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Birkhoff Interpolation
In mathematics, Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem of finding a polynomial ''p'' of degree ''d'' such that certain derivatives have specified values at specified points: : p^(x_i) = y_i \qquad\mbox i=1,\ldots,d, where the data points (x_i,y_i) and the nonnegative integers n_i are given. It differs from Hermite interpolation in that it is possible to specify derivatives of ''p'' at some points without specifying the lower derivatives or the polynomial itself. The name refers to George David Birkhoff, who first studied the problem. Existence and uniqueness of solutions In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution. For instance, there is no quadratic polynomial \textstyle p such that \textstyle p(-1)=p(1)=0 and \textstyle p'(0)=1. On the other hand, the Birkhoff interpolation problem where the values of \textstyle p'(-1), \textstyle p(0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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 of 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) is said to interpolate the data if p(x_j)=y_j for each j\in\. Two common explicit formulas for this polynomial are the Lagrange polynomials and Newton polynomials. Applications Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography, given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions: pick a few known data points, create a lookup table, and interpolate between those data points. This results in significantly faster computations. Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations and Secure Multi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...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 derivatives have the same values at given points as a given function and its first derivatives. Hermite's method of interpolation is closely related to the Newton's interpolation method, in that both are 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 writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see . Statement of the pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George David Birkhoff
George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during his time he was considered by many to be the preeminent American mathematician. The George D. Birkhoff House, his residence in Cambridge, Massachusetts, has been designated a National Historic Landmark. Personal life He was born in Overisel Township, Michigan, the son of David Birkhoff and Jane Gertrude Droppers. The mathematician Garrett Birkhoff (1911–1996) was his son. Career Birkhoff obtained his A.B. and A.M. from Harvard University. He completed his Ph.D. in 1907, on differential equations, at the University of Chicago. While E. H. Moore was his supervisor, he was most influenced by the writings of Henri Poincaré. After teaching at the University of Wisconsin–Madison and Princeton University, he taught at Harvard from 191 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Interpolation
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + 1 nodes \, which must all be distinct, x_j \neq x_m for indices j \neq m, the Lagrange basis for polynomials of degr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaac Jacob Schoenberg
Isaac Jacob Schoenberg (April 21, 1903 – February 21, 1990) was a Romanian-American mathematician, known for his invention of splines. Life and career Schoenberg was born in Galați. He studied at the University of Iași, receiving his M.A. in 1922. From 1922 to 1925 he studied at the Universities of Berlin and Göttingen, working on a topic in analytic number theory suggested by Issai Schur. He presented his thesis to the University of Iași, obtaining his Ph.D. in 1926. In Göttingen, he met Edmund Landau, who arranged a visit for Schoenberg to the Hebrew University of Jerusalem in 1928. During this visit, Schoenberg began his work on total positivity and variation-diminishing linear transformations. In 1930, he returned from Jerusalem, and married Landau's daughter Charlotte in Berlin. In 1930, he was awarded a Rockefeller Fellowship, which enabled him to go to the United States, visiting the University of Chicago, Harvard, and the Institute for Advanced Study in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Pólya
George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education. He has been described as one of The Martians, an informal category which included one of his most famous students at ETH Zurich, John Von Neumann. Life and works Pólya was born in Budapest, Austria-Hungary, to Anna Deutsch and Jakab Pólya, Hungarian Jews who had converted to Christianity in 1886. Although his parents were religious and he was baptized into the Catholic Church upon birth, George eventually grew up to be an agnostic. He was a professor of mathematics from 1914 to 1940 at ETH Zürich in Switzerland and from 1940 to 1953 at Stanford University. He remained a Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
