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} .
Racter
Racter is an artificial intelligence program that generates English language prose at random. It was published by Mindscape for IBM PC compatibles in 1984, then for the Apple II, Mac, and Amiga. An expanded version of the software, not the one released through Mindscape, was used to generate the text for the published book The Policeman's Beard Is Half Constructed. == History == Racter, short for raconteur, was written by William Chamberlain and Thomas Etter. Racter's initial creation was the short story Soft Ions, which appeared in the October 1981 issue of Omni (magazine). The publication's editors bought the story in January 1980, before it had even been written. In exchange for the rights, the editors offered financial support to Chamberlain and Etter so the two could refine Racter. In 1983, Racter produced a book called The Policeman's Beard Is Half Constructed (ISBN 0-446-38051-2). The program originally was written for an OSI which only supported file names at most six characters long, causing the name to be shorted to Racter and it was later adapted to run on a CP/M machine where it was written in "compiled ASIC on a Z80 microcomputer with 64K of RAM." This version, the program that allegedly wrote the book, was not released to the general public. The sophistication claimed for the program was likely exaggerated, as could be seen by investigation of the template system of text generation. In 1984, Mindscape released an interactive version of Racter, developed by Inrac Corporation, for IBM PC compatibles, and it was ported to the Apple II, Mac, and Amiga. The published Racter was similar to a chatterbot. The BASIC program that was released by Mindscape was far less sophisticated than anything that could have written the fairly sophisticated prose of The Policeman's Beard. The commercial version of Racter could be likened to a computerized version of Mad Libs, the game in which you fill in the blanks in advance and then plug them into a text template to produce a surrealistic tale. The commercial program attempted to parse text inputs, identifying significant nouns and verbs, which it would then regurgitate to create "conversations", plugging the input from the user into phrase templates which it then combined, along with modules that conjugated English verbs. By contrast, the text in The Policeman's Beard, apart from being edited from a large amount of output, would have been the product of Chamberlain's own specialized templates and modules, which were not included in the commercial release of the program. == Reception == The Boston Phoenix called the story Soft Ions "schematic nonsense. But the scheme is obvious enough and the nonsense accessible enough to an attentive reader that one can almost believe Chamberlain when he predicts that before long Racter will be ready to write for the pulp-reading public." PC Magazine described some of Policeman's Beard's scenes as "surprising for their frankness" and "reflective". It concluded that the book was "whimsical and wise and sometimes fun". Computer Gaming World described Racter as "a diversion into another dimension that might best be seen before paying the price of a ticket. (Try before you buy!)" A 1985 review of the program in The New York Times notes that, "As computers move ever closer to artificial intelligence, Racter is on the edge of artificial insanity." It also states that Racter's "always-changing sentences are grammatically correct, often funny and, for a computer, sometimes profound." The article includes examples showing interaction with Racter, most often Racter asking the user questions. == Reviews == Jeux & Stratégie #47
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SNNS
SNNS (Stuttgart Neural Network Simulator) is a neural network simulator originally developed at the University of Stuttgart. While it was originally built for X11 under Unix, there are Windows ports. Its successor JavaNNS never reached the same popularity. == Features == SNNS is written around a simulation kernel to which user written activation functions, learning procedures and output functions can be added. It has support for arbitrary network topologies and the standard release contains support for a number of standard neural network architectures and training algorithms. == Status == There is currently no ongoing active development of SNNS. In July 2008 the license was changed to the GNU LGPL.
Yi Zeng (AI researcher)
Yi Zeng (Chinese: 曾毅) is a Chinese artificial intelligence researcher and professor at the Chinese Academy of Sciences, who also serves as the founding director of Center for Long-term AI, and as a member of the United Nations Advisory Body on AI. == Career == On May 25, 2019, Zeng led the team that published the Beijing Artificial Intelligence Principles, proposed as an initiative for the long-term research, governance and planning of AI, and the "realization of beneficial AI for mankind and nature". He was named on the Time 100 AI list, a list featuring the hundred most influential figures in artificial intelligence of the year, in 2023. In July 2023, Zeng addressed the United Nations Security Council in a meeting on the risks posed by recent strides in artificial intelligence. He said that AI models “cannot be trusted as responsible agents that can help humans to make decisions,” and warned of the risk of extinction posed by both near-term and long-term AI, arguing that “in the long term, we haven’t given superintelligence any practical reasons why they should protect humans”. Zeng stated that humans should always be responsible for final decision-making on the use of nuclear weapons, and that the United Nations must produce an international framework on AI development and governance, to ensure global peace and security. In October 2023, UN Secretary-General António Guterres announced the creation of an advisory body on issues surrounding the international governance of AI, of which Zeng would be a member. He leads teams of researchers at the Institute of Philosophy and the Institute of Automation of the Chinese Academy of Sciences, including doctoral candidates, postdoctoral fellows, research fellows, assistant professors, and associate professors. Among them is his first international PhD student, Ammar Younas, a lawyer and arbitrator whose research focuses on cross-cultural dimensions of AI ethics and governance.
Interactions Corporation
Interactions LLC (also known as Interactions Corporation) is an American software company that develops voice and text-based virtual assistant applications for customer-service contact centers. Since September 2025, it has been a subsidiary of SoundHound AI. == History == Interactions was founded in 2004. In July 2011, the company announced a $12 million venture-capital funding round led by Sigma Partners. In November 2014, AT&T sold its "Watson" speech recognition platform and related patents to Interactions in exchange for equity. In May 2017, Interactions acquired the social media customer-engagement company Digital Roots; financial terms were not disclosed. On September 3, 2025, SoundHound AI completed its acquisition of Interactions Corporation, with the acquired company becoming a wholly owned subsidiary. == Products and services == Interactions' products have been described as automated voice portals and intelligent virtual assistants used for customer-service tasks. In 2011, Humana expanded the use of an Interactions voice portal for Medicare Part D enrollment.
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