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Hypre
The Parallel High Performance Preconditioners (hypre) is a library of routines for scalable (parallel) solution of linear systems. The built-in BLOPEX package in addition allows solving eigenvalue problems. The main strength of Hypre is availability of high performance parallel multigrid preconditioners for both structured and unstructured grid problems, see (Falgout et al., 2005, 2006). Currently, Hypre supports only real double-precision arithmetic. Hypre uses the Message Passing Interface (MPI) standard for all message-passing communication. PETSc has an interface to call Hypre preconditioners. Hypre is being developed and is supported by members of the Scalable Linear Solvers project within the Lawrence Livermore National Laboratory. Features hypre provides the following features: * Parallel vectors and matrices, using several different interfaces * Scalable parallel preconditioners * Built-in BLOPEX Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BLOPEX
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free methods, matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem :A x= \lambda B x, for a given pair (A, B) of complex Hermitian matrix, Hermitian or real Symmetric matrix, symmetric matrices, where the matrix B is also assumed positive-definite matrix, positive-definite. Background Kantorovich in 1948 proposed calculating the smallest eigenvalue \lambda_1 of a symmetric matrix A by steepest descent using a direction r = Ax-\lambda (x) x of a scaled gradient of a Rayleigh quotient \lambda(x) = (x, Ax)/(x, x) in a scalar product (x, y) = x'y, with the step size computed by minimizing the Rayleigh quotient in the linear span of the vectors x and w, i.e. in a locally optimal manner. Samokish proposed applying a preconditioner T to the residual vector r to generate the preconditioned direction w = T r and derived ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linux
Linux ( or ) is a family of open-source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991, by Linus Torvalds. Linux is typically packaged as a Linux distribution, which includes the kernel and supporting system software and libraries, many of which are provided by the GNU Project. Many Linux distributions use the word "Linux" in their name, but the Free Software Foundation uses the name "GNU/Linux" to emphasize the importance of GNU software, causing some controversy. Popular Linux distributions include Debian, Fedora Linux, and Ubuntu, the latter of which itself consists of many different distributions and modifications, including Lubuntu and Xubuntu. Commercial distributions include Red Hat Enterprise Linux and SUSE Linux Enterprise. Desktop Linux distributions include a windowing system such as X11 or Wayland, and a desktop environment such as GNOME or KDE Plasma. Distributions intended for ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preconditioning
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. Preconditioning for linear systems In linear algebra and numerical analysis, a preconditioner P of a matrix A is a matrix such that P^A has a smaller condition number than A. It is also common to call T=P^ the preconditioner, rather than P, since P itself is rarely explicitly available. In modern preconditioning, the application of T=P^, i.e., multiplication of a column vector, or a block of column vectors, by T=P^, is commonly performed in a matrix-free fashion, i.e., where neither P, nor T=P^ (and often not even A) are explicitly available in a matrix form. Preconditioners are useful in iterative methods to solve a linear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Libraries
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Numerical may refer to: * Number * Numerical digit * Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawrence Livermore National Laboratory
Lawrence Livermore National Laboratory (LLNL) is a federal research facility in Livermore, California, United States. The lab was originally established as the University of California Radiation Laboratory, Livermore Branch in 1952 in response to the detonation of the first atomic bomb by the Soviet Union during the Cold War. It later became autonomous in 1971 and was designated a national laboratory in 1981. A federally funded research and development center, Lawrence Livermore Lab is primarily funded by the U.S. Department of Energy and it is managed privately and operated by Lawrence Livermore National Security, LLC (a partnership of the University of California), Bechtel, BWX Technologies, AECOM, and Battelle Memorial Institute in affiliation with the Texas A&M University System. In 2012, the laboratory had the synthetic chemical element livermorium (element 116) named after it. Overview LLNL is self-described as a "premier research and development institution for sci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PETSc
The Portable, Extensible Toolkit for Scientific Computation (PETSc, pronounced PET-see; the S is silent), is a suite of data structures and routines developed by Argonne National Laboratory for the scalable (parallel) solution of scientific applications modeled by partial differential equations. It employs the Message Passing Interface (MPI) standard for all message-passing communication. PETSc is the world’s most widely used parallel numerical software library for partial differential equations and sparse matrix computations. PETSc received an R&D 100 Award in 2009. The PETSc Core Development Group won the SIAM/ACM Prize in Computational Science and Engineering for 2015. PETSc is intended for use in large-scale application projects, many ongoing computational science projects are built around the PETSc libraries. Its careful design allows advanced users to have detailed control over the solution process. PETSc includes a large suite of parallel linear and nonlinear equ ... [...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 dynamic range of numeric values by using a floating radix point. Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width), even if at the cost of precision. Double precision may be chosen when the range or precision of single precision would be insufficient. In the IEEE 754-2008 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. One of the first programming languages to provide single- and double-precision floating-point data types was Fortran. Before the widespread adoption of IEEE 754-1985, the representation and p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multigrid
In numerical analysis, a multigrid method (MG 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. The main idea of multigrid is to accelerate the convergence of a basic iterative method (known as relaxation, which generally reduces short-wavelength error) by a ''global'' correction of the fine grid solution approximation from time to time, accomplished by solving a coarse problem. The coarse problem, while cheaper to solve, is similar to the fine grid problem in that it also has short- and long-wavelength errors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unix
Unix (; trademarked as UNIX) is a family of multitasking, multiuser computer operating systems that derive from the original AT&T Unix, whose development started in 1969 at the Bell Labs research center by Ken Thompson, Dennis Ritchie, and others. Initially intended for use inside the Bell System, AT&T licensed Unix to outside parties in the late 1970s, leading to a variety of both academic and commercial Unix variants from vendors including University of California, Berkeley (Berkeley Software Distribution, BSD), Microsoft (Xenix), Sun Microsystems (SunOS/Solaris (operating system), Solaris), Hewlett-Packard, HP/Hewlett Packard Enterprise, HPE (HP-UX), and IBM (IBM AIX, AIX). In the early 1990s, AT&T sold its rights in Unix to Novell, which then sold the UNIX trademark to The Open Group, an industry consortium founded in 1996. The Open Group allows the use of the mark for certified operating systems that comply with the Single UNIX Specification (SUS). Unix systems are chara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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