AI App Make Video

AI App Make Video — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

List of .NET libraries and frameworks

This article contains a list of libraries that can be used in .NET languages. These languages require .NET Framework, Mono, or .NET, which provide a basis for software development, platform independence, language interoperability and extensive framework libraries. Standard Libraries (including the Base Class Library) are not included in this article. == Introduction == Apps created with .NET Framework or .NET run in a software environment known as the Common Language Runtime (CLR), an application virtual machine that provides services such as security, memory management, and exception handling. The framework includes a large class library called Framework Class Library (FCL). Thanks to the hosting virtual machine, different languages that are compliant with the .NET Common Language Infrastructure (CLI) can operate on the same kind of data structures. These languages can therefore use the FCL and other .NET libraries that are also written in one of the CLI compliant languages. When the source code of such languages are compiled, the compiler generates platform-independent code in the Common Intermediate Language (CIL, also referred to as bytecode), which is stored in CLI assemblies. When a .NET app runs, the just-in-time compiler (JIT) turns the CIL code into platform-specific machine code. To improve performance, .NET Framework also comes with the Native Image Generator (NGEN), which performs ahead-of-time compilation to machine code. This architecture provides language interoperability. Each language can use code written in other languages. Calls from one language to another are exactly the same as would be within a single programming language. If a library is written in one CLI language, it can be used in other CLI languages. Moreover, apps that consist only of pure .NET assemblies, can be transferred to any platform that contains an implementation of CLI and run on that platform. For example, apps written using .NET can run on Windows, macOS, and various versions of Linux. .NET apps or their libraries, however, may depend on native platform features, e.g. COM. As such, platform independence of .NET apps depends on the ability to transfer necessary native libraries to target platforms. In 2019, the Windows Forms and Windows Presentation Foundation portions of .NET Framework were made open source. === .NET implementations === There are four primary .NET implementations that are actively developed and maintained: .NET Framework: The original .NET implementation that has existed since 2002. While not yet discontinued, Microsoft does not plan on releasing its next major version, 5.0. Mono: A cross-platform implementation of .NET Framework by Ximian, introduced in 2004. It is free and open-source. It is now developed by Xamarin, a subsidiary of Microsoft. Universal Windows Platform (UWP): An implementation of .NET used for building UWP apps. It's designed to unify development for different targeted types of devices, including PCs, tablets, phablets, phones, and the Xbox. .NET: A cross-platform re-implementation of .NET Framework, introduced in 2016 and initially called .NET Core. It is free and open-source. .NET superseded .NET Framework with the release of .NET 5. Each implementation of .NET includes the following components: One or more runtime environments, e.g. Common Language Runtime (CLR) for .NET Framework and CoreCLR for .NET A class library The .NET Standard is a set of common APIs that are implemented in the Base Class Library of any .NET implementation. The class library of each implementation must implement the .NET Standard, but may also implement additional APIs. Traditionally, .NET apps targeted a certain version of a .NET implementation, e.g. .NET Framework 4.6. Starting with the .NET Standard, an app can target a version of the .NET Standard and then it could be used (without recompiling) by any implementation that supports that level of the standard. This enables portability across different .NET implementations. The following table lists the .NET implementations that adhere to the .NET Standard and the version number at which each implementation became compliant with a given version of .NET Standard. For example, according to this table, .NET Core 3.0 was the first version of .NET Core that adhered to .NET Standard 2.1. This means that any version of .NET Core bigger than 3.0 (e.g. .NET Core 3.1) also adheres to .NET Standard 2.1. == Web frameworks == === ASP.NET === First released in 2002, ASP.NET is an open-source server-side web application framework designed for web development to produce dynamic web pages. It is the successor to Microsoft's Active Server Pages (ASP) technology, built on the Common Language Runtime (CLR). === ASP.NET Core === ASP.NET was completely rewritten in 2016 as a modular web framework, together with other frameworks like Entity Framework. The re-written framework uses the new open-source .NET Compiler Platform (also known by its codename "Roslyn") and is cross platform. The programming models ASP.NET MVC, ASP.NET Web API, and ASP.NET Web Pages (a model using only Razor pages) were merged into a unified MVC 6. === Blazor === Blazor is a free and open-source web framework that enables developers to create Single-page Web apps using C# and HTML in ASP.NET Razor pages ("components"). Blazor is part of the ASP.NET Core framework. Blazor Server apps are hosted on a web server, while Blazor WebAssembly apps are downloaded to the client's web browser before running. In addition, a Blazor Hybrid framework is available with server-based and client-based application components. == Numerical libraries == === Open-source numerical libraries === ==== AForge.NET ==== This is a computer vision and artificial intelligence library. It implements a number of genetic, fuzzy logic and machine learning algorithms with several architectures of artificial neural networks with corresponding training algorithms. ==== ALGLIB ==== This is a cross-platform open source numerical analysis and data processing library. It consists of algorithm collections written in different programming languages (C++, C#, FreePascal, Delphi, VBA) and has dual licensing – commercial and GPL. ==== Math.NET Numerics ==== This library aims to provide methods and algorithms for numerical computations in science, engineering and everyday use. Covered topics include special functions, linear algebra, probability models, random numbers, interpolation, integral transforms and more. MIT/X11 license. ==== Meta.Numerics ==== This is a library for advanced scientific computation in the .NET Framework. ==== ML.NET ==== This is a free software machine learning library. The preview release of ML.NET included transforms for feature engineering like n-gram creation, and learners to handle binary classification, multi-class classification, and regression tasks. Additional ML tasks like anomaly detection and recommendation systems have since been added, and other approaches like deep learning will be included in future versions. === Proprietary numerical libraries === ==== ILNumerics.Net ==== This is a high performance, typesafe numerical array set of classes and functions for general math, FFT and linear algebra. The library, developed for .NET/Mono, aims to provide 32- and 64-bit script-like syntax in C#, 2D & 3D plot controls, and efficient memory management. It is released under GPLv3 or commercial license. ==== Measurement Studio ==== This is an integrated suite of UI controls and class libraries for use in developing test and measurement applications. The analysis class libraries provide various digital signal processing, signal filtering, signal generation, peak detection, and other general mathematical functionality. ==== NMath ==== This is a numerical component library for the .NET platform developed by CenterSpace Software. It includes signal processing (FFT) classes, a linear algebra (LAPACK & BLAS) framework, and a statistics package. == 3D graphics == === Open-source 3D graphics === ==== Open Toolkit (OpenTK) ==== This is a low-level C# binding for OpenGL, OpenGL ES and OpenAL. It runs on Windows, Linux, Mac OS X, BSD, Android and iOS. It can be used standalone or integrated into a GUI. ==== Windows Presentation Foundation (WPF) ==== This is a graphical subsystem for rendering user interfaces, developed by Microsoft. It also contains a 3D rendering engine. In addition, interactive 2D content can be overlaid on 3D surfaces natively. It only runs on Windows operating systems. === Proprietary 3D graphics === ==== Unity ==== This is a cross-platform game engine developed by Unity Technologies and used to develop video games for PC, consoles, mobile devices and websites. == Image processing == === AForge.NET === This is a computer vision and artificial intelligence library. It implements a number of image processing algorithms and filters. It is released under the LGPLv3 and partly GPLv3 license. Majority of the library is written in C# and th
Read more →
Information Rules

Information Rules is a 1999 book by Carl Shapiro and Hal Varian applying traditional economic theories to modern information-based technologies. The book examines commercial strategies appropriate to companies that deal in information, given the high "first copy" and low "subsequent copy" costs of information commodities, such as music CDs or original texts. == Content == The book examines competing standards, and how a company might influence widespread consumer acceptance of one over another, such as VHS versus Betamax, or HD DVD versus Blu-ray. The book mentions possible business strategies of such publishers as Encyclopædia Britannica who have to confront how to stay viable as technology changes the value and availability of information.
Read more →
Visual Peer Review

== Development and history == Visual Peer Review was first described in a 2017 classroom study by Friedman and Rosen, which examined how students evaluate peer-produced data visualizations using structured rubrics. Developed within the broader fields of data visualization, information visualization, and educational technology, the system emphasized clear labeling, visual integrity, and reduction of chartjunk. Students assigned rubric scores and provided written explanations, aligning the activity with established principles of peer review. Follow-up research expanded both the methodological and analytic dimensions of the framework. Friedman and colleagues applied natural language processing (NLP) to peer-review text to analyze part-of-speech patterns, sentence complexity, and comment length. These analyses offered insight into how students expressed critique and engaged with core design principles. Later studies incorporated advanced statistical modeling to evaluate system-level behavior, including peer review networks and reviewer typologies. Between 2021 and 2024, the framework underwent iterative refinement through a series of studies that explored interface design, behavioral nudges, reviewer engagement, and social network dynamics. The system was influenced by earlier work in computer-supported peer review—particularly My Reviewers, a rubric-based writing assessment platform developed by Joe Moxley at the University of South Florida. While Moxley's platform focused on text-based feedback, Visual Peer Review adapted its core structure to support critique of DataVis and visual analytics. To guide structured analysis and feedback, Friedman and Rosen also drew on the “what, why, and how” framework introduced by Liu and Stasko (2010), which emphasizes understanding a visualization's purpose, task alignment, and encoding strategy. == Framework and components == Visual Peer Review is designed to support critique, reflection, and learning in courses focusing on data visualization, visual analytics, and related fields in educational technology. The system consists of interconnected component. Core components include: Visual Artifacts: Students generate original visualizations using software such as R (e.g., ggplot2), Tableau, Python, or Adobe Illustrator. These artifacts may include statistical graphics, dashboards, or design-oriented infographics. Rubric-Based Assessment: Peer reviewers evaluate submitted visualizations using structured rubrics grounded in visualization theory and design heuristics. Rubric dimensions typically include: Use of labeling and axis scales Minimalization of chartjunk and clutter (following Tufte's principles) Optimization of the data–ink ratio Preservation of visual integrity through accurate representation (lie factor) Written Peer Comments: In addition to scoring, reviewers provide narrative feedback explaining their reasoning. These comments aim to improve design literacy, strengthen visual reasoning, and support the learning process common to peer review across educational contexts. Instructor Analytics Dashboard: Instructors access an analytics dashboard that displays peer-review activity across the course. Metrics include comment length, rubric coverage, participation patterns, and potential indicators of disengagement. These features position the framework within the domain of learning analytics, where visualized data helps instructors monitor student progress and identify support needs. == Ongoing development == Current work focuses on enhancing rubric structure, integrating principles from human–computer interaction, DataVis and expanding learning-analytics capabilities. Ongoing studies investigate how interface design, reviewer behavior, and classroom context influence the quality of feedback and overall engagement. Continuing development positions Visual Peer Review at the intersection of data visualization education, peer assessment, and educational technology.
Read more →
Timeline of algorithms

The following timeline of algorithms outlines the development of algorithms (mainly "mathematical recipes") since their inception. == Antiquity == Before – writing about "recipes" (on cooking, rituals, agriculture and other themes) c. 1700–2000 BC – Egyptians develop earliest known algorithms for multiplying two numbers c. 1600 BC – Babylonians develop earliest known algorithms for factorization and finding square roots c. 300 BC – Euclid's algorithm c. 200 BC – the Sieve of Eratosthenes 263 AD – Gaussian elimination described by Liu Hui == Medieval Period == 628 – Chakravala method described by Brahmagupta c. 820 – Al-Khawarizmi described algorithms for solving linear equations and quadratic equations in his Algebra; the word algorithm comes from his name 825 – Al-Khawarizmi described the algorism, algorithms for using the Hindu–Arabic numeral system, in his treatise On the Calculation with Hindu Numerals, which was translated into Latin as Algoritmi de numero Indorum, where "Algoritmi", the translator's rendition of the author's name gave rise to the word algorithm (Latin algorithmus) with a meaning "calculation method" c. 850 – cryptanalysis and frequency analysis algorithms developed by Al-Kindi (Alkindus) in A Manuscript on Deciphering Cryptographic Messages, which contains algorithms on breaking encryptions and ciphers c. 1025 – Ibn al-Haytham (Alhazen), was the first mathematician to derive the formula for the sum of the fourth powers, and in turn, he develops an algorithm for determining the general formula for the sum of any integral powers c. 1400 – Ahmad al-Qalqashandi gives a list of ciphers in his Subh al-a'sha which include both substitution and transposition, and for the first time, a cipher with multiple substitutions for each plaintext letter; he also gives an exposition on and worked example of cryptanalysis, including the use of tables of letter frequencies and sets of letters which can not occur together in one word == Before 1940 == 1540 – Lodovico Ferrari discovered a method to find the roots of a quartic polynomial 1545 – Gerolamo Cardano published Cardano's method for finding the roots of a cubic polynomial 1614 – John Napier develops method for performing calculations using logarithms 1671 – Newton–Raphson method developed by Isaac Newton 1690 – Newton–Raphson method independently developed by Joseph Raphson 1706 – John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places 1768 – Leonhard Euler publishes his method for numerical integration of ordinary differential equations in problem 85 of Institutiones calculi integralis 1789 – Jurij Vega improves Machin's formula and computes π to 140 decimal places, 1805 – FFT-like algorithm known by Carl Friedrich Gauss 1842 – Ada Lovelace writes the first algorithm for a computing engine 1903 – A fast Fourier transform algorithm presented by Carle David Tolmé Runge 1918 - Soundex 1926 – Borůvka's algorithm 1926 – Primary decomposition algorithm presented by Grete Hermann 1927 – Hartree–Fock method developed for simulating a quantum many-body system in a stationary state. 1934 – Delaunay triangulation developed by Boris Delaunay 1936 – Turing machine, an abstract machine developed by Alan Turing, with others developed the modern notion of algorithm. == 1940s == 1942 – A fast Fourier transform algorithm developed by G.C. Danielson and Cornelius Lanczos 1945 – Merge sort developed by John von Neumann 1947 – Simplex algorithm developed by George Dantzig == 1950s == 1950 – Hamming codes developed by Richard Hamming 1952 – Huffman coding developed by David A. Huffman 1953 – Simulated annealing introduced by Nicholas Metropolis 1954 – Radix sort computer algorithm developed by Harold H. Seward 1964 – Box–Muller transform for fast generation of normally distributed numbers published by George Edward Pelham Box and Mervin Edgar Muller. Independently pre-discovered by Raymond E. A. C. Paley and Norbert Wiener in 1934. 1956 – Kruskal's algorithm developed by Joseph Kruskal 1956 – Ford–Fulkerson algorithm developed and published by R. Ford Jr. and D. R. Fulkerson 1957 – Prim's algorithm developed by Robert Prim 1957 – Bellman–Ford algorithm developed by Richard E. Bellman and L. R. Ford, Jr. 1959 – Dijkstra's algorithm developed by Edsger Dijkstra 1959 – Shell sort developed by Donald L. Shell 1959 – De Casteljau's algorithm developed by Paul de Casteljau 1959 – QR factorization algorithm developed independently by John G.F. Francis and Vera Kublanovskaya 1959 – Rabin–Scott powerset construction for converting NFA into DFA published by Michael O. Rabin and Dana Scott == 1960s == 1960 – Karatsuba multiplication 1961 – CRC (Cyclic redundancy check) invented by W. Wesley Peterson 1962 – AVL trees 1962 – Quicksort developed by C. A. R. Hoare 1962 – Bresenham's line algorithm developed by Jack E. Bresenham 1962 – Gale–Shapley 'stable-marriage' algorithm developed by David Gale and Lloyd Shapley 1964 – Heapsort developed by J. W. J. Williams 1964 – multigrid methods first proposed by R. P. Fedorenko 1965 – Cooley–Tukey algorithm rediscovered by James Cooley and John Tukey 1965 – Levenshtein distance developed by Vladimir Levenshtein 1965 – Cocke–Younger–Kasami (CYK) algorithm independently developed by Tadao Kasami 1965 – Buchberger's algorithm for computing Gröbner bases developed by Bruno Buchberger 1965 – LR parsers invented by Donald Knuth 1966 – Dantzig algorithm for shortest path in a graph with negative edges 1967 – Viterbi algorithm proposed by Andrew Viterbi 1967 – Cocke–Younger–Kasami (CYK) algorithm independently developed by Daniel H. Younger 1968 – A graph search algorithm described by Peter Hart, Nils Nilsson, and Bertram Raphael 1968 – Risch algorithm for indefinite integration developed by Robert Henry Risch 1969 – Strassen algorithm for matrix multiplication developed by Volker Strassen == 1970s == 1970 – Dinic's algorithm for computing maximum flow in a flow network by Yefim (Chaim) A. Dinitz 1970 – Knuth–Bendix completion algorithm developed by Donald Knuth and Peter B. Bendix 1970 – BFGS method of the quasi-Newton class 1970 – Needleman–Wunsch algorithm published by Saul B. Needleman and Christian D. Wunsch 1972 – Edmonds–Karp algorithm published by Jack Edmonds and Richard Karp, essentially identical to Dinic's algorithm from 1970 1972 – Graham scan developed by Ronald Graham 1972 – Red–black trees and B-trees discovered 1973 – RSA encryption algorithm discovered by Clifford Cocks 1973 – Jarvis march algorithm developed by R. A. Jarvis 1973 – Hopcroft–Karp algorithm developed by John Hopcroft and Richard Karp 1974 – Pollard's p − 1 algorithm developed by John Pollard 1974 – Quadtree developed by Raphael Finkel and J.L. Bentley 1975 – Genetic algorithms popularized by John Holland 1975 – Pollard's rho algorithm developed by John Pollard 1975 – Aho–Corasick string matching algorithm developed by Alfred V. Aho and Margaret J. Corasick 1975 – Cylindrical algebraic decomposition developed by George E. Collins 1976 – Salamin–Brent algorithm independently discovered by Eugene Salamin and Richard Brent 1976 – Knuth–Morris–Pratt algorithm developed by Donald Knuth and Vaughan Pratt and independently by J. H. Morris 1977 – Boyer–Moore string-search algorithm for searching the occurrence of a string into another string. 1977 – RSA encryption algorithm rediscovered by Ron Rivest, Adi Shamir, and Len Adleman 1977 – LZ77 algorithm developed by Abraham Lempel and Jacob Ziv 1977 – multigrid methods developed independently by Achi Brandt and Wolfgang Hackbusch 1978 – LZ78 algorithm developed from LZ77 by Abraham Lempel and Jacob Ziv 1978 – Bruun's algorithm proposed for powers of two by Georg Bruun 1979 – Khachiyan's ellipsoid method developed by Leonid Khachiyan 1979 – ID3 decision tree algorithm developed by Ross Quinlan == 1980s == 1980 – Brent's Algorithm for cycle detection Richard P. Brendt 1981 – Quadratic sieve developed by Carl Pomerance 1981 – Smith–Waterman algorithm developed by Temple F. Smith and Michael S. Waterman 1983 – Simulated annealing developed by S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi 1983 – Classification and regression tree (CART) algorithm developed by Leo Breiman, et al. 1984 – LZW algorithm developed from LZ78 by Terry Welch 1984 – Karmarkar's interior-point algorithm developed by Narendra Karmarkar 1984 – ACORN PRNG discovered by Roy Wikramaratna and used privately 1985 – Simulated annealing independently developed by V. Cerny 1985 – Car–Parrinello molecular dynamics developed by Roberto Car and Michele Parrinello 1985 – Splay trees discovered by Sleator and Tarjan 1986 – Blum Blum Shub proposed by L. Blum, M. Blum, and M. Shub 1986 – Push relabel maximum flow algorithm by Andrew Goldberg and Robert Tarjan 1986 – Barnes–Hut tree method developed by Josh Barnes and Piet Hut for fast approximate simulation of n-body problems 1987 – Fast multipole method developed by Leslie Greengard and Vladimir
Read more →
Global serializability

In concurrency control of databases, transaction processing (transaction management), and other transactional distributed applications, global serializability (or modular serializability) is a property of a global schedule of transactions. A global schedule is the unified schedule of all the individual database (and other transactional object) schedules in a multidatabase environment (e.g., federated database). Complying with global serializability means that the global schedule is serializable, has the serializability property, while each component database (module) has a serializable schedule as well. In other words, a collection of serializable components provides overall system serializability, which is usually incorrect. A need in correctness across databases in multidatabase systems makes global serializability a major goal for global concurrency control (or modular concurrency control). With the proliferation of the Internet, Cloud computing, Grid computing, and small, portable, powerful computing devices (e.g., smartphones), as well as increase in systems management sophistication, the need for atomic distributed transactions and thus effective global serializability techniques, to ensure correctness in and among distributed transactional applications, seems to increase. In a federated database system or any other more loosely defined multidatabase system, which are typically distributed in a communication network, transactions span multiple (and possibly distributed) databases. Enforcing global serializability in such system, where different databases may use different types of concurrency control, is problematic. Even if every local schedule of a single database is serializable, the global schedule of a whole system is not necessarily serializable. The massive communication exchanges of conflict information needed between databases to reach conflict serializability globally would lead to unacceptable performance, primarily due to computer and communication latency. Achieving global serializability effectively over different types of concurrency control has been open for several years. == The global serializability problem == === Problem statement === The difficulties described above translate into the following problem: Find an efficient (high-performance and fault tolerant) method to enforce Global serializability (global conflict serializability) in a heterogeneous distributed environment of multiple autonomous database systems. The database systems may employ different concurrency control methods. No limitation should be imposed on the operations of either local transactions (confined to a single database system) or global transactions (span two or more database systems). === Quotations === Lack of an appropriate solution for the global serializability problem has driven researchers to look for alternatives to serializability as a correctness criterion in a multidatabase environment (e.g., see Relaxing global serializability below), and the problem has been characterized as difficult and open. The following two quotations demonstrate the mindset about it by the end of the year 1991, with similar quotations in numerous other articles: "Without knowledge about local as well as global transactions, it is highly unlikely that efficient global concurrency control can be provided... Additional complications occur when different component DBMSs [Database Management Systems] and the FDBMSs [Federated Database Management Systems] support different concurrency mechanisms... It is unlikely that a theoretically elegant solution that provides conflict serializability without sacrificing performance (i.e., concurrency and/or response time) and availability exists." === Proposed solutions === Several solutions, some partial, have been proposed for the global serializability problem. Among them: Global conflict graph (serializability graph, precedence graph) checking Distributed Two-phase locking (Distributed 2PL) Distributed Timestamp ordering Tickets (local logical timestamps which define local total orders, and are propagated to determine global partial order of transactions) == Relaxing global serializability == Some techniques have been developed for relaxed global serializability (i.e., they do not guarantee global serializability; see also Relaxing serializability). Among them (with several publications each): Quasi serializability Two-level serializability Another common reason nowadays for Global serializability relaxation is the requirement of availability of internet products and services. This requirement is typically answered by large scale data replication. The straightforward solution for synchronizing replicas' updates of a same database object is including all these updates in a single atomic distributed transaction. However, with many replicas such a transaction is very large, and may span several computers and networks that some of them are likely to be unavailable. Thus such a transaction is likely to end with abort and miss its purpose. Consequently, Optimistic replication (Lazy replication) is often utilized (e.g., in many products and services by Google, Amazon, Yahoo, and alike), while global serializability is relaxed and compromised for eventual consistency. In this case relaxation is done only for applications that are not expected to be harmed by it. Classes of schedules defined by relaxed global serializability properties either contain the global serializability class, or are incomparable with it. What differentiates techniques for relaxed global conflict serializability (RGCSR) properties from those of relaxed conflict serializability (RCSR) properties that are not RGCSR is typically the different way global cycles (span two or more databases) in the global conflict graph are handled. No distinction between global and local cycles exists for RCSR properties that are not RGCSR. RCSR contains RGCSR. Typically RGCSR techniques eliminate local cycles, i.e., provide local serializability (which can be achieved effectively by regular, known concurrency control methods); however, obviously they do not eliminate all global cycles (which would achieve global serializability).
Read more →
Ontology for Biomedical Investigations

The Ontology for Biomedical Investigations (OBI) is an open-access, integrated ontology for the description of biological and clinical investigations. OBI provides a model for the design of an investigation, the protocols and instrumentation used, the materials used, the data generated and the type of analysis performed on it. The project is being developed as part of the OBO Foundry and as such adheres to all the principles therein such as orthogonal coverage (i.e. clear delineation from other foundry member ontologies) and the use of a common formal language. In OBI the common formal language used is the Web Ontology Language (OWL). As of March 2008, a pre-release version of the ontology was made available at the project's SVN repository. == Scope == The Ontology for Biomedical Investigations (OBI) addresses the need for controlled vocabularies to support integration and joint ("cross-omics") analysis of experimental data, a need originally identified in the transcriptomics domain by the FGED Society, which developed the MGED Ontology as an annotation resource for microarray data.Smith B, Ashburner M, Rosse C, Bard J, Bug W, Ceusters W, et al. (November 2007). "The OBO Foundry: coordinated evolution of ontologies to support biomedical data integration". Nature Biotechnology. 25 (11): 1251–5. doi:10.1038/nbt1346. PMC 2814061. PMID 17989687. OBI uses the basic formal ontology upper-level ontology as a means of describing general entities that do not belong to a specific problem domain. As such, all OBI classes are a subclass of some BFO class. The ontology has the scope of modeling all biomedical investigations and as such contains ontology terms for aspects such as: biological material – for example blood plasma instrument (and parts of an instrument therein) – for example DNA microarray, centrifuge information content – such as an image or a digital information entity such as an electronic medical record design and execution of an investigation (and individual experiments therein) – for example study design, electrophoresis material separation data transformation (incorporating aspects such as data normalization and data analysis) – for example principal components analysis dimensionality reduction, mean calculation Less 'concrete' aspects such as the role a given entity may play in a particular scenario (for example the role of a chemical compound in an experiment) and the function of an entity (for example the digestive function of the stomach to nutriate the body) are also covered in the ontology. == OBI consortium == The MGED Ontology was originally identified in the transcriptomics domain by the FGED Society and was developed to address the needs of data integration. Following a mutual decision to collaborate, this effort later became a wider collaboration between groups such as FGED, PSI and MSI in response to the needs of areas such as transcriptomics, proteomics and metabolomics and the FuGO (Functional Genomics Investigation Ontology) was created. This later became the OBI covering the wider scope of all biomedical investigations. As an international, cross-domain initiative, the OBI consortium draws upon a pool of experts from a variety of fields, not limited to biology. The current list of OBI consortium members is available at the OBI consortium website. The consortium is made up of a coordinating committee which is a combination of two subgroups, the Community Representative (those representing a particular biomedical community) and the Core Developers (ontology developers who may or may not be members of any single community). Separate to the coordinating committee is the Developers Working Group which consists of developers within the communities collaborating in the development of OBI at the discretion of current OBI Consortium members. == Papers on OBI ==
Read more →
MarkLogic Server

MarkLogic Server is a document-oriented database developed by MarkLogic. It is a NoSQL multi-model database that evolved from an XML database to natively store JSON documents and RDF triples, the data model for semantics. MarkLogic is designed to be a data hub for operational and analytical data. == History == MarkLogic Server was built to address shortcomings with existing search and data products. The product first focused on using XML as the document markup standard and XQuery as the query standard for accessing collections of documents up to hundreds of terabytes in size. Currently the MarkLogic platform is widely used in publishing, government, finance and other sectors. MarkLogic's customers are mostly Global 2000 companies. == Technology == MarkLogic uses documents without upfront schemas to maintain a flexible data model. In addition to having a flexible data model, MarkLogic uses a distributed, scale-out architecture that can handle hundreds of billions of documents and hundreds of terabytes of data. It has received Common Criteria certification, and has high availability and disaster recovery. MarkLogic is designed to run on-premises and within public or private cloud environments like Amazon Web Services. == Features == Indexing MarkLogic indexes the content and structure of documents including words, phrases, relationships, and values in over 200 languages with tokenization, collation, and stemming for core languages. Functionality includes the ability to toggle range indexes, geospatial indexes, the RDF triple index, and reverse indexes on or off based on your data, the kinds of queries that you will run, and your desired performance. Full-text search MarkLogic supports search across its data and metadata using a word or phrase and incorporates Boolean logic, stemming, wildcards, case sensitivity, punctuation sensitivity, diacritic sensitivity, and search term weighting. Data can be searched using JavaScript, XQuery, SPARQL, and SQL. Semantics MarkLogic uses RDF triples to provide semantics for ease of storing metadata and querying. ACID Unlike other NoSQL databases, MarkLogic maintains ACID consistency for transactions. Replication MarkLogic provides high availability with replica sets. Scalability MarkLogic scales horizontally using sharding. MarkLogic can run over multiple servers, balancing the load or replicating data to keep the system up and running in the event of hardware failure. Security MarkLogic has built in security features such as element-level permissions and data redaction. Optic API for Relational Operations An API that lets developers view their data as documents, graphs or rows. Security MarkLogic provides redaction, encryption, and element-level security (allowing for control on read and write rights on parts of a document). == Applications == Banking Big Data Fraud prevention Insurance Claims Management and Underwriting Master data management Recommendation engines == Licensing == MarkLogic is available under various licensing and delivery models, namely a free Developer or an Essential Enterprise license.[3] Licenses are available from MarkLogic or directly from cloud marketplaces such as Amazon Web Services and Microsoft Azure. == Releases == 2001 – Cerisent XQE 1: ACID transactions, Full-text search, XML Storage, XQuery, Role-based security 2004 – Cerisent XQE 2: Scale-out architecture, Enhanced search (stemming, thesaurus, wildcard), Backup and restore 2005 – MarkLogic Server 3: Continuing search improvements, Content Processing Framework (including PDF, Word, Excel, PPT), Failover 2008 – MarkLogic Server 4: Geospatial search, entity extraction, advanced XQuery, performance, scalability enhancements, auditing 2011 – MarkLogic Server 5: Flexible replication / DDIL, real-time indexing, advanced search, improved analytics, concurrency enhancements 2012 – MarkLogic Server 6: REST and Java APIs, App Builder, enhanced UI, improved search 2013 – MarkLogic Server 7: Semantic graph, bitemporal data, tiered storage, improved search, better management 2015 – MarkLogic Server 8: A Native JSON storage, Server-side JavaScript, Bitemporal, Node.js client API, Incremental backup, Flexible replication[16] 2017 – MarkLogic Server 9: Data integration across Relational and Non-Relational data, Advanced Encryption, Element Level Security, Redaction 2019 – MarkLogic Server 10: Enhanced Data Hub, improved SQL, security, analytics performance, cloud support 2022 – MarkLogic Server 11: MarkLogic Ops Director (Monitoring and Administration Improvements), expanded PKI 2025 – MarkLogic Server 12: Generative AI and Native Vector Search, Graph Algorithm Support, Virtual TDEs (relational views on the fly)
Read more →
Informationist

An informationist (or information specialist in context) provides research and knowledge management services in the context of clinical care or biomedical research. Although there is no one educational pathway or formalized set of skills or knowledge for informationists, one way to think of the informationist is as one who possesses the knowledge and skill of a medical librarian with extensive research specialization and some formal clinical or public health education that goes beyond on-the-job osmosis. Medical librarians and other biomedical professional organizations have been exploring the possibilities for evaluating how informationists are being used and whether their activities supplement or replace medical library activity. More generally, an informationist is a professional who works with information within a particular business, analytic or scientific context to drive toward outcomes based on evidence, analysis, prediction and execution. For example, an extension of the term is increasingly emerging in financial services, life sciences and health care industries. Though still nascently in use, its adoption applies to individuals with extensive industry expertise, acute familiarity with organizational structures and processes, deep domain level information mastery and information systems technical savvy. Informationists in this context support transformational initiatives within and across functional areas of an enterprise as architects, governance experts, continuous improvement advocates and strategists. == Background == The term was proposed in 2000 by Davidoff & Florance. Their editorial suggested that physicians should be delegating their information needs to informationists, just as they currently order CT scans from radiologists or cardiac catheterizations from cardiologists. They conceived of an information professional who was embedded in (and indeed, supported by) the clinical departments. Supporters of the concept see it as a means for librarians to reinvigorate connections with the faculty/clinicians, as well as provide superior service by dint of informationists' biomedical training. Critics complained that the idea is nothing new; librarians already provide in-depth, high quality information services and clinical medical librarians have been working alongside physicians, nurses and other clinicians for years. Large informationist programs in the U.S. exist at the National Institutes of Health and at Vanderbilt University. Welch Medical Library at Johns Hopkins University (JHU) is developing an informationist service model in which its 10 clinical and public health librarians are moving from serving as liaison librarians for assigned departments toward becoming embedded informationists within their departments. To prepare for the embedded informationist role, librarians are undertaking education as needed to supplement their backgrounds. For example, librarians bring experience in clinical behavior counseling, public health, nursing, and more. Informationist training can then focus upon filling gaps in research methods knowledge more so than on gaining additional knowledge in the librarian's area of expertise. Courses, seminars and workshops being undertaken include those covering systematic reviews, evidence-based medicine, critical appraisal, medical language, anatomy and physiology, biostatistics, and clinical research. The term informationist is related to that of informatician—also informaticist—and many informationists do possess skills in clinical topics, bioinformatics, and biomedical informatics. Harvard University, the University of Pittsburgh, and Washington University in St. Louis are examples of institutional libraries which have hired PhD-level scientists (who may or may not have library degrees) to provide informatics support for biomedical research.
Read more →
Ideonomy

Ideonomy is a combinatorial "science of ideas" developed by American independent scholar Patrick M. Gunkel (1947–2017). Specifically, Ideonomy is concerned with the systematic organization of ideas and the discovery of the rules behind how ideas combine, diverge, and transform. Gunkel defined ideonomy as "the science of the laws of ideas and of the application of such laws to the generation of all possible ideas in connection with any subject, idea, or thing." In his 1992 book A History of Knowledge, Charles Van Doren compared ideonomy to a "mining operation" that excavates meanings and thought to discover treasures hidden deep within language. Sources from the 1980s and 1990s demonstrate that ideonomy was useful to academic researchers in fields including biology, toxicology, and nursing/patient care. Beginning in the 2010s, academics in a wide range of fields including machine learning, marketing, computational modeling, and cybersecurity have relied on materials generated for ideonomy to provide methodological support for their research. == Etymology and definition == The word "ideonomy" combines the Greek roots ideo- (from idea, meaning pattern or form) and -nomy (from nomos, meaning law or custom). The suffix -nomy suggests the laws concerning or the totality of knowledge about a given subject, as in astronomy or taxonomy. In a note posted on the MIT ideonomy website, Gunkel states that the word was supposedly first coined by the French Encyclopedists to refer to a science of ideas. No evidence is provided for this statement, however. The concept bears some relationship to Antoine Destutt de Tracy's "ideology" (1796), which originally meant a systematic science of ideas before acquiring its modern political connotations. Gunkel provided several metaphorical descriptions of ideonomy: An "idea bank": a computer network enabling systematic exploration of infinite possible ideas A "kaleidoscope" that can exhibit all possible combinations and transformations of ideas A "prism" capable of diffracting any idea into its cognitive components A "gigantic microscope for magnifying the ideocosm" == History and development == In 1984, Gunkel received a five-year unsolicited grant from the Richard Lounsbery Foundation of New York to develop ideonomy. A June 1, 1987 article on the front page of The Wall Street Journal brought Gunkel and ideonomy to wider public attention. Some academics were interested in using ideonomy's techniques, including biologist Betsey Dyer, who published several contemporaneous peer-reviewed studies citing ideonomy. Academic researchers in the field of toxicology and nursing/patient care also used ideonomy. However, ideonomy's broadest contribution to date came beginning in the 2010s, as a list of personality traits generated for combinatorial matching was used by researchers in artificial intelligence to code human emotions for machine-learning tasks, develop computational models related to personality, develop a measurement framework for influencer-brand recommender systems, and aid information awareness/cybersecurity assessment. == Methodology == The foundational empirical method of ideonomy involves the systematic creation of extensive lists. Gunkel's apartment reportedly contained thousands of lists on every conceivable topic. Gunkel termed each list an "organon," which he described as expanding through "combination, permutation, transformation, generalization, specialization, intersection, interaction, reapplication, recursive use, etc. of existing organons." The ideonomic process follows a progressive structure. The ideonomist begins with a simple list of examples of a particular idea, concept, or thing. The list need not be exhaustive. By studying this list, the ideonomist isolates and identifies types. This categorical analysis then reveals missing items, allowing the primary list to be improved and refined. Gunkel emphasized that list items must not only cover genuine categories of nature but also be formulated in ways that yield the largest possible number of syntactically coherent possibilities when combined. The core technique of ideonomy is "ideocombinatorics"—the systematic intersection and combination of items from different lists to generate novel composite concepts. Gunkel developed computer programs to automate this process. For example, combining a list of 230 Universal Elementary Shapes (pits, pyramids, trenches, hemispheres, needles) with a list of 74 Types of Order (recurrence, identity, likeness of parts) yields 17,020 possible "shapes of order." These combinations, when phrased as questions ("Can there be pits of recurrence?"), could suggest new categories of phenomena worthy of investigation. The computer-generated output is typically repetitive and often meaningless. However, with sufficient frequency, the combinations yield results that are unexpectedly interesting and fruitful. In one documented case, Gunkel's programs generated 45,540 questions about toxins for microbiologist David Bermudes. One question—"Can hierarchies of cell process be used as a basis for classifying toxic action?"—prompted Bermudes to develop a novel approach to classifying biological toxins by the type of molecule they attack, rather than by chemical structure or physiological system affected. According to one contemporaneous account of ideonomy, "Gunkel takes for his field all fields and all ideas about anything. He uses a computer to generate lists of words and phrases and by juxtaposition reviews the resultant patterns for novel ideas. The computer is ideal for this task because the mind would rebel at the formidable processing task ideonomy involves. What we have here is computer generated originality." == Applications == Gunkel and his supporters identified several practical applications for ideonomic methods: Scientific research: Biologist Betsey Dyer of Wheaton College published research crediting ideonomy for helping to generate ideas. Medical science: When Austin pathologist Michael T. O'Brien was presented with the ideonomically-generated question "Can arteries have rashes?", he initially dismissed it as nonsense. Upon reflection, he realized that large arteries are supplied with blood by tiny vessels that might become inflamed and dilated, analogous to skin vessels in a rash—a phenomenon potentially worth researching. Analogical thinking: Harvard law professor Robert Clark used ideonomic analogies to write a research paper comparing plant structure with human hierarchies. Artificial intelligence: Douglas Lenat, a researcher at Microelectronics and Computer Technology Corporation (MCC) in Austin, suggested that Gunkel's lists enumerating types of human mistakes could help design AI systems capable of recognizing and correcting their own errors. == Reception and criticism == Ideonomy received mixed reactions from the academic and scientific communities. Prominent supporters included: Edward Fredkin, former director of MIT's computer science laboratory, who praised Gunkel's "provocative ideas on artificial intelligence." Marvin Minsky, AI scientist and MIT professor, who described ideonomy as "perhaps the most extensive study of ways to generate ideas." Frederick Seitz, president emeritus of Rockefeller University, who noted Gunkel's "encyclopedic scope" Robert C. Clark, Harvard law professor, who called Gunkel "the most intelligent person I ever met" However, skeptics questioned whether ideonomy constituted a genuine science. Fredkin himself noted that Gunkel "pours out about 60 ideas a minute, and 59 of them are bad," though he added that "even with one good idea out of 60, it's still an amazing accomplishment." Douglas Lenat observed that brainstorming with Gunkel was "a bit like being hit over the head by the muse with a sledgehammer" and that "he puts people off." Gunkel himself acknowledged that ideonomy was in its infancy and might seem "absurdly utopian." His planned magnum opus on ideonomy remained incomplete, and was posted on an MIT website thanks to faculty advisor Whitman Richards. Gunkel wrote: "Pioneering in a completely new field, yes in a new science, is almost unreal. It is heartbreaking, it is pitiable, it is almost inhuman. Honestly, it is a hell. There is nothing heroic about it." == Related concepts == Gunkel identified several historical precedents for ideonomic thinking: Gottfried Wilhelm Leibniz (1646–1716): The philosopher's work on a universal characteristic (characteristica universalis) and calculus of reasoning Peter Mark Roget (1779–1869): Creator of Roget's Thesaurus, which organized concepts into a systematic taxonomy Dmitri Mendeleev (1834–1907): Developer of the periodic table, demonstrating how combining lists of element families could reveal previously unseen connections Fritz Zwicky (1898–1974): The Caltech astrophysicist whom Gunkel called the "grandfather of ideonomy" for his development of "morphological research"—systematic exploration of all possible solutions t
Read more →
Ontology merging

Ontology merging defines the act of bringing together two conceptually divergent ontologies or the instance data associated to two ontologies. This is similar to work in database merging (schema matching). This merging process can be performed in a number of ways, manually, semi automatically, or automatically. Manual ontology merging although ideal is extremely labour-intensive and current research attempts to find semi or entirely automated techniques to merge ontologies. These techniques are statistically driven often taking into account similarity of concepts and raw similarity of instances through textual string metrics and semantic knowledge. These techniques are similar to those used in information integration employing string metrics from open source similarity libraries.
Read more →
Collaborative diffusion

Collaborative Diffusion is a type of pathfinding algorithm which uses the concept of antiobjects, objects within a computer program that function opposite to what would be conventionally expected. Collaborative Diffusion is typically used in video games, when multiple agents must path towards a single target agent. For example, the ghosts in Pac-Man. In this case, the background tiles serve as antiobjects, carrying out the necessary calculations for creating a path and having the foreground objects react accordingly, whereas having foreground objects be responsible for their own pathing would be conventionally expected. Collaborative Diffusion is favored for its efficiency over other pathfinding algorithms, such as A, when handling multiple agents. Also, this method allows elements of competition and teamwork to easily be incorporated between tracking agents. Notably, the time taken to calculate paths remains constant as the number of agents increases.
Read more →
Semantic translation

Semantic translation is the process of using semantic information to aid in the translation of data in one representation or data model to another representation or data model. Semantic translation takes advantage of semantics that associate meaning with individual data elements in one dictionary to create an equivalent meaning in a second system. An example of semantic translation is the conversion of XML data from one data model to a second data model using formal ontologies for each system such as the Web Ontology Language (OWL). This is frequently required by intelligent agents that wish to perform searches on remote computer systems that use different data models to store their data elements. The process of allowing a single user to search multiple systems with a single search request is also known as federated search. Semantic translation should be differentiated from data mapping tools that do simple one-to-one translation of data from one system to another without actually associating meaning with each data element. Semantic translation requires that data elements in the source and destination systems have "semantic mappings" to a central registry or registries of data elements. The simplest mapping is of course where there is equivalence. There are three types of Semantic equivalence: Class Equivalence - indicating that class or "concepts" are equivalent. For example: "Person" is the same as "Individual" Property Equivalence - indicating that two properties are equivalent. For example: "PersonGivenName" is the same as "FirstName" Instance Equivalence - indicating that two individual instances of objects are equivalent. For example: "Dan Smith" is the same person as "Daniel Smith" Semantic translation is very difficult if the terms in a particular data model do not have direct one-to-one mappings to data elements in a foreign data model. In that situation, an alternative approach must be used to find mappings from the original data to the foreign data elements. This problem can be alleviated by centralized metadata registries that use the ISO-11179 standards such as the National Information Exchange Model (NIEM).
Read more →
Xara Designer Pro+

Xara Designer Pro+ is an image editing program incorporating photo editing and vector illustration tools created by British software company Xara. Xara Xtreme LX was an early open source version for Linux. The Windows version was previously sold under the names Xara Studio, Xara X and Xara Xtreme, and traces its origin in the late 1980s to a title called ArtWorks for the Acorn Archimedes line of computers using RISC OS. There is a pro version called Xara Designer Pro (formerly Xara Xtreme Pro). The current commercial version of Xara Photo & Graphic Designer runs only on Windows, although Xara documents can be edited in a web browser on any platform using the Xara Cloud service. Versions up to 4.x can be run on Linux using Wine. == History == ArtWorks, the predecessor of Xara Photo and Graphic Designer, was developed on Acorn Archimedes and Risc PC 32-bit RISC computers running RISC OS by Computer Concepts during the late 1980s. The first version, developed for Microsoft Windows was initially called Xara Studio. It was licensed to Corel Corporation before wide-scale public availability, and from 1995 to 2000 was released as CorelXARA. Corel ceded the licensing rights back to Xara in 2000. The first Xara X version released in 2000 by its original owner. The next version, Xara X¹, was released in 2004. Xara Xtreme was released in 2005. In November 2006, Xara Xtreme PRO (an enhanced version of Xara Xtreme) was released. Xara Xtreme 3.2 and Xtreme Pro 3.2 were released in May 2007. 3.2 Pro included Xara3D, and both versions had more robust typography. In April 2008, Xara Xtreme 4.0 was released. Xara Xtreme and Xara Xtreme Pro 5.1 were released in June 2009. Features included more text-area enhancements, content-aware scaling of bitmap images, improved file import and export, master-page (repeated) objects, an object gallery (replacing the layer gallery), website-creation tools, and multi-stage graduated transparency. In June 2010, Xara Photo & Graphic Designer 6 and Xara Designer Pro 6 were released. Xtreme was renamed Photo & Graphic Designer, and Xtreme Pro was renamed Designer Pro. In May 2011, Xara Photo & Graphic Designer 7 and Xara Designer Pro 7 were released. Features included "magic" photo erase, user interface improvements to docking galleries and snapping alignment, and (in Pro) new webpage and website-design features. In May 2012, Xara Photo & Graphic Designer 2013 and Xara Designer Pro X (v8) were released. Xara Photo & Graphic Designer 9 was released in May 2013. In July of that year, Xara Designer Pro X9 was released. Xara Photo & Graphic Designer 10 was released on 16 July 2014, and Xara Designer Pro X10 on 23 July. Xara Photo & Graphic Designer 11 was released on 29 June 2015, and Xara Designer Pro X11 was released the following month. In 2016, the delivery model was changed to an update service which can be renewed annually. Users are entitled to any updates released while the update service is active. The first update-service updates were in May 2016 for Xara Photo & Graphic Designer, and July 2016 for Xara Designer Pro X. == Features == Xara Photo & Graphic Designer is known for its usability and fast renderer. It provides a fully anti-aliased display, advanced gradient fill, and transparency tools. Among vector editors, Xara Photo & Graphic Designer is considered to be fairly easy to learn, with similarities to CorelDRAW and Inkscape in terms of interface. Alongside the vector illustration tools, Xara Photo & Graphic Designer also includes an integrated photo tool offering manual and automatic photo enhance, cropping, adjustment of brightness levels, red-eye fix, 'magic' erase, photo healing, color and background erase, panoramas and content aware resizing. Designer Pro includes a wider range of tools for other design tasks including the creation of web pages and websites, and text and page layout tools for DTP with the aim of providing a single solution for all graphic and web design tasks.
Read more →
OpenWSN

OpenWSN aims to build an open standard-based and open source implementation of a complete constrained network protocol stack for wireless sensor networks and Internet of Things. The project was created at the University of California Berkeley and extended at the INRIA and at the Open University of Catalonia (UOC). The root of OpenWSN is a deterministic MAC layer implementing the IEEE 802.15.4e TSCH based on the concept of Time Slotted Channel Hopping (TSCH). Above the MAC layer, the Low Power Lossy Network stack is based on IETF standards including the IETF 6TiSCH management and adaptation layer (a minimal configuration profile, 6top protocol and different scheduling functions). The stack is complemented by an implementation of 6LoWPAN, RPL in non-storing mode, UDP and CoAP, enabling access to devices running the stack from the native IPv6 through open standards. OpenWSN is related to other projects including the following: RIOT OpenMote OpenWSN is available for Linux, Windows and OS X platforms. Current release of OpenWSN is 1.14.0.
Read more →
Bartels–Stewart algorithm

In numerical linear algebra, the Bartels–Stewart algorithm is used to numerically solve the Sylvester matrix equation A X − X B = C {\displaystyle AX-XB=C} . Developed by R.H. Bartels and G.W. Stewart in 1971, it was the first numerically stable method that could be systematically applied to solve such equations. The algorithm works by using the real Schur decompositions of A {\displaystyle A} and B {\displaystyle B} to transform A X − X B = C {\displaystyle AX-XB=C} into a triangular system that can then be solved using forward or backward substitution. In 1979, G. Golub, C. Van Loan and S. Nash introduced an improved version of the algorithm, known as the Hessenberg–Schur algorithm. It remains a standard approach for solving Sylvester equations when X {\displaystyle X} is of small to moderate size. == The algorithm == Let X , C ∈ R m × n {\displaystyle X,C\in \mathbb {R} ^{m\times n}} , and assume that the eigenvalues of A {\displaystyle A} are distinct from the eigenvalues of B {\displaystyle B} . Then, the matrix equation A X − X B = C {\displaystyle AX-XB=C} has a unique solution. The Bartels–Stewart algorithm computes X {\displaystyle X} by applying the following steps: 1.Compute the real Schur decompositions R = U T A U , {\displaystyle R=U^{T}AU,} S = V T B T V . {\displaystyle S=V^{T}B^{T}V.} The matrices R {\displaystyle R} and S {\displaystyle S} are block-upper triangular matrices, with diagonal blocks of size 1 × 1 {\displaystyle 1\times 1} or 2 × 2 {\displaystyle 2\times 2} . 2. Set F = U T C V . {\displaystyle F=U^{T}CV.} 3. Solve the simplified system R Y − Y S T = F {\displaystyle RY-YS^{T}=F} , where Y = U T X V {\displaystyle Y=U^{T}XV} . This can be done using forward substitution on the blocks. Specifically, if s k − 1 , k = 0 {\displaystyle s_{k-1,k}=0} , then ( R − s k k I ) y k = f k + ∑ j = k + 1 n s k j y j , {\displaystyle (R-s_{kk}I)y_{k}=f_{k}+\sum _{j=k+1}^{n}s_{kj}y_{j},} where y k {\displaystyle y_{k}} is the k {\displaystyle k} th column of Y {\displaystyle Y} . When s k − 1 , k ≠ 0 {\displaystyle s_{k-1,k}\neq 0} , columns [ y k − 1 ∣ y k ] {\displaystyle [y_{k-1}\mid y_{k}]} should be concatenated and solved for simultaneously. 4. Set X = U Y V T . {\displaystyle X=UYV^{T}.} === Computational cost === Using the QR algorithm, the real Schur decompositions in step 1 require approximately 10 ( m 3 + n 3 ) {\displaystyle 10(m^{3}+n^{3})} flops, so that the overall computational cost is 10 ( m 3 + n 3 ) + 2.5 ( m n 2 + n m 2 ) {\displaystyle 10(m^{3}+n^{3})+2.5(mn^{2}+nm^{2})} . === Simplifications and special cases === In the special case where B = − A T {\displaystyle B=-A^{T}} and C {\displaystyle C} is symmetric, the solution X {\displaystyle X} will also be symmetric. This symmetry can be exploited so that Y {\displaystyle Y} is found more efficiently in step 3 of the algorithm. == The Hessenberg–Schur algorithm == The Hessenberg–Schur algorithm replaces the decomposition R = U T A U {\displaystyle R=U^{T}AU} in step 1 with the decomposition H = Q T A Q {\displaystyle H=Q^{T}AQ} , where H {\displaystyle H} is an upper-Hessenberg matrix. This leads to a system of the form H Y − Y S T = F {\displaystyle HY-YS^{T}=F} that can be solved using forward substitution. The advantage of this approach is that H = Q T A Q {\displaystyle H=Q^{T}AQ} can be found using Householder reflections at a cost of ( 5 / 3 ) m 3 {\displaystyle (5/3)m^{3}} flops, compared to the 10 m 3 {\displaystyle 10m^{3}} flops required to compute the real Schur decomposition of A {\displaystyle A} . == Software and implementation == The subroutines required for the Hessenberg-Schur variant of the Bartels–Stewart algorithm are implemented in the SLICOT library. These are used in the MATLAB control system toolbox. == Alternative approaches == For large systems, the O ( m 3 + n 3 ) {\displaystyle {\mathcal {O}}(m^{3}+n^{3})} cost of the Bartels–Stewart algorithm can be prohibitive. When A {\displaystyle A} and B {\displaystyle B} are sparse or structured, so that linear solves and matrix vector multiplies involving them are efficient, iterative algorithms can potentially perform better. These include projection-based methods, which use Krylov subspace iterations, methods based on the alternating direction implicit (ADI) iteration, and hybridizations that involve both projection and ADI. Iterative methods can also be used to directly construct low rank approximations to X {\displaystyle X} when solving A X − X B = C {\displaystyle AX-XB=C} .
Read more →