−1 x−x j 1 x−xn l j x= ∏ = ⋯ ⋯ i=0,i≠ j x j −xi x j−x0 x j−x j−1 x j−x j1 x j−xn Properties of Lagrange interpolation polynomial and Lagrange basis They are the lj polynomials which verify the following property: l x = = 1 i= j , ∀ i=0,...,n. j i ji {0 i≠ j They form a basis of the vector space Pn of polynomials of degree at most equal to n n ∑ j l j x=0 j=0 By setting: x = xi, we obtain: n n ∑ j l j xi =∑ j ji=0 ⇒ i =0 j=0 j=0 The set (lj)0≤j≤n is linearly independent and consists of n + 1 vectors. -
The Newton Fractal's Leonardo Sequence Study with the Google
INTERNATIONAL ELECTRONIC JOURNAL OF MATHEMATICS EDUCATION e-ISSN: 1306-3030. 2020, Vol. 15, No. 2, em0575 OPEN ACCESS https://doi.org/10.29333/iejme/6440 The Newton Fractal’s Leonardo Sequence Study with the Google Colab Francisco Regis Vieira Alves 1*, Renata Passos Machado Vieira 1 1 Federal Institute of Science and Technology of Ceara - IFCE, BRAZIL * CORRESPONDENCE: [email protected] ABSTRACT The work deals with the study of the roots of the characteristic polynomial derived from the Leonardo sequence, using the Newton fractal to perform a root search. Thus, Google Colab is used as a computational tool to facilitate this process. Initially, we conducted a study of the Leonardo sequence, addressing it fundamental recurrence, characteristic polynomial, and its relationship to the Fibonacci sequence. Following this study, the concept of fractal and Newton’s method is approached, so that it can then use the computational tool as a way of visualizing this technique. Finally, a code is developed in the Phyton programming language to generate Newton’s fractal, ending the research with the discussion and visual analysis of this figure. The study of this subject has been developed in the context of teacher education in Brazil. Keywords: Newton’s fractal, Google Colab, Leonardo sequence, visualization INTRODUCTION Interest in the study of homogeneous, linear and recurrent sequences is becoming more and more widespread in the scientific literature. Therefore, the Fibonacci sequence is the best known and studied in works found in the literature, such as Oliveira and Alves (2019), Alves and Catarino (2019). In these works, we can see an investigation and exploration of the Fibonacci sequence and the complexification of its mathematical model and the generalizations that result from it. -
Mathematical Topics
A Mathematical Topics This chapter discusses most of the mathematical background needed for a thor ough understanding of the material presented in the book. It has been mentioned in the Preface, however, that math concepts which are only used once (such as the mediation operator and points vs. vectors) are discussed right where they are introduced. Do not worry too much about your difficulties in mathematics, I can assure you that mine a.re still greater. - Albert Einstein. A.I Fourier Transforms Our curves are functions of an arbitrary parameter t. For functions used in science and engineering, time is often the parameter (or the independent variable). We, therefore, say that a function g( t) is represented in the time domain. Since a typical function oscillates, we can think of it as being similar to a wave and we may try to represent it as a wave (or as a combination of waves). When this is done, we have the function G(f), where f stands for the frequency of the wave, and we say that the function is represented in the frequency domain. This turns out to be a useful concept, since many operations on functions are easy to carry out in the frequency domain. Transforming a function between the time and frequency domains is easy when the function is periodic, but it can also be done for certain non periodic functions. The present discussion is restricted to periodic functions. 1IIIt Definition: A function g( t) is periodic if (and only if) there exists a constant P such that g(t+P) = g(t) for all values of t (Figure A.la). -
Lecture 1: What Is Calculus?
Math1A:introductiontofunctionsandcalculus OliverKnill, 2012 Lecture 1: What is Calculus? Calculus formalizes the process of taking differences and taking sums. Differences measure change, sums explore how things accumulate. The process of taking differences has a limit called derivative. The process of taking sums will lead to the integral. These two pro- cesses are related in an intimate way. In this first lecture, we look at these two processes in the simplest possible setup, where functions are evaluated only on integers and where we do not take any limits. About 25’000 years ago, numbers were represented by units like 1, 1, 1, 1, 1, 1,... for example carved in the fibula bone of a baboon1. It took thousands of years until numbers were represented with symbols like today 0, 1, 2, 3, 4,.... Using the modern concept of function, we can say f(0) = 0, f(1) = 1, f(2) = 2, f(3) = 3 and mean that the function f assigns to an input like 1001 an output like f(1001) = 1001. Lets call Df(n) = f(n + 1) − f(n) the difference between two function values. We see that the function f satisfies Df(n) = 1 for all n. We can also formalize the summation process. If g(n) = 1 is the function which is constant 1, then Sg(n) = g(0) + g(1) + ... + g(n − 1) = 1 + 1 + ... + 1 = n. We see that Df = g and Sg = f. Now lets start with f(n) = n and apply summation on that function: Sf(n) = f(0) + f(1) + f(2) + .. -
Fractal Newton Basins
FRACTAL NEWTON BASINS M. L. SAHARI AND I. DJELLIT Received 3 June 2005; Accepted 14 August 2005 The dynamics of complex cubic polynomials have been studied extensively in the recent years. The main interest in this work is to focus on the Julia sets in the dynamical plane, and then is consecrated to the study of several topics in more detail. Newton’s method is considered since it is the main tool for finding solutions to equations, which leads to some fantastic images when it is applied to complex functions and gives rise to a chaotic sequence. Copyright © 2006 M. L. Sahari and I. Djellit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Isaac Newton discovered what we now call Newton’s method around 1670. Although Newton’s method is an old application of calculus, it was discovered relatively recently that extending it to the complex plane leads to a very interesting fractal pattern. We have focused on complex analysis, that is, studying functions holomorphic on a domain in the complex plane or holomorphic mappings. It is a very exciting field, in which many new phenomena wait to be discovered (and have been discovered). It is very closely linked with fractal geometry, as the basins of attraction for Newton’s method have fractal character in many instances. We are interested in geometric problems, for example, the boundary behavior of conformal mappings. In mathematics, dynamical systems have become popular for another reason: beautiful pictures.