Conjugate gradient methodIn mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.
Newton's method in optimizationIn calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function F, which are solutions to the equation F (x) = 0. As such, Newton's method can be applied to the derivative f ′ of a twice-differentiable function f to find the roots of the derivative (solutions to f ′(x) = 0), also known as the critical points of f. These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article.
Gradient descentIn mathematics, gradient descent (also often called steepest descent) is a iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent.
Iterative methodIn 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. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of the iterative method.
Extended precisionExtended precision refers to floating-point number formats that provide greater precision than the basic floating-point formats. Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values of expressions on the base format. In contrast to extended precision, arbitrary-precision arithmetic refers to implementations of much larger numeric types (with a storage count that usually is not a power of two) using special software (or, rarely, hardware).
Biconjugate gradient methodIn mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations Unlike the conjugate gradient method, this algorithm does not require the matrix to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose A*. Choose initial guess , two other vectors and and a preconditioner for do In the above formulation, the computed and satisfy and thus are the respective residuals corresponding to and , as approximate solutions to the systems is the adjoint, and is the complex conjugate.
Newton's methodIn numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f′, and an initial guess x0 for a root of f. If the function satisfies sufficient assumptions and the initial guess is close, then is a better approximation of the root than x0.
Secant methodIn numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method. However, the secant method predates Newton's method by over 3000 years. For finding a zero of a function f, the secant method is defined by the recurrence relation. As can be seen from this formula, two initial values x0 and x1 are required.
Accuracy and precisionAccuracy and precision are two measures of observational error. Accuracy is how close a given set of measurements (observations or readings) are to their true value, while precision is how close the measurements are to each other. In other words, precision is a description of random errors, a measure of statistical variability. Accuracy has two definitions: More commonly, it is a description of only systematic errors, a measure of statistical bias of a given measure of central tendency; low accuracy causes a difference between a result and a true value; ISO calls this trueness.
Precision and recallIn pattern recognition, information retrieval, object detection and classification (machine learning), precision and recall are performance metrics that apply to data retrieved from a collection, corpus or sample space. Precision (also called positive predictive value) is the fraction of relevant instances among the retrieved instances. Written as a formula:. Recall (also known as sensitivity) is the fraction of relevant instances that were retrieved. Written as a formula: . Both precision and recall are therefore based on relevance.
Quadruple-precision floating-point formatIn computing, quadruple precision (or quad precision) is a binary floating point–based computer number format that occupies 16 bytes (128 bits) with precision at least twice the 53-bit double precision. This 128-bit quadruple precision is designed not only for applications requiring results in higher than double precision, but also, as a primary function, to allow the computation of double precision results more reliably and accurately by minimising overflow and round-off errors in intermediate calculations and scratch variables.
Cyrillic scriptThe Cyrillic script (sᵻˈɹɪlᵻk ), Slavonic script or the Slavic script is a writing system used for various languages across Eurasia. It is the designated national script in various Slavic, Turkic, Mongolic, Uralic, Caucasian and Iranic-speaking countries in Southeastern Europe, Eastern Europe, the Caucasus, Central Asia, North Asia, and East Asia, and used by many other minority languages. around 250 million people in Eurasia use Cyrillic as the official script for their national languages, with Russia accounting for about half of them.
The Dream of the Fisherman's Wife, also known as Girl Diver and Octopi, Diver and Two Octopi, etc., is a woodblock-printed design by the Japanese artist Hokusai. It is included in ('Young Pines'), a three-volume book of erotica first published in 1814, and has become Hokusai's most famous design. Playing with themes popular in Japanese art, it depicts a young diver entwined sexually with a pair of octopuses. The Dream of the Fisherman's Wife is the most famous image in , published in three volumes from 1814. The book is a work of (erotic art) within the ukiyo-e genre.
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.
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.
Marco EvaristtiMarco Evaristti (born 1963) is a Chilean artist who has lived in Denmark since the 1980s. While a trained and practicing architect, he is best known for hosting a dinner party where the main course was agnolotti pasta that was topped with a meatball made with his own fat, removed earlier in the year in a liposuction operation. Though raised a Catholic, in his teenage years Evaristti found out he was born to a Jewish mother, which some account for the philosophical and religious themes in his work.
Flash memoryFlash memory is an electronic non-volatile computer memory storage medium that can be electrically erased and reprogrammed. The two main types of flash memory, NOR flash and NAND flash, are named for the NOR and NAND logic gates. Both use the same cell design, consisting of floating gate MOSFETs. They differ at the circuit level depending on whether the state of the bit line or word lines is pulled high or low: in NAND flash, the relationship between the bit line and the word lines resembles a NAND gate; in NOR flash, it resembles a NOR gate.
Random-access memoryRandom-access memory (RAM; ræm) 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.
Linear systemIn systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.
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).
