|
Remez Algorithm
The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are the best in the uniform norm ''L''∞ sense. It is sometimes referred to as Remes algorithm or Reme algorithm. A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order ''n'' in the space of real continuous functions on an interval, ''C'' 'a'', ''b'' The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function. In this case, the form of the solution is precised by the equioscillation theorem. Procedure The Remez algorithm starts with the function f to be approximated and a set X of n + 2 sample points x_1, x_2, ...,x_ in the approximation interval, usually the extrema of Chebyshev polynomial line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evgeny Yakovlevich Remez
Evgeny Yakovlevich Remez (sometimes spelled as Evgenii Yakovlevich Remez, ; 1895 in Mstislavl, now Belarus – 1975 in Kyiv, now Ukraine) was a Soviet mathematician. He is known for his work in the constructive function theory, in particular, for the Remez algorithm The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are the ... and the Remez inequality. His doctoral students include Boris Korenblum. References *V K Dzyadyk, Yu A Mitropol'skii and A M Samoilenko, Evgenii Yakovlevich Remez (on the centenary of his birth) (Ukrainian), Ukrain. Mat. Zh. 48 (2) (1996), 285-286. *Yu A Mitropol'skii, V K Dzyadyk and V T Gavrilyuk, Evgenii Yakovlevich Remez (on the occasion of the ninetieth anniversary of his birth) (Russian), Ukrain. Mat. Zh. 38 (4) (1986), 539-540. *Yu A Mitropol'skii, V K Dzjadyk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Constant
In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are fixed). The Lebesgue constant for polynomials of degree at most and for the set of nodes is generally denoted by . These constants are named after Henri Lebesgue. Definition We fix the interpolation nodes x_0, ..., x_nand an interval ,\,b/math> containing all the interpolation nodes. The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space ''C''( 'a'', ''b'' of all continuous functions on 'a'', ''b''to itself. The map ''X'' is linear and it is a projection on the subspace of polynomials of degree or less. The Lebesgue constant \Lambda_n(T) is defined as the operator norm of ''X''. This definition requires us to specify a norm on ''C''( 'a'', ''b''. The un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomials
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boost (C++ Libraries)
Boost, boosted or boosting may refer to: Science, technology and mathematics * Boost, positive manifold pressure in Turbocharger, turbocharged engines * Boost (C++ libraries), a set of free peer-reviewed portable C++ libraries * Boost (material), a material branded and used by Adidas in the midsoles of shoes. * Boost, a loose term for turbo or supercharger * Boost converter, an electrical circuit variation of a DC to DC converter, which increases (boosts) the voltage * Boosted fission weapon, a type of nuclear bomb that uses a small amount of fusion fuel to increase the rate, and thus yield, of a fission reaction * Boosting (behavioral science), a technique to improve human decisions * Boosting (machine learning), a supervised learning algorithm * Intel Turbo Boost, a technology that enables a processor to run above its base operating frequency * Jump start (vehicle), to start a vehicle * Lorentz boost, a type of Lorentz transformation Arts, entertainment, and media Fictional ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating Point
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form are called floating-point numbers. For example, the number 2469/200 is a floating-point number in base ten with five digits: 2469/200 = 12.345 = \! \underbrace_\text \! \times \! \underbrace_\text\!\!\!\!\!\!\!\overbrace^ However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Jean De La Vallée Poussin
Charles-Jean Étienne Gustave Nicolas, baron de la Vallée Poussin (; 14 August 18662 March 1962) was a Belgian mathematician. He is best known for proving the prime number theorem. The King of Belgium ennobled him with the title of baron. Biography De la Vallée Poussin was born in Leuven, Belgium. He studied mathematics at the Catholic University of Leuven under his uncle Louis-Philippe Gilbert, after he had earned his bachelor's degree in engineering. De la Vallée Poussin was encouraged to study for a doctorate in physics and mathematics, and in 1891, at the age of just 25, he became an assistant professor in mathematical analysis. De la Vallée Poussin became a professor at the same university (as was his father, Charles Louis de la Vallée Poussin, who taught mineralogy and geology) in 1892. De la Vallée Poussin was awarded with Gilbert's chair when Gilbert died. While he was a professor there, de la Vallée Poussin carried out research in mathematical analysis and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Mascheroni Constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by : \begin \gamma &= \lim_\left(-\log n + \sum_^n \frac1\right)\\ px&=\int_1^\infty\left(-\frac1x+\frac1\right)\,\mathrm dx. \end Here, represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: History The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43), where he described it as "worthy of serious consideration". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations and for the constant. The Italian mathematician Lorenzo Mascheroni attempted to calculate the constant to 32 dec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Approximation Theory
The ''Journal of Approximation Theory'' is "devoted to advances in pure and applied approximation theory and related areas." References External links ''Journal of Approximation Theory'' web site''Journal of Approximation Theory'' home page at Elsevier Approximation theory journals Approximation theory Academic journals established in 1968 Elsevier academic journals English-language journals Monthly journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimax Approximation Algorithm
A minimax approximation algorithm (or L∞ approximation or uniform approximation) is a method to find an approximation of a mathematical function In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. ... that minimizes maximum error. For example, given a function f defined on the interval ,b/math> and a degree bound n, a minimax polynomial approximation algorithm will find a polynomial p of degree at most n to minimize ::\max_, f(x)-p(x), . Polynomial approximations The Weierstrass approximation theorem states that every continuous function defined on a closed interval ,bcan be uniformly approximated as closely as desired by a polynomial function. For practical work it is often desirable to minimize the maximum absolute or relative error of a polynomial fit for any given number of ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Space
In approximation theory, a Haar space or Chebyshev space is a finite-dimensional subspace V of \mathcal C(X, \mathbb K), where X is a compact space and \mathbb K either the real numbers or the complex numbers, such that for any given f \in \mathcal C(X, \mathbb K) there is exactly one element of V that approximates f "best", i.e. with minimum distance to f in supremum norm In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when t .... References Approximation theory {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equioscillation Theorem
In mathematics, the equioscillation theorem concerns the approximation of continuous functions using polynomials when the merit function is the maximum difference (uniform norm). Its discovery is attributed to Chebyshev. Statement Let f be a continuous function from ,b/math> to \mathbb. Among all the polynomials of degree \le n, the polynomial g minimizes the uniform norm of the difference \, f - g \, _\infty if and only if there are n+2 points a \le x_0 < x_1 < \cdots < x_ \le b such that where is either -1 or +1. Variants The equioscillation theorem is also valid when polynomials are replaced by rational functions: among all rational functions whose numerator has degree and denominator has degree , the rational function , with and being relativel ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Difference
The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y, and is a special case of the Lp distance for all 1\le p\le\infty. Its applications in statistics include the absolute deviation from a central tendency. Properties Absolute difference has the following properties: * For x\ge 0, , x-0, =x (zero is the identity element on non-negative numbers) * For all x, , x-x, =0 (every element is its own inverse element) * , x-y, \ge 0 (non-negativity) * , x-y, = 0 if and only if x=y (nonzero for distinct arguments). * , x-y, =, y-x, (''symmetry'' or ''commutativity''). * , x-z, \le, x-y, +, y-z, (the ''triangle inequality''); equality holds if and only if x\le y\le z or x\ge y\ge z. Because it is non-negative, nonzero for distinct arguments, symmetric, and obeys the triangle inequality, the real numbers form a metric space with the absolute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
