|
Bisection Method
In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method, the binary search method, or the dichotomy method. For polynomials, more elaborate methods exist for testing the existence of a root in an interval ( Descartes' rule of signs, Sturm's theorem, Budan's theorem). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see Real-root isolation. The method ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). # The image of a continuous function over an interval is itself an interval. Motivation This captures an intuitive property of continuous functions over the real numbers: given ''f'' continuous on ,2/math> with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Theorem The intermediate value theorem states the following: Consider the closed interval I = ,b/math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nested Intervals
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of Interval (mathematics), intervals I_n on the Interval (mathematics), real number line with natural number, natural numbers n=1,2,3,\dots as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met: # Every interval in the sequence is contained in the previous one (I_ is always a subset of I_n). # The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold \varepsilon after a certain index N). In other words, the left bound of the interval I_n can only increase (a_\geq a_n), and the right bound can only decrease (b_\leq b_n). Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonia, Babylonians discovered a Methods of computing square roots, metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lehmer–Schur Algorithm
In mathematics, the Lehmer–Schur algorithm (named after Derrick Henry Lehmer and Issai Schur) is a root-finding algorithm for complex polynomials, extending the idea of enclosing roots like in the one-dimensional bisection method to the complex plane. It uses the Schur-Cohn test to test increasingly smaller disks for the presence or absence of roots. Schur-Cohn algorithm This algorithm allows one to find the distribution of the roots of a complex polynomial with respect to the unit circle in the complex plane. It is based on two auxiliary polynomials, introduced by Schur. For a complex polynomial p of degree n its ''reciprocal adjoint polynomial'' p^ is defined by p^(z) = z^\overline and its ''Schur Transform'' Tp by : Tp =\overlinep-\overlinep^*, where a bar denotes complex conjugation. So, if p(z) = a_z^+\cdots+a_z +a_ with a_n\neq 0, then p^(z) = \bar_z^+ \bar_z^+\cdots+\bar_, with leading zero-terms, if any, removed. The coefficients of Tp can therefore be directl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices A, B, C and D is sometimes denoted as \square ABCD. Quadrilaterals are either simple polygon, simple (not self-intersecting), or complex polygon, complex (self-intersecting, or crossed). Simple quadrilaterals are either convex polygon, convex or concave polygon, concave. The Internal and external angle, interior angles of a simple (and Plane (geometry), planar) quadrilateral ''ABCD'' add up to 360 Degree (angle), degrees, that is :\angle A+\angle B+\angle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Continuous Mapping
In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations. The degree of a map between general manifolds was first defined by Brouwer, who showed that the degree is homotopy invariant and used it to prove the Brouwer fixed point theorem. Less general forms of the concept existed before Brouwer, such as the winding number and the Kronecker characteristic (or Kronecker integral). In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number. Definitions of the degree From ''S''''n'' to ''S''''n'' The simplest and most important ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''Polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a ''surface'' as shown in the illustration. Surface integrals have applications in physics, particularly in the classical theories of electromagnetism and fluid mechanics. Surface integrals of scalar fields Assume that ''f'' is a scalar, vector, or tensor field defined on a surface ''S''. To find an explicit formula for the surface integral of ''f'' over ''S'', we need to parameterize ''S'' by defining a system of curvilinear coordinates on ''S'', like the latitude and longitude on a sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sign Function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented as \sgn x or \sgn (x). Definition The signum function of a real number x is a piecewise function which is defined as follows: \sgn x :=\begin -1 & \text x 0. \end The law of trichotomy states that every real number must be positive, negative or zero. The signum function denotes which unique category a number falls into by mapping it to one of the values , or which can then be used in mathematical expressions or further calculations. For example: \begin \sgn(2) &=& +1\,, \\ \sgn(\pi) &=& +1\,, \\ \sgn(-8) &=& -1\,, \\ \sgn(-\frac) &=& -1\,, \\ \sgn(0) &=& 0\,. \end Basic properties Any real number can be expressed as the product of its absolute value and its sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi. The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally used f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Degree
In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can often be used to prove existence of a second, nontrivial, solution. There are different types of degree for different types of maps: e.g. for maps between Banach spaces there is the Brouwer degree in R''n'', the Leray-Schauder degree for compact mappings in normed spaces, the coincidence degree and various other types. There is also a degree for continuous maps between manifolds. Topological degree theory has applications in complementarity problems, differential equations, differential inclusions and dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double-precision
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide range of numeric values by using a floating radix point. Double precision may be chosen when the range or precision of single precision would be insufficient. In the IEEE 754 standard, the 64-bit base-2 format is officially referred to as binary64; it was called double in IEEE 754-1985. IEEE 754 specifies additional floating-point formats, including 32-bit base-2 ''single precision'' and, more recently, base-10 representations (decimal floating point). One of the first programming languages to provide floating-point data types was Fortran. Before the widespread adoption of IEEE 754-1985, the representation and properties of floating-point data types depended on the computer manufacturer and computer model, and upon decisions made by programming-language implementers. E.g., GW-BASIC's double-precision ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
