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Polynomial Interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no two x_j the same, a polynomial function p(x)=a_0+a_1x+\cdots+a_nx^n is said to interpolate the data if p(x_j)=y_j for each j\in\. There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials. Applications The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions. Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point. Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toom–Cook Multiplication
Toom–Cook, sometimes known as Toom-3, named after Andrei Toom, who introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers. Given two large integers, ''a'' and ''b'', Toom–Cook splits up ''a'' and ''b'' into ''k'' smaller parts each of length ''l'', and performs operations on the parts. As ''k'' grows, one may combine many of the multiplication sub-operations, thus reducing the overall computational complexity of the algorithm. The multiplication sub-operations can then be computed recursively using Toom–Cook multiplication again, and so on. Although the terms "Toom-3" and "Toom–Cook" are sometimes incorrectly used interchangeably, Toom-3 is only a single instance of the Toom–Cook algorithm, where ''k'' = 3. Toom-3 reduces 9 multiplications to 5, and runs in Θ(''n''log(5)/log(3)) ≈ Θ(''n''1.46). In general, Toom-''k'' runs in , where , ''ne'' is the time spent on sub-mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vandermonde Matrix
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \dots & x_2^\\ 1 & x_3 & x_3^2 & \dots & x_3^\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 1 & x_m & x_m^2 & \dots & x_m^ \end, or :V_ = x_i^ \, for all indices and . Some authors define the Vandermonde matrix as the transpose of the above matrix. The determinant of a square Vandermonde matrix is called a '' Vandermonde polynomial'' or ''Vandermonde determinant''. Its value is the polynomial :\det(V) = \prod_ (x_j - x_i) which is non-zero if and only if all x_i are distinct. The Vandermonde determinant was sometimes called the ''discriminant'', although, presently, the discriminant of a polynomial is the square of the Vandermonde determinant of the roots of the polynomial. The Vandermonde determinant is an alternating form in the x_i, meaning that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices and is denoted as . Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. Computing matrix products is a central operation in all computational applications of linear algebra. Notation This article will use the following n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor Theorem
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial f(x) has a factor (x - \alpha) if and only if f(\alpha)=0 (i.e. \alpha is a root). Factorization of polynomials Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent. The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:. # Deduce the candidate of zero a of the polynomial f from its leading coefficient a_n and constant term a_0. (See Rational Root Theorem.) # Use the factor theorem to conclude that (x-a) is a factor of f(x). # Compute the polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Polynomials
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 degr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unisolvent Functions
In mathematics, a set of ''n'' functions ''f''1, ''f''2, ..., ''f''''n'' is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors : \beginf_1(x_1) \\ f_1(x_2) \\ \vdots \\ f_1(x_n)\end, \beginf_2(x_1) \\ f_2(x_2) \\ \vdots \\ f_2(x_n)\end, \dots, \beginf_n(x_1) \\ f_n(x_2) \\ \vdots \\ f_n(x_n)\end are linearly independent for any choice of ''n'' distinct points ''x''1, ''x''2 ... ''x''''n'' in Ω. Equivalently, the collection is unisolvent if the matrix ''F'' with entries ''f''''i''(''x''''j'') has nonzero determinant: det(''F'') ≠ 0 for any choice of distinct ''x''''j'''s in Ω. Unisolvency is a property of vector spaces, not just particular sets of functions. That is, a vector space of functions of dimension ''n'' is unisolvent if given any basis (equivalently, a linearly independent set of ''n'' functions), the basis is unisolvent (as a set of functions). This is because any two bases are related by an invertible matrix (the change of basis matrix), so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
