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Newton Polynomials
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method. Definition Given a set of ''k'' + 1 data points :(x_0, y_0),\ldots,(x_j, y_j),\ldots,(x_k, y_k) where no two ''x''''j'' are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials :N(x) := \sum_^ a_ n_(x) with the Newton basis polynomials defined as :n_j(x) := \prod_^ (x - x_i) for ''j'' > 0 and n_0(x) \equiv 1. The coefficients are defined as :a_j := _0,\ldots,y_j/math> where _0,\ldots,y_j/math> are the divided differences defined as \begin \mathopen _k&:= y_k, && k \in \ \\ \mathopen _k,\ldots,y_&:= \frac, && k\in\,\ j\in\. \end Thus the Newton polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vandermonde Matrix
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an (m + 1) \times (n + 1) matrix :V = V(x_0, x_1, \cdots, x_m) = \begin 1 & x_0 & x_0^2 & \dots & x_0^n\\ 1 & x_1 & x_1^2 & \dots & x_1^n\\ 1 & x_2 & x_2^2 & \dots & x_2^n\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 1 & x_m & x_m^2 & \dots & x_m^n \end with entries V_ = x_i^j , the ''j''th power of the number x_i, for all zero-based indices i and j . Some authors define the Vandermonde matrix as the transpose of the above matrix. The determinant of a square Vandermonde matrix (when n=m) is called a Vandermonde determinant or Vandermonde polynomial. Its value is: :\det(V) = \prod_ (x_j - x_i). This is non-zero if and only if all x_i are distinct (no two are equal), making the Vandermonde matrix invertible. Applications The polynomial interpolation problem is to find a polynomial p(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Differences
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly denoted \Delta, is the operator (mathematics), operator that maps a function to the function \Delta[f] defined by \Delta[f](x) = f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations. Certain Recurrence relation#Relationship to difference equations narrowly defined, recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for #Relation with derivatives, approximating derivatives, and the term "finite difference" is often used a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation
In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling (statistics), sampling or experimentation, which represent the values of a function for a limited number of values of the Dependent and independent variables, independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the function approximation, approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Table Of Newtonian Series
In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a_n written in the form :f(s) = \sum_^\infty (-1)^n a_n = \sum_^\infty \frac a_n where : is the binomial coefficient and (s)_n is the rising factorial, falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus. List The generalized binomial theorem gives : (1+z)^s = \sum_^z^n = 1+z+z^2+\cdots. A proof for this identity can be obtained by showing that it satisfies the differential equation : (1+z) \frac = s (1+z)^s. The digamma function: :\psi(s+1)=-\gamma-\sum_^\infty \frac . The Stirling numbers of the second kind are given by the finite sum :\left\ =\frac\sum_^(-1)^ j^n. This formula is a special case of the ''k''th forward difference of the monomial ''x''''n'' evaluated at ''x'' = 0: : \Delta^k x^n = \sum_^(-1)^ (x+j)^n. A related identity forms the basis of the Nörlund–Rice integral: :\sum_^n \frac = \frac = \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlson's Theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem. Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for other expansions. Statement Assume that satisfies the following three conditions. The first two conditions bound the growth of at infinity, whereas the third one states that vanishes on the non-negative integers. # is an entire function of exponential type, meaning that , f(z), \leq C e^, \quad z \in \mathbb for some real values , . # There exists such that , f(iy), \leq C e^, \quad y \in \mathbb # for every non-negative integer . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Interpolation
In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than that takes the same value at given points as a given function. Instead, Hermite interpolation computes a polynomial of degree less than such that the polynomial and its first few derivatives have the same values at (fewer than ) given points as the given function and its first few derivatives at those points. The number of pieces of information, function values and derivative values, must add up to n. Hermite's method of interpolation is closely related to the Newton's interpolation method, in that both can be derived from the calculation of divided differences. However, there are other methods for computing a Hermite interpolating polynomial. One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernstein Polynomial
In the mathematics, mathematical field of numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of #Bernstein basis polynomials, Bernstein basis polynomials. The idea is named after mathematician Sergei Natanovich Bernstein. Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Stone–Weierstrass theorem, Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves. A numerical stability, numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm. Definition Bernstein basis polynomials The \ n + 1\ Bernstein basis polynomials of degree \ n\ are defined as : \ b_(x)\ \equiv\ \binom\ x^ \left( 1 - x \right)^\ , ~~ for ~~ \nu = 0\ ,\ \ldots\ , n\ , where \ \tbinom\ is a binomial coefficient. So, for example, \ b_(x)\ =\ \tbinomx^2(1-x)^3\ =\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points in the dataset. Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no two x_j the same, a polynomial function p(x)=a_0+a_1x+\cdots+a_nx^n is said to interpolate the data if p(x_j)=y_j for each j\in\. There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials. Applications The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions. Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point. Polynomial interpolation also forms the basis for algorithms in numerical quadrature ( Simpson's rule) and numerical ordinary differential equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neville's Schema
In mathematics, Neville's algorithm is an algorithm used for polynomial interpolation that was derived by the mathematician Eric Harold Neville in 1934. Given ''n'' + 1 points, there is a unique polynomial of degree ''≤ n'' which goes through the given points. Neville's algorithm evaluates this polynomial. Neville's algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences. It is similar to Aitken's algorithm (named after Alexander Aitken Alexander Craig "Alec" Aitken (1 April 1895 – 3 November 1967) was one of New Zealand's most eminent mathematicians. In a 1935 paper he introduced the concept of generalized least squares, along with now standard vector/matrix notation ...), which is nowadays not used. The algorithm Given a set of ''n''+1 data points (''x''''i'', ''y''''i'') where no two ''x''''i'' are the same, the interpolating polynomial is the polynomial ''p'' of degree at most ''n'' with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Harriot
Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his contributions in navigational techniques, working closely with John White to create advanced maps for navigation. While Harriot worked extensively on numerous papers on the subjects of astronomy, mathematics and navigation, he remains obscure because he published little of it, namely only ''The Briefe and True Report of the New Found Land of Virginia'' (1588). This book includes descriptions of English settlements and financial issues in Virginia at the time. He is sometimes credited with the introduction of the potato to the British Isles. Harriot invented binary notation and arithmetic several decades before Gottfried Wilhelm Leibniz, but this remained unknown until the 1920s. He was also the first person to make a drawing of the Moon thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magisteria Magna
The magisterium of the Catholic Church is the church's authority or office to give authentic interpretation of the word of God, "whether in its written form or in the form of Tradition". According to the 1992 ''Catechism of the Catholic Church'', the task of interpretation is vested uniquely in the Pope and the bishops, though the concept has a complex history of development. Scripture and Tradition "make up a single sacred deposit of the Word of God, which is entrusted to the Church", and the magisterium is not independent of this, since "all that it proposes for belief as being divinely revealed is derived from this single deposit of faith." Solemn and ordinary The exercise of the Catholic Church's magisterium is sometimes, but only rarely, expressed in the solemn form of an ''ex cathedra'' papal declaration, "when, in the exercise of his office as shepherd and teacher of all Christians, in virtue of his supreme apostolic authority, he Bishop of Romedefines a doctrine conce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
