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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''. The unif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue's Lemma
''For Lebesgue's lemma for open covers of compact spaces in topology see Lebesgue's number 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 . See also * 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 ... [...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. Weisstein continuously improved the notes and accepted corrections and comments from online readers. 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 Mathem ... [...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. In linear regression the condition number of the moment matrix can be used as a diagnostic for multicollinearity. The condition number is an application of the derivative, 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 b ... [...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 mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ... in \Pi^m_n of lowest possible degree which interpolates 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 {{Matha ... [...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 programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as Computer program, programs. These programs enable computers to perform a wide range of tasks. A computer system is a nominally complete computer that includes the Computer hardware, hardware, operating system (main software), and peripheral equipment needed and used for full operation. This term may also refer 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. Simple special-purpose devices like microwave ovens and remote controls are included, as are factory devices like industrial robots and computer-aided design, as well as general-purpose devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Exponential Growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. 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). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay since the function values form a geometric progression. The formula for exponential growth of a variable at the growth rate , as time goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is x_t = x_0(1+r)^t where is the value of ... [...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 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] |
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 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 the triangle, with no side being greater than , then the triangle inequality states that :z \leq x + y , 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 distances, and it is written using vectors and vector lengths ( norms): :\, \mathbf x + \mathbf y\, \leq \, \mathbf x\, + \, \mathbf y\, , where the length of the third side has been replaced by the vector sum . When and are real numbers, they can be viewed as vectors in , and the tria ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
