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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] |
Approximation Theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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] |
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] |
