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Brahmagupta's Interpolation Formula
Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta (598–668 CE) in the early 7th century CE. The Sanskrit couplet describing the formula can be found in the supplementary part of ''Khandakadyaka'' a work of Brahmagupta completed in 665 CE. The same couplet appears in Brahmagupta's earlier ''Dhyana-graha-adhikara'', which was probably written "near the beginning of the second quarter of the 7th century CE, if not earlier." Brahmagupta was the one of the first to describe and use an interpolation formula using second-order differences. Brahmagupta's interpolation formula is equivalent to modern-day second-order Newton–Stirling interpolation formula. Mathematicians prior to Brahmagupta used a simple linear interpolation formula. The linear interpolation formula to compute is : f(a)=f_r+ t D_r where t=\frac. For the computation of , Brahmagupta replaces with another expr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation Formula
In the 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 or experimentation, which represent the values of a function for a limited number of values of the 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 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 gain in simplicity may outweigh the loss from interpolation error and give better performance in ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Interpolation
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known points are given by the coordinates (x_0,y_0) and (x_1,y_1), the linear interpolant is the straight line between these points. For a value in the interval (x_0, x_1), the value along the straight line is given from the equation of slopes \frac = \frac, which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with . Solving this equation for , which is the unknown value at , gives \begin y &= y_0 + (x-x_0)\frac \\ &= \frac + \frac\\ &= \frac \\ &= \frac, \end which is the formula for linear interpolation in the interval (x_0,x_1). Outside this interval, the formula is identical to linear extrapolation. This formula can also be understood as a weighted average. The weights are inv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation
In the 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 or experimentation, which represent the values of a function for a limited number of values of the 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 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 gain in simplicity may outweigh the loss from interpolation error and give better performance in ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta–Fibonacci Identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says :\begin \left(a^2 + b^2\right)\left(c^2 + d^2\right) & = \left(ac-bd\right)^2 + \left(ad+bc\right)^2 & & (1) \\ & = \left(ac+bd\right)^2 + \left(ad-bc\right)^2. & & (2) \end For example, :(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2. The identity is also known as the Diophantus identity,Daniel Shanks, Solved and unsolved problems in number theory, p.209, American Mathematical Society, Fourth edition 1993. as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity. Brahmagupta proved and used a more general identity (the Brahmagupta identity), equivalent to :\begin \left(a^2 + nb^2\right)\left(c^2 + nd^2\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta Matrix
In mathematics, the following matrix was given by Indian mathematician Brahmagupta: :B(x,y) = \begin x & y \\ \pm ty & \pm x \end. It satisfies :B(x_1,y_1) B(x_2,y_2) = B(x_1 x_2 \pm ty_1 y_2,x_1 y_2 \pm y_1 x_2).\, Powers of the matrix are defined by :B^n = \begin x & y \\ ty & x \end^n = \begin x_n & y_n \\ ty_n & x_n \end \equiv B_n. The \ x_n and \ y_n are called Brahmagupta polynomials. The Brahmagupta matrices can be extended to negative integers: :B^ = \begin x & y \\ ty & x \end^ = \begin x_ & y_ \\ ty_ & x_ \end \equiv B_. See also *Brahmagupta's identity * Brahmagupta's function References External links * Eric Weisstein. Brahmagupta Matrix', MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ..., 1999. * Brahmagupta Matrices {{math-hist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta's Identity
In algebra, Brahmagupta's identity says that, for given n, the product of two numbers of the form a^2+nb^2 is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically: :\begin \left(a^2 + nb^2\right)\left(c^2 + nd^2\right) & = \left(ac-nbd\right)^2 + n\left(ad+bc\right)^2 & & & (1) \\ & = \left(ac+nbd\right)^2 + n\left(ad-bc\right)^2, & & & (2) \end Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be obtained from (1), or (1) from (2), by changing ''b'' to −''b''. This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring. History The identity is a generalization of the so-called Fibonacci identity (where ''n''=1) which is actually found in Diophantus' '' Arithmetica'' (III, 19). That identity was rediscovered by Brahmagupta (598–668), an Indian mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plus Or Minus
Plus may refer to: Mathematics * Addition * +, the mathematical sign Music * ''+'' (Ed Sheeran album), (pronounced "plus"), 2011 * ''Plus'' (Cannonball Adderley Quintet album), 1961 * ''Plus'' (Matt Nathanson EP), 2003 * ''Plus'' (Martin Garrix EP), 2018 * Plus (band), a Japanese pop boy band * ''Plus'' (Autechre album), 2020 Companies * Plus Communication Sh.A, a cellphone company in Albania * Plus (telecommunications Poland), a mobile phone brand * Plus (British TV channel), run by Granada Sky Broadcasting * Plus (Slovak TV channel) * Plus (interbank network), Visa's ATM and debit card network * PLUS Markets, a small stock exchange in London, UK * PLUS Expressway Berhad, concessionaire of the North-South Expressway, Malaysia * PLUS (Dutch supermarket) * Plus (German supermarket) * Plus (autonomous trucking) * Plus Development, a defunct American computer storage manufacturer Other * +, the international call prefix * PLUS Loan, a United States Federal student loan * ''P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted \Delta is the operator that maps a function to the function \Delta /math> defined by :\Delta 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, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast, it shares land borders with Pakistan to the west; China, Nepal, and Bhutan to the north; and Bangladesh and Myanmar to the east. In the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives; its Andaman and Nicobar Islands share a maritime border with Thailand, Myanmar, and Indonesia. Modern humans arrived on the Indian subcontinent from Africa no later than 55,000 years ago., "Y-Chromosome and Mt-DNA data support the colonization of South Asia by modern humans originating in Africa. ... Coalescence dates for most non-European populations average to between 73–55 ka.", "Modern human beings—''Homo sapiens''—originated in Africa. Then, int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sanskrit
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late Bronze Age. Sanskrit is the sacred language of Hinduism, the language of classical Hindu philosophy, and of historical texts of Buddhism and Jainism. It was a link language in ancient and medieval South Asia, and upon transmission of Hindu and Buddhist culture to Southeast Asia, East Asia and Central Asia in the early medieval era, it became a language of religion and high culture, and of the political elites in some of these regions. As a result, Sanskrit had a lasting impact on the languages of South Asia, Southeast Asia and East Asia, especially in their formal and learned vocabularies. Sanskrit generally connotes several Old Indo-Aryan language varieties. The most archaic of these is the Vedic Sanskrit found in the Rig Veda, a colle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Common Era
Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the original Anno Domini (AD) and Before Christ (BC) notations used for the same calendar era. The two notation systems are numerically equivalent: " CE" and "AD " each describe the current year; "400 BCE" and "400 BC" are the same year. The expression traces back to 1615, when it first appeared in a book by Johannes Kepler as the la, annus aerae nostrae vulgaris (), and to 1635 in English as " Vulgar Era". The term "Common Era" can be found in English as early as 1708, and became more widely used in the mid-19th century by Jewish religious scholars. Since the later 20th century, BCE and CE have become popular in academic and scientific publications because BCE and CE are religiously neutral terms. They are used by others who wish to be sensit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
