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Bareiss Algorithm
In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder). The method can also be used to compute the determinant of matrices with (approximated) real entries, avoiding the introduction of any round-off errors beyond those already present in the input. Overview Determinant definition has only multiplication, addition and subtraction operations. Obviously the determinant is integer if all matrix entries are integer. However actual computation of the determinant using the definition or Leibniz formula is impractical, as it requires O(''n!'') operations. Gaussian elimination has O(''n''3) complexity, but introduces division, which results in round-off errors when implemented using floating point numbers. Round-off errors can be avoided if all the numbers are k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Erwin Bareiss
Erwin may refer to: People Given name * Erwin Chargaff (1905–2002), Austrian biochemist * Erwin Chemerinsky (born 1953), American legal scholar * Erwin Dold (1919–2012), German concentration camp commandant in World War 2 * Erwin Hauer (1926–2017), Austrian-born American sculptor * Egon Erwin Kisch (1885–1948), Czechoslovak writer and journalist * Erwin Emata (born 1973), Filipino mountain climber * Erwin James (born 1957), British writer and journalist * Erwin Josi (born 1955), Swiss alpine skier * Erwin Klein (died 1992), American table tennis player * Erwin Koeman (born 1961), Dutch footballer and coach * Erwin Kramer (1902–1979), East German politician * Erwin Kreyszig (1922–2008), American academic * Erwin Neutzsky-Wulff (born 1949), Danish author and philosopher * Erwin Osen (1891–1970), Austrian painter and mime artist * Erwin Panofsky (1892-1968), German-Jewish art historian * Erwin Ramírez (born 1971), Ecuadorian football player * Erwin Rommel (1891– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and Other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the '' Journal Citation Reports'', the journal has a 2020 impact factor of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Academic journals established in 1943 American Mathematical Society academic journals Bimonthly journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Numerical Linear Algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible. Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social scienc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Determinants
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end = aei + bfg + cdh - ceg - bdi - afh. The determinant of an matrix can be defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fast Multiplication
A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Numerous algorithms are known and there has been much research into the topic. The oldest and simplest method, known since antiquity as long multiplication or grade-school multiplication, consists of multiplying every digit in the first number by every digit in the second and adding the results. This has a time complexity of O(n^2), where ''n'' is the number of digits. When done by hand, this may also be reframed as grid method multiplication or lattice multiplication. In software, this may be called "shift and add" due to bitshifts and addition being the only two operations needed. In 1960, Anatoly Karatsuba discovered Karatsuba multiplication, unleashing a flood of research into fast multiplication algorithms. This method uses three multiplications rather than four to multiply two two-digit numbers. (A vari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Computational Complexity
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory. Both areas are highly related, as the complexity of an algorithm is always an upper bound on the complexity of the problem solved by this algorithm. Moreover, for designing efficient algorithms, it is often fundamental to compare the complexity of a specific algorithm to the complexity of the problem to be solved. Also, in most cases, the only thing that is known about the complexity of a problem is that it is lower than the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member of a #Related asymptotic notations, family of notations invented by German mathematicians Paul Gustav Heinrich Bachmann, Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '':wikt:Ordnung#German, Ordnung'', meaning the order of approximation. In computer science, big O notation is used to Computational complexity theory, classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetic function, arithmetical function and a better understood approximation; one well-known exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gaussian Elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855). To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: * Swapping two rows, * Multiplying a row by a nonzero number, * Adding a multiple of one row to another row. Using these operations, a matrix can always be transformed into an upper triangular matrix (possibly bordered by rows or columns of zeros), and in fact one that is in row echelon form. Once all of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hadamard Inequality
In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893.Maz'ya & Shaposhnikova It is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of ''n'' dimensions marked out by ''n'' vectors ''vi'' for 1 ≤ ''i'' ≤ ''n'' in terms of the lengths of these vectors , , ''vi'', , . Specifically, Hadamard's inequality states that if ''N'' is the matrix having columns ''vi'', then :\left, \det(N) \ \le \prod_^n \, v_i\, . If the ''n'' vectors are non-zero, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal. Alternate forms and corollaries A corollary is that if the entries of an ''n'' by ''n'' matrix ''N'' are bounded by ''B'', so , ''Nij'', ≤ ''B'' for all ''i'' and ''j'', then :\left, \det(N) \ \le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
In-place Algorithm
In computer science, an in-place algorithm is an algorithm that operates directly on the input data structure without requiring extra space proportional to the input size. In other words, it modifies the input in place, without creating a separate copy of the data structure. An algorithm which is not in-place is sometimes called not-in-place or out-of-place. In-place can have slightly different meanings. In its strictest form, the algorithm can only have a constant amount of extra space, counting everything including function calls and pointers. However, this form is very limited as simply having an index to a length array requires bits. More broadly, in-place means that the algorithm does not use extra space for manipulating the input but may require a small though nonconstant extra space for its operation. Usually, this space is , though sometimes anything in is allowed. Note that space complexity also has varied choices in whether or not to count the index lengths as part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Minor (linear Algebra)
In linear algebra, a minor of a matrix (mathematics), matrix is the determinant of some smaller square matrix generated from by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and Inverse matrix, inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition. Definition and illustration First minors If is a square matrix, then the ''minor'' of the entry in the -th row and -th column (also called the ''minor'', or a ''first minor'') is the determinant of the submatrix formed by deleting the -th row and -th column. This number is often denoted . The ''cofactor'' is obtained by multiplying the minor by . To illustrate these definitions, consider the following matrix, \begin 1 & 4 & 7 \\ 3 & 0 & 5 \\ -1 & 9 & 11 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Sylvester's Determinant Identity
In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851. Cited in Given an ''n''-by-''n'' matrix A, let \det(A) denote its determinant. Choose a pair :u =(u_1, \dots, u_m), v =(v_1, \dots, v_m) \subset (1, \dots, n) of ''m''-element ordered subsets of (1, \dots, n), where ''m'' ≤ ''n''. Let A^u_v denote the (''n''−''m'')-by-(''n''−''m'') submatrix of A obtained by deleting the rows in u and the columns in v. Define the auxiliary ''m''-by-''m'' matrix \tilde^u_v whose elements are equal to the following determinants : (\tilde^u_v)_ := \det(A^_), where uhat/math>, vhat A hat is a Headgear, head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorpor .../ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
