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Thiele's Interpolation Formula
In mathematics, Thiele's interpolation formula is a formula that defines a rational function f(x) from a finite set of inputs x_i and their function values f(x_i). The problem of generating a function whose graph passes through a given set of function values is called interpolation. This interpolation formula is named after the Danish mathematician Thorvald N. Thiele. It is expressed as a continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ..., where ρ represents the reciprocal difference: : f(x) = f(x_1) + \cfrac Note that the n-th level in Thiele's interpolation formula is :\rho_n(x_1,x_2,\cdots,x_)-\rho_(x_1,x_2,\cdots,x_)+\cfrac, while the n-th reciprocal difference is defined to be :\rho_n(x_1,x_2,\ldots,x_)=\frac+\rho_(x_2,\ldots,x_). The two \rho_ terms ... [...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] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is . The set of rational functions over a field is a field, the field of fractions of the ring of the polynomial functions over . Definitions A function f is called a rational function if it can be written in the form : f(x) = \frac where P and Q are polynomial functions of x and Q is not the zero function. The domain of f is the set of all values of x for which the denominator Q(x) is not zero. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...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] |
Danish People
Danes (, ), or Danish people, are an ethnic group and nationality native to Denmark and a modern nation identified with the country of Denmark. This connection may be ancestral, legal, historical, or cultural. History Early history Denmark has been inhabited by various Germanic peoples since ancient times, including the Angles (tribe), Angles, Cimbri, Jutes, Herules, Teutones and others. A 2025 study in ''Nature'' found genetic evidence of an influx of central European population after about 500 ce into the region later ruled by the Danes. Viking Age The first mention of Danes within Denmark is on the Jelling stones#Runestone of Harald Bluetooth, Jelling Rune Stone, which mentions the conversion of the Danes to Christianity by Harald Bluetooth in the 10th century. Between and the early 980s, Bluetooth established a kingdom in the lands of the Danes, stretching from Jutland to Scania. Around the same time, he received a visit from a German missionary who, by surviving an t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thorvald N
Thorvald is from the ''Old Norse'' name ''Þórvaldr'', which means "Thor's ruler". Despite this pagan origin, the name survived the conversion of Scandinavians to Christianity and remains popular up to the present. Thorvald may refer to: * Thorvald Aagaard (1877–1937), Danish composer, organist and college teacher * Thorvald Astrup (1876–1940), Norwegian architect, known for industrial architecture * Thorvald Asvaldsson, father of the colonizer of Greenland, Erik the Red (Eiríkr Rauði) * Thorwald Bergquist (1899–1972), Swedish politician *Thorvald Bindesbøll (1846–1908), Danish architect and designer * Thorvald Eigenbrod (1892–1977), Danish field hockey player * Thorvald Ellegaard (1877–1954), Danish track racing cyclist * Thorvald Eriksson, son of Eric the Red and brother of Leif Ericsson * Thorvald Hansen, Norwegian Nordic combined skierThorvald Hedgehog(2000-2016), the world's oldest European hedgehog * Thorvald Jørgensen (June 1867 – 1946), Danish architect * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continued Fraction
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite. Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article simple continued fraction. The present article treats the case where numerators and denominators are sequences \,\ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction". Formulation A continued fraction is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Difference
In mathematics, the reciprocal difference of a finite sequence of numbers (x_0, x_1, ..., x_n) on a function f(x) is defined inductively by the following formulas: :\rho_1(x_1, x_2) = \frac :\rho_2(x_1, x_2, x_3) = \frac + f(x_2) :\rho_n(x_1,x_2,\ldots,x_)=\frac+\rho_(x_2,\ldots,x_) See also *Divided differences In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its ... References * *{{cite book, last=Abramowitz, first=Milton, author2=Irene A. Stegun, title= Handbook of Mathematical Functions, orig-date=1964, year=1972, publisher=Dover, edition=ninth Dover printing, tenth GPO printing, isbn=0-486-61272-4, pag878} Finite differences ... [...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] |
Articles With Example ALGOL 68 Code
Article often refers to: * Article (grammar), a grammatical element used to indicate definiteness or indefiniteness * Article (publishing), a piece of nonfictional prose that is an independent part of a publication Article(s) may also refer to: Government and law * Elements of treaties of the European Union * Articles of association, the regulations governing a company, used in India, the UK and other countries; called articles of incorporation in the US * Articles of clerkship, the contract accepted to become an articled clerk * Articles of Confederation, the predecessor to the current United States Constitution * Article of impeachment, a formal document and charge used for impeachment in the United States * Article of manufacture, in the United States patent law, a category of things that may be patented * Articles of organization, for limited liability organizations, a US equivalent of articles of association Other uses * Article element , in HTML * "Articles", a song on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
