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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 de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lagrange Polynomial
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 degre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Indeterminate Form
In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. More specifically, an indeterminate form is a mathematical expression involving at most two of 0~, 1 or \infty, obtained by applying the algebraic limit theorem in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being sought. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity). The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century. There are seven indeterminate forms which are typically c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Digital Signal Processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor. Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include audio and speech processing, sonar, radar and other sensor array processing, spectral density estimation, statistical signal processing, digital image processing, data compression, video coding, audio coding, image compression, signal processing for telecommunications, control systems, biomedical engineering, and seismology, among others. DSP can involve l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Chebyshev Nodes
In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the effect of Runge's phenomenon. Definition For a given positive integer ''n'' the Chebyshev nodes in the interval (−1, 1) are :x_k = \cos\left(\frac\pi\right), \quad k = 1, \ldots, n. These are the roots of the Chebyshev polynomial of the first kind of degree ''n''. For nodes over an arbitrary interval 'a'', ''b''an affine transformation can be used: :x_k = \frac (a + b) + \frac (b - a) \cos\left(\frac\pi\right), \quad k = 1, \ldots, n. Approximation The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. Given a function ƒ on the interval 1,+1 Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The popul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Horner Scheme
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials. The algorithm is based on Horner's rule: :\begin a_0 &+ a_1x + a_2x^2 + a_3x^3 + \cdots + a_nx^n \\ &= a_0 + x \bigg(a_1 + x \Big(a_2 + x \big(a_3 + \cdots + x(a_ + x \, a_n) \cdots \big) \Big) \bigg). \end This allows the evaluation of a polynomial of degree with only n multiplications and n additions. This is optimal, since there are polynomials of degree that cannot be evaluated with fewer arithmetic operations. Alternatively, Horner's method also refers to a method for approximating the roots of polynomials, described by H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |