The actor-critic algorithm (AC) is a family of reinforcement learning (RL) algorithms that combine policy-based RL algorithms such as policy gradient methods, and value-based RL algorithms such as value iteration, Q-learning, SARSA, and TD learning. An AC algorithm consists of two main components: an "actor" that determines which actions to take according to a policy function, and a "critic" that evaluates those actions according to a value function. Some AC algorithms are on-policy, some are off-policy. Some apply to either continuous or discrete action spaces. Some work in both cases. == Overview == The actor-critic methods can be understood as an improvement over pure policy gradient methods like REINFORCE via introducing a baseline. === Actor === The actor uses a policy function π ( a | s ) {\displaystyle \pi (a|s)} , while the critic estimates either the value function V ( s ) {\displaystyle V(s)} , the action-value Q-function Q ( s , a ) , {\displaystyle Q(s,a),} the advantage function A ( s , a ) {\displaystyle A(s,a)} , or any combination thereof. The actor is a parameterized function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are the parameters of the actor. The actor takes as argument the state of the environment s {\displaystyle s} and produces a probability distribution π θ ( ⋅ | s ) {\displaystyle \pi _{\theta }(\cdot |s)} . If the action space is discrete, then ∑ a π θ ( a | s ) = 1 {\displaystyle \sum _{a}\pi _{\theta }(a|s)=1} . If the action space is continuous, then ∫ a π θ ( a | s ) d a = 1 {\displaystyle \int _{a}\pi _{\theta }(a|s)da=1} . The goal of policy optimization is to improve the actor. That is, to find some θ {\displaystyle \theta } that maximizes the expected episodic reward J ( θ ) {\displaystyle J(\theta )} : J ( θ ) = E π θ [ ∑ t = 0 T γ t r t ] {\displaystyle J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\gamma ^{t}r_{t}\right]} where γ {\displaystyle \gamma } is the discount factor, r t {\displaystyle r_{t}} is the reward at step t {\displaystyle t} , and T {\displaystyle T} is the time-horizon (which can be infinite). The goal of policy gradient method is to optimize J ( θ ) {\displaystyle J(\theta )} by gradient ascent on the policy gradient ∇ J ( θ ) {\displaystyle \nabla J(\theta )} . As detailed on the policy gradient method page, there are many unbiased estimators of the policy gradient: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ j ≤ T ∇ θ ln π θ ( A j | S j ) ⋅ Ψ j | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{0\leq j\leq T}\nabla _{\theta }\ln \pi _{\theta }(A_{j}|S_{j})\cdot \Psi _{j}{\Big |}S_{0}=s_{0}\right]} where Ψ j {\textstyle \Psi _{j}} is a linear sum of the following: ∑ 0 ≤ i ≤ T ( γ i R i ) {\textstyle \sum _{0\leq i\leq T}(\gamma ^{i}R_{i})} . γ j ∑ j ≤ i ≤ T ( γ i − j R i ) {\textstyle \gamma ^{j}\sum _{j\leq i\leq T}(\gamma ^{i-j}R_{i})} : the REINFORCE algorithm. γ j ∑ j ≤ i ≤ T ( γ i − j R i ) − b ( S j ) {\textstyle \gamma ^{j}\sum _{j\leq i\leq T}(\gamma ^{i-j}R_{i})-b(S_{j})} : the REINFORCE with baseline algorithm. Here b {\displaystyle b} is an arbitrary function. γ j ( R j + γ V π θ ( S j + 1 ) − V π θ ( S j ) ) {\textstyle \gamma ^{j}\left(R_{j}+\gamma V^{\pi _{\theta }}(S_{j+1})-V^{\pi _{\theta }}(S_{j})\right)} : TD(1) learning. γ j Q π θ ( S j , A j ) {\textstyle \gamma ^{j}Q^{\pi _{\theta }}(S_{j},A_{j})} . γ j A π θ ( S j , A j ) {\textstyle \gamma ^{j}A^{\pi _{\theta }}(S_{j},A_{j})} : Advantage Actor-Critic (A2C). γ j ( R j + γ R j + 1 + γ 2 V π θ ( S j + 2 ) − V π θ ( S j ) ) {\textstyle \gamma ^{j}\left(R_{j}+\gamma R_{j+1}+\gamma ^{2}V^{\pi _{\theta }}(S_{j+2})-V^{\pi _{\theta }}(S_{j})\right)} : TD(2) learning. γ j ( ∑ k = 0 n − 1 γ k R j + k + γ n V π θ ( S j + n ) − V π θ ( S j ) ) {\textstyle \gamma ^{j}\left(\sum _{k=0}^{n-1}\gamma ^{k}R_{j+k}+\gamma ^{n}V^{\pi _{\theta }}(S_{j+n})-V^{\pi _{\theta }}(S_{j})\right)} : TD(n) learning. γ j ∑ n = 1 ∞ λ n − 1 1 − λ ⋅ ( ∑ k = 0 n − 1 γ k R j + k + γ n V π θ ( S j + n ) − V π θ ( S j ) ) {\textstyle \gamma ^{j}\sum _{n=1}^{\infty }{\frac {\lambda ^{n-1}}{1-\lambda }}\cdot \left(\sum _{k=0}^{n-1}\gamma ^{k}R_{j+k}+\gamma ^{n}V^{\pi _{\theta }}(S_{j+n})-V^{\pi _{\theta }}(S_{j})\right)} : TD(λ) learning, also known as GAE (generalized advantage estimate). This is obtained by an exponentially decaying sum of the TD(n) learning terms. === Critic === In the unbiased estimators given above, certain functions such as V π θ , Q π θ , A π θ {\displaystyle V^{\pi _{\theta }},Q^{\pi _{\theta }},A^{\pi _{\theta }}} appear. These are approximated by the critic. Since these functions all depend on the actor, the critic must learn alongside the actor. The critic is learned by value-based RL algorithms. For example, if the critic is estimating the state-value function V π θ ( s ) {\displaystyle V^{\pi _{\theta }}(s)} , then it can be learned by any value function approximation method. Let the critic be a function approximator V ϕ ( s ) {\displaystyle V_{\phi }(s)} with parameters ϕ {\displaystyle \phi } . The simplest example is TD(1) learning, which trains the critic to minimize the TD(1) error: δ i = R i + γ V ϕ ( S i + 1 ) − V ϕ ( S i ) {\displaystyle \delta _{i}=R_{i}+\gamma V_{\phi }(S_{i+1})-V_{\phi }(S_{i})} The critic parameters are updated by gradient descent on the squared TD error: ϕ ← ϕ − α ∇ ϕ ( δ i ) 2 = ϕ + α δ i ∇ ϕ V ϕ ( S i ) {\displaystyle \phi \leftarrow \phi -\alpha \nabla _{\phi }(\delta _{i})^{2}=\phi +\alpha \delta _{i}\nabla _{\phi }V_{\phi }(S_{i})} where α {\displaystyle \alpha } is the learning rate. Note that the gradient is taken with respect to the ϕ {\displaystyle \phi } in V ϕ ( S i ) {\displaystyle V_{\phi }(S_{i})} only, since the ϕ {\displaystyle \phi } in γ V ϕ ( S i + 1 ) {\displaystyle \gamma V_{\phi }(S_{i+1})} constitutes a moving target, and the gradient is not taken with respect to that. This is a common source of error in implementations that use automatic differentiation, and requires "stopping the gradient" at that point. Similarly, if the critic is estimating the action-value function Q π θ {\displaystyle Q^{\pi _{\theta }}} , then it can be learned by Q-learning or SARSA. In SARSA, the critic maintains an estimate of the Q-function, parameterized by ϕ {\displaystyle \phi } , denoted as Q ϕ ( s , a ) {\displaystyle Q_{\phi }(s,a)} . The temporal difference error is then calculated as δ i = R i + γ Q θ ( S i + 1 , A i + 1 ) − Q θ ( S i , A i ) {\displaystyle \delta _{i}=R_{i}+\gamma Q_{\theta }(S_{i+1},A_{i+1})-Q_{\theta }(S_{i},A_{i})} . The critic is then updated by θ ← θ + α δ i ∇ θ Q θ ( S i , A i ) {\displaystyle \theta \leftarrow \theta +\alpha \delta _{i}\nabla _{\theta }Q_{\theta }(S_{i},A_{i})} The advantage critic can be trained by training both a Q-function Q ϕ ( s , a ) {\displaystyle Q_{\phi }(s,a)} and a state-value function V ϕ ( s ) {\displaystyle V_{\phi }(s)} , then let A ϕ ( s , a ) = Q ϕ ( s , a ) − V ϕ ( s ) {\displaystyle A_{\phi }(s,a)=Q_{\phi }(s,a)-V_{\phi }(s)} . Although, it is more common to train just a state-value function V ϕ ( s ) {\displaystyle V_{\phi }(s)} , then estimate the advantage by A ϕ ( S i , A i ) ≈ ∑ j ∈ 0 : n − 1 γ j R i + j + γ n V ϕ ( S i + n ) − V ϕ ( S i ) {\displaystyle A_{\phi }(S_{i},A_{i})\approx \sum _{j\in 0:n-1}\gamma ^{j}R_{i+j}+\gamma ^{n}V_{\phi }(S_{i+n})-V_{\phi }(S_{i})} Here, n {\displaystyle n} is a positive integer. The higher n {\displaystyle n} is, the more lower is the bias in the advantage estimation, but at the price of higher variance. The Generalized Advantage Estimation (GAE) introduces a hyperparameter λ {\displaystyle \lambda } that smoothly interpolates between Monte Carlo returns ( λ = 1 {\displaystyle \lambda =1} , high variance, no bias) and 1-step TD learning ( λ = 0 {\displaystyle \lambda =0} , low variance, high bias). This hyperparameter can be adjusted to pick the optimal bias-variance trade-off in advantage estimation. It uses an exponentially decaying average of n-step returns with λ {\displaystyle \lambda } being the decay strength. == Variants == Asynchronous Advantage Actor-Critic (A3C): Parallel and asynchronous version of A2C. Soft Actor-Critic (SAC): Incorporates entropy maximization for improved exploration. Deep Deterministic Policy Gradient (DDPG): Specialized for continuous action spaces.
Clean Email
Clean Email is an automated software as a service email management application which identifies and clears junk mail from inboxes. The service uses a subscription business model with a free trial for the first 1,000 emails. and is available on macOS, iOS, Android, and on the web. == History == Clean Email is a self-funded company headquartered in Los Angeles, California. Initially developed by the founder for personal use, the service was designed to address the growing issue of inbox clutter and privacy concerns. In 2017, John Gruber recognized Clean Email as a trustworthy alternative to Unroll.me after the latter was found to be selling user data. == Features == Clean Email uses algorithms to identify and categorize emails, enabling users to group, remove, label, and archive email messages in bulk. Its Unsubscriber tool consolidates all subscriptions and newsletters into a single view for quick management, allowing users to bulk unsubscribe or temporarily pause mail. Its Screener feature transforms the inbox into an "opt-in" system, enabling users to pre-approve mail from new senders. Cleaning Suggestions identifies frequently cleaned mail, recommending actions accordingly. Additional functionalities include automatic deletion of aging emails, delivery of messages to specified folders, and options to mute or block senders.
Tertiary source
A tertiary source is an index or textual consolidation of already published primary and secondary sources that does not provide additional interpretations or analysis of the sources. Some tertiary sources can be used as an aid to find key (seminal) sources, key terms, general common knowledge and established mainstream science on a topic. The exact definition of tertiary varies by academic field. Academic research standards generally do not accept tertiary sources such as encyclopedias as citations, although survey articles are frequently cited rather than the original publication. == Overlap with secondary sources == As is also the case with distinguishing primary and secondary sources in some disciplines, there is not always a clear distinguishing line between secondary and tertiary sources. Depending on the topic of research, a scholar may use a bibliography, dictionary, or encyclopedia as either a tertiary or a secondary source. This causes some difficulty in defining many sources as either one type or the other. In some academic disciplines, the differentiation between a secondary and tertiary source is relative. In the United Nations International Scientific Information System (UNISIST) model, a secondary source is a bibliography, whereas a tertiary source is a synthesis of primary sources. == Types of tertiary sources == Tertiary sources can come in book form or as an online resource. Tertiary sources in book form are frequently organised in alphabetical order, whereas an online tertiary source may be searchable by keyword. Examples of tertiary sources include; reference books, encyclopedias, dictionaries, some textbooks, abstracts, directories, factbooks, handbooks, manuals and compendia. Indexes, bibliographies, concordances, and databases are aggregates of primary and secondary sources and therefore often considered tertiary sources. They may also serve as a point of access to the full or partial text of primary and secondary sources. Almanacs, travel guides, field guides, and timelines are also examples of tertiary sources. Tertiary sources attempt to summarize, collect, and consolidate the source materials into an overview without adding analysis and synthesis of new conclusions. Wikipedia is a tertiary source.
Artificial intelligence industry in China
The roots of the development of artificial intelligence in the People's Republic of China started in the late 1970s following Deng Xiaoping's reform and opening up emphasizing science and technology as the country's primary productive force. The initial stages of China's AI development were slow and encountered significant challenges due to lack of resources and talent. At the beginning China was behind most Western countries in terms of AI development. A majority of the research was led by scientists who had received higher education abroad. Since 2006, the Chinese government has steadily developed a national agenda for artificial intelligence development and emerged as one of the leading nations in artificial intelligence research and development. In 2016, the Chinese Communist Party (CCP) released its 13th Five-Year Plan in which it aimed to become a global AI leader by 2030. As of 2025, China is considered to be a world leader in AI technology along with the United States. The State Council has a list of "national AI teams" including fifteen China-based companies, including Baidu, Tencent, Alibaba, SenseTime, and iFlytek. Each company should lead the development of a designated specialized AI sector in China, such as facial recognition, software/hardware, and speech recognition. China's rapid AI development has significantly impacted Chinese society in many areas, including the socio-economic, military, intelligence, and political spheres. Agriculture, transportation, accommodation and food services, and manufacturing are the top industries that would be the most impacted by further AI deployment. The private sector, university laboratories, and the military are working collaboratively in many aspects as there are few current existing boundaries. In 2021, China published the Data Security Law of the People's Republic of China, its first national law addressing AI-related ethical concerns. In October 2022, the United States federal government announced a series of export controls and trade restrictions intended to restrict China's access to advanced computer chips for AI applications. In 2023, the Cyberspace Administration of China issued guidelines requiring that AI content upholds the ideology of the CCP including Core Socialist Values, avoids discrimination, respects intellectual property rights, and safeguards user data. In 2025, the Chinese government issued a document regarding training data, requiring companies to use as little as data deemed "unsafe" as possible, as well as requiring companies to test models regularly. Concerns have been raised about the effects of the Chinese government's censorship regime on the development of generative artificial intelligence and long-term talent acquisition with state of the country's demographics. Others have noted that official notions of AI safety require following the priorities of the CCP and are antithetical to standards in democratic societies and raised concerns about the extension of China's system of mass surveillance and censorship abroad. == History == The Chinese term for artificial intelligence (réngōngzhìnéng 人工智能) connotes "humanmade" intelligence. The term developed as mid-20th century localisation of the Japanese term jinko chino. The research and development of artificial intelligence in China started in the 1980s, with the announcement by Deng Xiaoping of the importance of science and technology for China's economic growth. === Late 1970s to early 2010s === Chinese artificial intelligence research and development began in late 1970s after Deng Xiaoping's reform and opening up. China's first national conference on AI occurred in 1979. Academic journals in the late 1970s began publishing literature reviews of Western research on AI topics. In the 1980s, a group of Chinese scientists launched AI research led by Qian Xuesen and Wu Wenjun. However, during the time, China's society still had a generally conservative view towards AI. In the early 1980s, Science Press published translated versions of Western textbooks such as Patrick Winston's Artificial Intelligence and Nils John Nilsson's Principles of Artificial Intelligence. In 1980, a journal of the Chinese Academy of Sciences convened its first annual National Symposium on Artificial Intelligence, which included national and international scholars like Herbert A. Simon. The Chinese Association for Artificial Intelligence (CAAI) was founded in September 1981 and was authorized by the Ministry of Civil Affairs. CAAI has continued to be the largest AI association in China as of 2025. In 1982, CAAI began publishing the Artificial Intelligence Journal, which published early AI research by Chinese academics. In the 1980s, Chinese research on AI was influenced by the field of cybernetics, particularly the work of Norbert Weiner and his text Cybernetics: Or Control and Communication in the Animal and the Machine. Chinese researchers at the time sought to situate AI as part of a broader "Intelligence Science" field which would include disciplines like mathematics, computer science, cognitive science, social sciences, and philosophy. In 1987, Tsinghua University began a research publication on AI. Beginning in 1993, smart automation and intelligence have been part of China's national technology plan. Since the 2000s, the Chinese government has further expanded its research and development funds for AI and the number of government-sponsored research projects has dramatically increased. In 2006, China announced a policy priority for the development of artificial intelligence, which was included in the National Medium and Long Term Plan for the Development of Science and Technology (2006–2020), released by the State Council. In the same year, artificial intelligence was also mentioned in the 11th Five-Year Plan. In 2011, the Association for the Advancement of Artificial Intelligence (AAAI) established a branch in Beijing, China. At same year, the Wu Wenjun Artificial Intelligence Science and Technology Award was founded in honor of Chinese mathematician Wu Wenjun, and it became the highest award for Chinese achievements in the field of artificial intelligence. The first award ceremony was held on May 14, 2012. In 2013, the International Joint Conferences on Artificial Intelligence (IJCAI) was held in Beijing, marking the first time the conference was held in China. This event coincided with the Chinese government's announcement of the "Chinese Intelligence Year," a significant milestone in China's development of artificial intelligence. === Late 2010s to early 2020s === AI became a major issue of commercial, public, and political focus in China in the latter half of the 2010s. Various interpretations of the primary cause for this increased focus exist, with some analyses focusing on the 2016 Go match between Google's AlphaGo and Lee Sedol, others emphasising the U.S. increasing trade restrictions on China's technology industries and the desire to achieve national technological self-sufficiency. The State Council of China issued "A Next Generation Artificial Intelligence Development Plan" (State Council Document [2017] No. 35) on 20 July 2017. In the document, the CCP Central Committee and the State Council urged governing bodies in China to promote the development of artificial intelligence. Specifically, the plan described AI as a strategic technology that has become a "focus of international competition".:2 The document urged significant investment in a number of strategic areas related to AI and called for close cooperation between the state and private sectors. It set the goal of China becoming the preeminent country for AI research and application by 2030. During the general secretaryship of Xi Jinping, artificial intelligence has been a focus of the CCP's military-civil fusion efforts. On the occasion of Xi's speech at the first plenary meeting of the Central Military-Civil Fusion Development Committee (CMCFDC), scholars from the National Defense University wrote in the PLA Daily that the "transferability of social resources" between economic and military ends is an essential component to being a great power. During the Two Sessions 2017,"artificial intelligence plus" was proposed to be elevated to a strategic level. The same year witnessed the emergence of multiple application-level usages in the medical field according to reports. In 2018, Xinhua News Agency, in partnership with Tencent's subsidiary Sogou, launched its first artificial intelligence-generated news anchor. In 2018, the State Council budgeted $2.1 billion for an AI industrial park in Mentougou district. In order to achieve this the State Council stated the need for massive talent acquisition, theoretical and practical developments, as well as public and private investments. Some of the stated motivations that the State Council gave for pursuing its AI strategy include the potential of artificial intelligence for industrial transformation, better social
Kullback–Leibler Upper Confidence Bound
In multi-armed bandit problems, KL-UCB (for Kullback–Leibler Upper Confidence Bound) is a UCB-type algorithm that is asymptotically optimal, in the sense that its regret matches the problem-dependent Lai-Robbins lower bound. == Multi-armed bandit problem == The Multi-armed bandit problem is a sequential game where one player has to choose at each turn between K {\displaystyle K} actions (arms). Behind every arm a {\displaystyle a} there is an unknown distribution ν a {\displaystyle \nu _{a}} that lies in a set D {\displaystyle {\mathcal {D}}} known by the player (for example, D {\displaystyle {\mathcal {D}}} can be the set of Gaussian distributions or Bernoulli distributions). At each turn t {\displaystyle t} the player chooses (pulls) an arm a t {\displaystyle a_{t}} , he then gets an observation X t {\displaystyle X_{t}} of the distribution ν a t {\displaystyle \nu _{a_{t}}} . === Regret minimization === The goal is to minimize the regret at time T {\displaystyle T} that is defined as R T := ∑ a = 1 K Δ a E [ N a ( T ) ] {\displaystyle R_{T}:=\sum _{a=1}^{K}\Delta _{a}\mathbb {E} [N_{a}(T)]} where μ a := E [ ν a ] {\displaystyle \mu _{a}:=\mathbb {E} [\nu _{a}]} is the mean of arm a {\displaystyle a} μ ∗ := max a μ a {\displaystyle \mu ^{}:=\max _{a}\mu _{a}} is the highest mean Δ a := μ ∗ − μ a {\displaystyle \Delta _{a}:=\mu ^{}-\mu _{a}} N a ( t ) {\displaystyle N_{a}(t)} is the number of pulls of arm a {\displaystyle a} up to turn t {\displaystyle t} The player has to find an algorithm that chooses at each turn t {\displaystyle t} which arm to pull based on the previous actions and observations ( a s , X s ) s < t {\displaystyle (a_{s},X_{s})_{s
Semantic neural network
Semantic neural network (SNN) is based on John von Neumann's neural network [von Neumann, 1966] and Nikolai Amosov M-Network. There are limitations to a link topology for the von Neumann’s network but SNN accept a case without these limitations. Only logical values can be processed, but SNN accept that fuzzy values can be processed too. All neurons into the von Neumann network are synchronized by tacts. For further use of self-synchronizing circuit technique SNN accepts neurons can be self-running or synchronized. In contrast to the von Neumann network there are no limitations for topology of neurons for semantic networks. It leads to the impossibility of relative addressing of neurons as it was done by von Neumann. In this case an absolute readdressing should be used. Every neuron should have a unique identifier that would provide a direct access to another neuron. Of course, neurons interacting by axons-dendrites should have each other's identifiers. An absolute readdressing can be modulated by using neuron specificity as it was realized for biological neural networks. There’s no description for self-reflectiveness and self-modification abilities into the initial description of semantic networks [Dudar Z.V., Shuklin D.E., 2000]. But in [Shuklin D.E. 2004] a conclusion had been drawn about the necessity of introspection and self-modification abilities in the system. For maintenance of these abilities a concept of pointer to neuron is provided. Pointers represent virtual connections between neurons. In this model, bodies and signals transferring through the neurons connections represent a physical body, and virtual connections between neurons are representing an astral body. It is proposed to create models of artificial neuron networks on the basis of virtual machine supporting the opportunity for paranormal effects. SNN is generally used for natural language processing. == Related models == Computational creativity Semantic hashing Semantic Pointer Architecture Sparse distributed memory
Zassenhaus algorithm
In mathematics, the Zassenhaus algorithm is a method to calculate a basis for the intersection and sum of two subspaces of a vector space. It is named after Hans Zassenhaus, but no publication of this algorithm by him is known. It is used in computer algebra systems. == Algorithm == === Input === Let V be a vector space and U, W two finite-dimensional subspaces of V with the following spanning sets: U = ⟨ u 1 , … , u n ⟩ {\displaystyle U=\langle u_{1},\ldots ,u_{n}\rangle } and W = ⟨ w 1 , … , w k ⟩ . {\displaystyle W=\langle w_{1},\ldots ,w_{k}\rangle .} Finally, let B 1 , … , B m {\displaystyle B_{1},\ldots ,B_{m}} be linearly independent vectors so that u i {\displaystyle u_{i}} and w i {\displaystyle w_{i}} can be written as u i = ∑ j = 1 m a i , j B j {\displaystyle u_{i}=\sum _{j=1}^{m}a_{i,j}B_{j}} and w i = ∑ j = 1 m b i , j B j . {\displaystyle w_{i}=\sum _{j=1}^{m}b_{i,j}B_{j}.} === Output === The algorithm computes the base of the sum U + W {\displaystyle U+W} and a base of the intersection U ∩ W {\displaystyle U\cap W} . === Algorithm === The algorithm creates the following block matrix of size ( ( n + k ) × ( 2 m ) ) {\displaystyle ((n+k)\times (2m))} : ( a 1 , 1 a 1 , 2 ⋯ a 1 , m a 1 , 1 a 1 , 2 ⋯ a 1 , m ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a n , 1 a n , 2 ⋯ a n , m a n , 1 a n , 2 ⋯ a n , m b 1 , 1 b 1 , 2 ⋯ b 1 , m 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ b k , 1 b k , 2 ⋯ b k , m 0 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,m}&a_{1,1}&a_{1,2}&\cdots &a_{1,m}\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,m}&a_{n,1}&a_{n,2}&\cdots &a_{n,m}\\b_{1,1}&b_{1,2}&\cdots &b_{1,m}&0&0&\cdots &0\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\b_{k,1}&b_{k,2}&\cdots &b_{k,m}&0&0&\cdots &0\end{pmatrix}}} Using elementary row operations, this matrix is transformed to the row echelon form. Then, it has the following shape: ( c 1 , 1 c 1 , 2 ⋯ c 1 , m ∙ ∙ ⋯ ∙ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ c q , 1 c q , 2 ⋯ c q , m ∙ ∙ ⋯ ∙ 0 0 ⋯ 0 d 1 , 1 d 1 , 2 ⋯ d 1 , m ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 d ℓ , 1 d ℓ , 2 ⋯ d ℓ , m 0 0 ⋯ 0 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}c_{1,1}&c_{1,2}&\cdots &c_{1,m}&\bullet &\bullet &\cdots &\bullet \\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\c_{q,1}&c_{q,2}&\cdots &c_{q,m}&\bullet &\bullet &\cdots &\bullet \\0&0&\cdots &0&d_{1,1}&d_{1,2}&\cdots &d_{1,m}\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\0&0&\cdots &0&d_{\ell ,1}&d_{\ell ,2}&\cdots &d_{\ell ,m}\\0&0&\cdots &0&0&0&\cdots &0\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\0&0&\cdots &0&0&0&\cdots &0\end{pmatrix}}} Here, ∙ {\displaystyle \bullet } stands for arbitrary numbers, and the vectors ( c p , 1 , c p , 2 , … , c p , m ) {\displaystyle (c_{p,1},c_{p,2},\ldots ,c_{p,m})} for every p ∈ { 1 , … , q } {\displaystyle p\in \{1,\ldots ,q\}} and ( d p , 1 , … , d p , m ) {\displaystyle (d_{p,1},\ldots ,d_{p,m})} for every p ∈ { 1 , … , ℓ } {\displaystyle p\in \{1,\ldots ,\ell \}} are nonzero. Then ( y 1 , … , y q ) {\displaystyle (y_{1},\ldots ,y_{q})} with y i := ∑ j = 1 m c i , j B j {\displaystyle y_{i}:=\sum _{j=1}^{m}c_{i,j}B_{j}} is a basis of U + W {\displaystyle U+W} and ( z 1 , … , z ℓ ) {\displaystyle (z_{1},\ldots ,z_{\ell })} with z i := ∑ j = 1 m d i , j B j {\displaystyle z_{i}:=\sum _{j=1}^{m}d_{i,j}B_{j}} is a basis of U ∩ W {\displaystyle U\cap W} . === Proof of correctness === First, we define π 1 : V × V → V , ( a , b ) ↦ a {\displaystyle \pi _{1}:V\times V\to V,(a,b)\mapsto a} to be the projection to the first component. Let H := { ( u , u ) ∣ u ∈ U } + { ( w , 0 ) ∣ w ∈ W } ⊆ V × V . {\displaystyle H:=\{(u,u)\mid u\in U\}+\{(w,0)\mid w\in W\}\subseteq V\times V.} Then π 1 ( H ) = U + W {\displaystyle \pi _{1}(H)=U+W} and H ∩ ( 0 × V ) = 0 × ( U ∩ W ) {\displaystyle H\cap (0\times V)=0\times (U\cap W)} . Also, H ∩ ( 0 × V ) {\displaystyle H\cap (0\times V)} is the kernel of π 1 | H {\displaystyle {\pi _{1}|}_{H}} , the projection restricted to H. Therefore, dim ( H ) = dim ( U + W ) + dim ( U ∩ W ) {\displaystyle \dim(H)=\dim(U+W)+\dim(U\cap W)} . The Zassenhaus algorithm calculates a basis of H. In the first m columns of this matrix, there is a basis y i {\displaystyle y_{i}} of U + W {\displaystyle U+W} . The rows of the form ( 0 , z i ) {\displaystyle (0,z_{i})} (with z i ≠ 0 {\displaystyle z_{i}\neq 0} ) are obviously in H ∩ ( 0 × V ) {\displaystyle H\cap (0\times V)} . Because the matrix is in row echelon form, they are also linearly independent. All rows which are different from zero ( ( y i , ∙ ) {\displaystyle (y_{i},\bullet )} and ( 0 , z i ) {\displaystyle (0,z_{i})} ) are a basis of H, so there are dim ( U ∩ W ) {\displaystyle \dim(U\cap W)} such z i {\displaystyle z_{i}} s. Therefore, the z i {\displaystyle z_{i}} s form a basis of U ∩ W {\displaystyle U\cap W} . == Example == Consider the two subspaces U = ⟨ ( 1 − 1 0 1 ) , ( 0 0 1 − 1 ) ⟩ {\displaystyle U=\left\langle \left({\begin{array}{r}1\\-1\\0\\1\end{array}}\right),\left({\begin{array}{r}0\\0\\1\\-1\end{array}}\right)\right\rangle } and W = ⟨ ( 5 0 − 3 3 ) , ( 0 5 − 3 − 2 ) ⟩ {\displaystyle W=\left\langle \left({\begin{array}{r}5\\0\\-3\\3\end{array}}\right),\left({\begin{array}{r}0\\5\\-3\\-2\end{array}}\right)\right\rangle } of the vector space R 4 {\displaystyle \mathbb {R} ^{4}} . Using the standard basis, we create the following matrix of dimension ( 2 + 2 ) × ( 2 ⋅ 4 ) {\displaystyle (2+2)\times (2\cdot 4)} : ( 1 − 1 0 1 1 − 1 0 1 0 0 1 − 1 0 0 1 − 1 5 0 − 3 3 0 0 0 0 0 5 − 3 − 2 0 0 0 0 ) . {\displaystyle \left({\begin{array}{rrrrrrrr}1&-1&0&1&&1&-1&0&1\\0&0&1&-1&&0&0&1&-1\\\\5&0&-3&3&&0&0&0&0\\0&5&-3&-2&&0&0&0&0\end{array}}\right).} Using elementary row operations, we transform this matrix into the following matrix: ( 1 0 0 0 ∙ ∙ ∙ ∙ 0 1 0 − 1 ∙ ∙ ∙ ∙ 0 0 1 − 1 ∙ ∙ ∙ ∙ 0 0 0 0 1 − 1 0 1 ) {\displaystyle \left({\begin{array}{rrrrrrrrr}1&0&0&0&&\bullet &\bullet &\bullet &\bullet \\0&1&0&-1&&\bullet &\bullet &\bullet &\bullet \\0&0&1&-1&&\bullet &\bullet &\bullet &\bullet \\\\0&0&0&0&&1&-1&0&1\end{array}}\right)} (Some entries have been replaced by " ∙ {\displaystyle \bullet } " because they are irrelevant to the result.) Therefore ( ( 1 0 0 0 ) , ( 0 1 0 − 1 ) , ( 0 0 1 − 1 ) ) {\displaystyle \left(\left({\begin{array}{r}1\\0\\0\\0\end{array}}\right),\left({\begin{array}{r}0\\1\\0\\-1\end{array}}\right),\left({\begin{array}{r}0\\0\\1\\-1\end{array}}\right)\right)} is a basis of U + W {\displaystyle U+W} , and ( ( 1 − 1 0 1 ) ) {\displaystyle \left(\left({\begin{array}{r}1\\-1\\0\\1\end{array}}\right)\right)} is a basis of U ∩ W {\displaystyle U\cap W} .