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Frontal Solver
A frontal solver is an approach to solving sparse linear systems which is used extensively in finite element analysis. Algorithms of this kind are variants of Gauss elimination that automatically avoids a large number of operations involving zero terms due to the fact that the matrix is only sparse. The development of frontal solvers is usually considered as dating back to work by Bruce Irons. A frontal solver builds a LU or Cholesky decomposition of a sparse matrix. Frontal solvers start with one or a few diagonal entries of the matrix, then consider all of those diagonal entries that are coupled to the first set via off-diagonal entries, and so on. In the finite element context, these consecutive sets form "fronts" that march through the domain (and consequently through the matrix, if one were to permute rows and columns of the matrix in such a way that the diagonal entries are ordered by the wave they are part of). Processing the front involves dense matrix operations, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sparse Matrix
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly equal to the number of rows or columns. By contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements (e.g., ''m'' × ''n'' for an ''m'' × ''n'' matrix) is sometimes referred to as the sparsity of the matrix. Conceptually, sparsity corresponds to systems with few pairwise interactions. For example, consider a line of balls connected by springs from one to the next: this is a sparse system, as only adjacent balls are coupled. By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iain S
Ian or Iain is a name of Scottish Gaelic origin, which is derived from the Hebrew given name (Yohanan, ') and corresponds to the English name John. The spelling Ian is an Anglicization of the Scottish Gaelic forename ''Iain''. This name is a popular name in Scotland, where it originated, as well as in other English-speaking countries. The name has fallen out of the top 100 male baby names in the United Kingdom, having peaked in popularity as one of the top 10 names throughout the 1960s. In 1900, Ian ranked as the 180th most popular male baby name in England and Wales. , the name has been in the top 100 in the United States every year since 1982, peaking at 65 in 2003. Other Gaelic forms of the name "John" include " Seonaidh" ("Johnny" from Lowland Scots), "Seon" (from English), "Seathan", and "Seán" and "Eoin" (from Irish). The Welsh equivalent is Ioan, the Cornish counterpart is Yowan and the Breton equivalent is Yann. Notable people named Ian Given name * Ian Agol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banded Matrix
In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal ''band'', comprising the main diagonal and zero or more diagonals on either side. Band matrix Bandwidth Formally, consider an ''n''×''n'' matrix ''A''=(''a''''i,j'' ). If all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''1 and ''k''2: :a_=0 \quad\mbox\quad ji+k_2; \quad k_1, k_2 \ge 0.\, then the quantities ''k''1 and ''k''2 are called the and , respectively. The of the matrix is the maximum of ''k''1 and ''k''2; in other words, it is the number ''k'' such that a_=0 if , i-j, > k . Examples *A band matrix with ''k''1 = ''k''2 = 0 is a diagonal matrix, with bandwidth 0. *A band matrix with ''k''1 = ''k''2 = 1 is a tridiagonal matrix, with bandwidth 1. *For ''k''1 = ''k''2 = 2 one has a pentadiagonal matrix and so on. * Triangular matrices **For ''k''1 = 0, ''k''2 = ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skyline Matrix
In scientific computing, skyline matrix storage, or SKS, or a variable band matrix storage, or envelope storage scheme is a form of a sparse matrix storage format matrix that reduces the storage requirement of a matrix more than band matrix, banded storage. In banded storage, all entries within a fixed distance from the diagonal (called half-bandwidth) are stored. In column-oriented skyline storage, only the entries from the first nonzero entry to the last nonzero entry in each column are stored. There is also row oriented skyline storage, and, for symmetric matrices, only one triangle is usually stored. Skyline storage has become very popular in the finite element codes for structural mechanics, because the skyline is preserved by Cholesky decomposition (a method of solving systems of linear equations with a symmetric, positive-definite matrix; all Sparse_matrix#Reducing_fill-in, fill-in falls within the skyline), and systems of equations from finite elements have a relatively sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UMFPACK
UMFPACK () is a set of routines for solving unsymmetric sparse linear systems of the form Ax=b, using the Unsymmetric MultiFrontal method (Matrix A is not required to be symmetric). Written in ANSI/ISO C and interfaces with * MATLAB version 6.0 and later * SciPy, and thus SciPy-relied software FuncDesigner, SageMath, PythonXY It appears as a built-in routine (for lu, backslash, and forward slash) in MATLAB, and includes a MATLAB interface, a C-callable interface, and a Fortran-callable interface. Note that "UMFPACK" is pronounced in two syllables, "Umph Pack". It is not "You Em Ef Pack" . UMFPACK has installation options to use the many versions of the BLAS, or no BLAS at all. BLAS is what UMFPACK relies on, to get high performance on a wide range of computers. Versions 1 and 1.1 were in Fortran 77 (Jan. 1995) and are licensed for non-commercial only. Version 2.2 appears as the Fortran package MA38 in the Harwell Subroutine Library. Versions 3 (March 2001) to 5.1 (May 2007 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MUMPS (software)
MUMPS (MUltifrontal Massively Parallel sparse direct Solver) is a software application for the solution of large sparse systems of linear algebraic equations on distributed memory parallel computers. It was developed in European project PARASOL (1996–1999) by CERFACS, IRIT- ENSEEIHT and RAL. The software implements the multifrontal method, which is a version of Gaussian elimination for large sparse systems of equations, especially those arising from the finite element method. It is written in Fortran 90 with parallelism by MPI and it uses BLAS and ScaLAPACK kernels for dense matrix computations. Since 1999, MUMPS has been supported by CERFACS, IRIT- ENSEEIHT, and INRIA The National Institute for Research in Digital Science and Technology (Inria) () is a French national research institution focusing on computer science and applied mathematics. It was created under the name French Institute for Research in Comp .... The importance of MUMPS lies in the fact that it i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Computing
Parallel computing is a type of computing, computation in which many calculations or Process (computing), processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different forms of parallel computing: Bit-level parallelism, bit-level, Instruction-level parallelism, instruction-level, Data parallelism, data, and task parallelism. Parallelism has long been employed in high-performance computing, but has gained broader interest due to the physical constraints preventing frequency scaling.S.V. Adve ''et al.'' (November 2008)"Parallel Computing Research at Illinois: The UPCRC Agenda" (PDF). Parallel@Illinois, University of Illinois at Urbana-Champaign. "The main techniques for these performance benefits—increased clock frequency and smarter but increasingly complex architectures—are now hitting the so-called power wall. The computer industry has accepted that future performance inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer File
A computer file is a System resource, resource for recording Data (computing), data on a Computer data storage, computer storage device, primarily identified by its filename. Just as words can be written on paper, so too can data be written to a computer file. Files can be shared with and transferred between computers and Mobile device, mobile devices via removable media, Computer networks, networks, or the Internet. Different File format, types of computer files are designed for different purposes. A file may be designed to store a written message, a document, a spreadsheet, an Digital image, image, a Digital video, video, a computer program, program, or any wide variety of other kinds of data. Certain files can store multiple data types at once. By using computer programs, a person can open, read, change, save, and close a computer file. Computer files may be reopened, modified, and file copying, copied an arbitrary number of times. Files are typically organized in a file syst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Analysis
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical models, mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Computers are usually used to perform the calculations required. With high-speed supercomputers, better solutions can be achieved and are often required to solve the largest and most complex problems. FEM is a general numerical analysis, numerical method for solving partial differential equations in two- or three-space variables (i.e., some boundary value problems). There are also studies about using FEM to solve high-dimensional problems. To solve a problem, FEM subdivides a large system into smaller, simpler parts called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the constructio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Memory
Computer memory stores information, such as data and programs, for immediate use in the computer. The term ''memory'' is often synonymous with the terms ''RAM,'' ''main memory,'' or ''primary storage.'' Archaic synonyms for main memory include ''core'' (for magnetic core memory) and ''store''. Main memory operates at a high speed compared to mass storage which is slower but less expensive per bit and higher in capacity. Besides storing opened programs and data being actively processed, computer memory serves as a Page cache, mass storage cache and write buffer to improve both reading and writing performance. Operating systems borrow RAM capacity for caching so long as it is not needed by running software. If needed, contents of the computer memory can be transferred to storage; a common way of doing this is through a memory management technique called ''virtual memory''. Modern computer memory is implemented as semiconductor memory, where data is stored within memory cell (com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Matrix
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly equal to the number of rows or columns. By contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements (e.g., ''m'' × ''n'' for an ''m'' × ''n'' matrix) is sometimes referred to as the sparsity of the matrix. Conceptually, sparsity corresponds to systems with few pairwise interactions. For example, consider a line of balls connected by springs from one to the next: this is a sparse system, as only adjacent balls are coupled. By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
