Richardson extrapolationIn numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value . In essence, given the value of for several values of , we can estimate by extrapolating the estimates to . It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century, though the idea was already known to Christiaan Huygens in his calculation of π. In the words of Birkhoff and Rota, "its usefulness for practical computations can hardly be overestimated.
Mathematical optimizationMathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
ComputationA computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computations are mathematical equations and computer algorithms. Mechanical or electronic devices (or, historically, people) that perform computations are known as computers. The study of computation is the field of computability, itself a sub-field of computer science. The notion that mathematical statements should be ‘well-defined’ had been argued by mathematicians since at least the 1600s, but agreement on a suitable definition proved elusive.
Neural networkA neural network can refer to a neural circuit of biological neurons (sometimes also called a biological neural network), a network of artificial neurons or nodes in the case of an artificial neural network. Artificial neural networks are used for solving artificial intelligence (AI) problems; they model connections of biological neurons as weights between nodes. A positive weight reflects an excitatory connection, while negative values mean inhibitory connections. All inputs are modified by a weight and summed.
Deep learningDeep learning is part of a broader family of machine learning methods, which is based on artificial neural networks with representation learning. The adjective "deep" in deep learning refers to the use of multiple layers in the network. Methods used can be either supervised, semi-supervised or unsupervised.
Complexity classIn computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements.
Theory of computationIn theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., approximate solutions versus precise ones). The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".
Model of computationIn computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology.
Online machine learningIn computer science, online machine learning is a method of machine learning in which data becomes available in a sequential order and is used to update the best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once. Online learning is a common technique used in areas of machine learning where it is computationally infeasible to train over the entire dataset, requiring the need of out-of-core algorithms.
Types of artificial neural networksThere are many types of artificial neural networks (ANN). Artificial neural networks are computational models inspired by biological neural networks, and are used to approximate functions that are generally unknown. Particularly, they are inspired by the behaviour of neurons and the electrical signals they convey between input (such as from the eyes or nerve endings in the hand), processing, and output from the brain (such as reacting to light, touch, or heat). The way neurons semantically communicate is an area of ongoing research.
Mathematics educationIn contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge. Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods.
Aitken's delta-squared processIn numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. Its early form was known to Seki Kōwa (end of 17th century) and was found for rectification of the circle, i.e. the calculation of π. It is most useful for accelerating the convergence of a sequence that is converging linearly.
