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Stone Method
In numerical analysis, Stone's method, also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations. The method uses an incomplete LU decomposition, which approximates the exact LU decomposition, to get an iterative solution of the problem. The method is named after Harold S. Stone, who proposed it in 1968. The LU decomposition is an excellent general-purpose linear equation solver. The biggest disadvantage is that it fails to take advantage of coefficient matrix to be a sparse matrix. The LU decomposition of a sparse matrix is usually not sparse, thus, for a large system of equations, LU decomposition may require a prohibitive amount of memory and number of arithmetical operations. In the preconditioned iterative methods, if the preconditioner matrix ''M'' is a good approximation of coefficient matrix ''A'' then the convergence is faster. This brings one to idea of using approximate factorization ''LU'' of ''A'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joel H
Joel or Yoel is a name meaning "Yahweh Is God" and may refer to: * Joel (given name), origin of the name including a list of people with the first name. * Joel (surname), a surname * Joel (footballer, born 1904), Joel de Oliveira Monteiro, Brazilian football goalkeeper * Joel (footballer, born 1980), Joel Bertoti Padilha, Brazilian football centre-back * Joel (prophet), a prophet of ancient Israel ** Book of Joel, a book in the Jewish Tanakh, and in the Christian Bible, ascribed to the prophet * Joel, Georgia Joel is an unincorporated community in Carroll County, in the U.S. state of Georgia Georgia most commonly refers to: * Georgia (country), a country in the Caucasus region of Eurasia * Georgia (U.S. state), a state in the Southeast United Stat ..., a community in the United States * Joel, Wisconsin, a community in the United States {{disambiguation, hn, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , 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 \Delta is the operator that maps a function to the function \Delta /math> defined by :\Delta x)= f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface, for example the hyperbolic plane and elliptic plane. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane. Euclidean geometry Euclid set forth the first great landmark of mathematical thought, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. Definitions Let L be linear differential operator of order ''m'' on a domain \Omega in R''n'' given by Lu = \sum_ a_\alpha(x)\partial^\alpha u where \alpha = (\alpha_1, \dots, \alpha_n) denotes a multi-index, and \partial^\alpha u = \partial^_1 \cdots \partial_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentadiagonal Matrix
In linear algebra, a pentadiagonal matrix is a special case of band matrices. Its only nonzero entries are on the main diagonal and the first two upper and two lower diagonals. So, it is of the form. :\begin c_1 & d_1 & e_1 & 0 & \cdots & \cdots & 0 \\ b_1 & c_2 & d_2 & e_2 & \ddots & & \vdots \\ a_1 & b_2 & \ddots & \ddots & \ddots & \ddots & \vdots \\ 0 & a_2 & \ddots & \ddots & \ddots & e_ & 0 \\ \vdots & \ddots & \ddots & \ddots & \ddots & d_ & e_ \\ \vdots & & \ddots & a_ & b_ & c_ & d_ \\ 0 & \cdots & \cdots & 0 & a_ & b_ & c_n \end \in\; \R^ \,. It follows that a pentadiagonal matrix has at most 5n-6 nonzero entries, where ''n'' is the size of the matrix. Hence, pentadiagonal matrices are sparse, making them useful in numerical analysis. See also * Tridiagonal matrix In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instruction (computer Science)
In computer science, an instruction set architecture (ISA), also called computer architecture, is an abstract model of a computer. A device that executes instructions described by that ISA, such as a central processing unit (CPU), is called an ''implementation''. In general, an ISA defines the supported instructions, data types, registers, the hardware support for managing main memory, fundamental features (such as the memory consistency, addressing modes, virtual memory), and the input/output model of a family of implementations of the ISA. An ISA specifies the behavior of machine code running on implementations of that ISA in a fashion that does not depend on the characteristics of that implementation, providing binary compatibility between implementations. This enables multiple implementations of an ISA that differ in characteristics such as performance, physical size, and monetary cost (among other things), but that are capable of running the same machine code ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Access Memory
Random-access memory (RAM; ) is a form of computer memory that can be read and changed in any order, typically used to store working data and machine code. A random-access memory device allows data items to be read or written in almost the same amount of time irrespective of the physical location of data inside the memory, in contrast with other direct-access data storage media (such as hard disks, CD-RWs, DVD-RWs and the older magnetic tapes and drum memory), where the time required to read and write data items varies significantly depending on their physical locations on the recording medium, due to mechanical limitations such as media rotation speeds and arm movement. RAM contains multiplexing and demultiplexing circuitry, to connect the data lines to the addressed storage for reading or writing the entry. Usually more than one bit of storage is accessed by the same address, and RAM devices often have multiple data lines and are said to be "8-bit" or "16-bit", etc. devices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold S
Harold may refer to: People * Harold (given name), including a list of persons and fictional characters with the name * Harold (surname), surname in the English language * András Arató, known in meme culture as "Hide the Pain Harold" Arts and entertainment * ''Harold'' (film), a 2008 comedy film * ''Harold'', an 1876 poem by Alfred, Lord Tennyson * ''Harold, the Last of the Saxons'', an 1848 book by Edward Bulwer-Lytton, 1st Baron Lytton * ''Harold or the Norman Conquest'', an opera by Frederic Cowen * ''Harold'', an 1885 opera by Eduard Nápravník * Harold, a character from the cartoon ''The Grim Adventures of Billy & Mandy'' *Harold & Kumar, a US movie; Harold/Harry is the main actor in the show. Places ;In the United States * Alpine, Los Angeles County, California, an erstwhile settlement that was also known as Harold * Harold, Florida, an unincorporated community * Harold, Kentucky, an unincorporated community * Harold, Missouri, an unincorporated community ;E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
