|
Birkhoff Interpolation
In mathematics, Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem of finding a polynomial P(x) of degree d such that ''only 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(x) at some points without specifying the lower derivatives or the polynomial itself. The name refers to George David Birkhoff, who first studied the problem in 1906. 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 P(x) such that P(-1)=P(1)=0 and P^(0)=1. On the other hand, the Birkhoff interpolation problem where the values of P^(-1), P(0) and P^(1) are given always has a un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
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] |
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] |
Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. 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. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. '' Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leib ... [...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] |
George David Birkhoff
George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body problem, and general relativity. Today, Birkhoff is best remembered for the ergodic theorem. The George D. Birkhoff House, his residence in Cambridge, Massachusetts, has been designated a National Historic Landmark. Early life He was born in Overisel Township, Michigan, the son of two Dutch immigrants, David Birkhoff, who arrived in the United States in 1870, and Jane Gertrude Droppers. Birkhoff's father worked as a physician in Chicago while he was a child. From 1896 to 1902, he would attend the Lewis Institute as a teenager. Career Birkhoff was part of a generation of American mathematicians who were the first to study entirely within the United States and not participate in academics within Europe. Following his time at the Lewis Insti ... [...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, Shamir's secret sharing scheme in cryptography, and Reed–Solomon error correction in coding theory. 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 indic ... [...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 to a Jewish family, the youngest of four children. 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 Chicag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Pólya
George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American 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 (scientists), 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, History of the Jews in Hungary, 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 received a PhD under Lipót Fejér in 1912, at Eötvös Loránd University. He was a professor o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completing The Square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By subsequently isolating and taking the square root, a quadratic problem can be reduced to a linear problem. The name ''completing the square'' comes from a geometrical picture in which represents an unknown length. Then the quantity represents the area of a square of side and the quantity represents the area of a pair of Congruence (geometry), congruent rectangles with sides and . To this square and pair of rectangles one more square is added, of side length . This crucial step ''completes'' a larger square of side length . Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian Empire, Old Babylonian clay tablets dating from 1800–1600 BCE, and is still taught in elementary algebra c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Polynomial
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 polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
