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Lebesgue Constant (interpolation)
In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are fixed). The Lebesgue constant for polynomials of degree at most and for the set of nodes is generally denoted by . These constants are named after Henri Lebesgue. Definition We fix the interpolation nodes x_0, ..., x_nand an interval ,\,b/math> containing all the interpolation nodes. The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space ''C''( 'a'', ''b'' of all continuous functions on 'a'', ''b''to itself. The map ''X'' is linear and it is a projection on the subspace of polynomials of degree or less. The Lebesgue constant \Lambda_n(T) is defined as the operator norm of ''X''. This definition requires us to specify a norm on ''C''( 'a'', ''b''. Th ... [...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] |
Lebesgue's Lemma
In mathematics, Lebesgue's lemma is an important statement in approximation theory. It provides a bound for the projection error, controlling the error of approximation by a linear subspace based on a linear projection relative to the optimal error together with the operator norm of the projection. Statement Let be a normed vector space, a subspace of , and a linear projector on . Then for each in : : \, v-Pv\, \leq (1+\, P\, )\inf_\, v-u\, . The proof is a one-line application of the triangle inequality: for any in , by writing as , it follows that :\, v-Pv\, \leq\, v-u\, +\, u-Pu\, +\, P(u-v)\, \leq(1+\, P\, )\, u-v\, where the last inequality uses the fact that together with the definition of the operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Inform ... . See ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled ''CRC Concise Encyclopedia of Mathematics''. The free online version became only partially accessible to the public. In 1999 Weisstein we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condition Number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given f(x) = y, one is solving for ''x,'' and thus the condition number of the (local) inverse must be used. The condition number is derived from the theory of propagation of uncertainty, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unisolvent Point Set
In approximation theory, a finite collection of points X \subset R^n is often called unisolvent for a space W if any element w \in W is uniquely determined by its values on X. X is unisolvent for \Pi^m_n (polynomials in n variables of degree at most m) if there exists a unique polynomial in \Pi^m_n of lowest possible degree which interpolate 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 ...s the data X. Simple examples in R would be the fact that two distinct points determine a line, three points determine a parabola, etc. It is clear that over R, any collection of ''k'' + 1 distinct points will uniquely determine a polynomial of lowest possible degree in \Pi^k. See also * Padua points External linksNumerical Methods / Interpolation Approximation theory {{Mathan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padua Points
In polynomial interpolation of two variables, the Padua points are the first known example (and up to now the only one) of a unisolvent point set (that is, the interpolating polynomial is unique) with ''minimal growth'' of their Lebesgue constant, proven to be O(\log^2 n). Their name is due to the University of Padua, where they were originally discovered. The points are defined in the domain 1,1\times 1,1\subset \mathbb^2. It is possible to use the points with four orientations, obtained with subsequent 90-degree rotations: this way we get four different families of Padua points. The four families We can see the Padua point as a " sampling" of a parametric curve, called ''generating curve'', which is slightly different for each of the four families, so that the points for interpolation degree n and family s can be defined as :\text_n^s=\lbrace\mathbf=(\xi_1,\xi_2)\rbrace=\left\lbrace\gamma_s\left(\frac\right),k=0,\ldots,n(n+1)\right\rbrace. Actually, the Padua points l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer
A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic sets of operations known as Computer program, ''programs'', which enable computers to perform a wide range of tasks. The term computer system may refer to a nominally complete computer that includes the Computer hardware, hardware, operating system, software, and peripheral equipment needed and used for full operation; or to a group of computers that are linked and function together, such as a computer network or computer cluster. A broad range of Programmable logic controller, industrial and Consumer electronics, consumer products use computers as control systems, including simple special-purpose devices like microwave ovens and remote controls, and factory devices like industrial robots. Computers are at the core of general-purpose devices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Nodes
In numerical analysis, Chebyshev nodes (also called Chebyshev points or a Chebyshev grid) are a set of specific algebraic numbers used as nodes for polynomial interpolation and numerical integration. They are the Projection (linear algebra), projection of a set of equispaced points on the unit circle onto the real interval [-1, 1], the circle's diameter. There are two kinds of Chebyshev nodes. The ''Chebyshev nodes of the first kind'', also called the Chebyshev–Gauss nodes or Chebyshev zeros, are the Zero of a function, zeros of a Chebyshev polynomial of the first kind, . The corresponding ''Chebyshev nodes of the second kind'', also called the Chebyshev–Lobatto nodes or Chebyshev extrema, are the Maximum and minimum, extrema of , which are also the zeros of a Chebyshev polynomial of the second kind, , along with the two endpoints of the interval. Both types of numbers are commonly referred to as ''Chebyshev nodes'' or ''Chebyshev points'' in literature. They are named aft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now. In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function f(x) = x^3 grows at an ever increasing rate, but is much slower than growing exponentially. For example, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, 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 ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#Triangle, degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If , , and are the lengths of the sides of a triangle then the triangle inequality states that :c \leq a + b , with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (Norm (mathematics), norms): :\, \mathbf u + \mathbf v\, \leq \, \mathbf u\, + \, \mathbf v\, , where the length of the third side has been replaced by the length of the vector sum . When and are real numbers, they can be viewed as vectors in \R^1, and the triang ... [...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] |
