Finite difference methodIn numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.
Spectrum (functional analysis)In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number is said to be in the spectrum of a bounded linear operator if either has no set-theoretic inverse; or the set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, is the identity operator. By the closed graph theorem, is in the spectrum if and only if the bounded operator is non-bijective on .
Self-adjointIn mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if then since in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In functional analysis, a linear operator on a Hilbert space is called self-adjoint if it is equal to its own adjoint A^∗.
Operator algebraIn functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic. Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory.
Finite element methodThe finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).
Numerical methods for linear least squaresNumerical methods for linear least squares entails the numerical analysis of linear least squares problems. A general approach to the least squares problem can be described as follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem is given by simply because is exactly a sought for orthogonal projection of onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X).
Finite differenceA finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives.
Decomposition of spectrum (functional analysis)The spectrum of a linear operator that operates on a Banach space is a fundamental concept of functional analysis. The spectrum consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts: a point spectrum, consisting of the eigenvalues of ; a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of a proper dense subset of the space; a residual spectrum, consisting of all other scalars in the spectrum.
ChemistryChemistry is the scientific study of the properties and behavior of matter. It is a physical science under natural sciences that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo during a reaction with other substances. Chemistry also addresses the nature of chemical bonds in chemical compounds. In the scope of its subject, chemistry occupies an intermediate position between physics and biology.
RoundingRounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing with, the fraction 312/937 with 1/3, or the expression with . Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement, or estimate; for example, a quantity that was computed as but is known to be accurate only to within a few hundred units is usually better stated as "about ". 