In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.
In numerical analysis, a multigrid method is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and long-wavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid. MG methods can be used as solvers as well as preconditioners.
In linear algebra, the order-rKrylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A, that is,
The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the "most useful" eigenvalues and eigenvectors of an Hermitian matrix, where is often but not necessarily much smaller than . Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability.
Gene Howard Golub, was an American numerical analyst who taught at Stanford University as Fletcher Jones Professor of Computer Science and held a courtesy appointment in electrical engineering.
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method.
In linear algebra, if are complex matrices for some nonnegative integer , and , then the matrix pencil of degree is the matrix-valued function defined on the complex numbers
In computational mathematics, a matrix-free method is an algorithm for solving a linear system of equations or an eigenvalue problem that does not store the coefficient matrix explicitly, but accesses the matrix by evaluating matrix-vector products. Such methods can be preferable when the matrix is so big that storing and manipulating it would cost a lot of memory and computing time, even with the use of methods for sparse matrices. Many iterative methods allow for a matrix-free implementation, including:
In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues , left eigenvectors and right eigenvectors such that
The image segmentation problem is concerned with partitioning an image into multiple regions according to some homogeneity criterion. This article is primarily concerned with graph theoretic approaches to image segmentation applying graph partitioning via minimum cut or maximum cut. Segmentation-based object categorization can be viewed as a specific case of spectral clustering applied to image segmentation.
Hendrik "Henk" Albertus van der Vorst is a Dutch mathematician and Emeritus Professor of Numerical Analysis at Utrecht University. According to the Institute for Scientific Information (ISI), his paper on the BiCGSTAB method was the most cited paper in the field of mathematics in the 1990s. He is a member of the Royal Netherlands Academy of Arts and Sciences (KNAW) since 2002 and the Netherlands Academy of Technology and Innovation. In 2006 he was awarded a knighthood of the Order of the Netherlands Lion. Henk van der Vorst is a Fellow of Society for Industrial and Applied Mathematics (SIAM).
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form
An algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time.
ARPACK, the ARnoldi PACKage, is a numerical software library written in FORTRAN 77 for solving large scale eigenvalue problems in the matrix-free fashion.
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem
