|
Jacobi Eigenvalue Algorithm
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization). It is named after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, but only became widely used in the 1950s with the advent of computers. Description Let S be a symmetric matrix, and G=G(i,j,\theta) be a Givens rotation matrix. Then: :S'=G S G^\top \, is symmetric and similar to S. Furthermore, S^\prime has entries: :\begin S'_ &= c^2\, S_ - 2\, s c \,S_ + s^2\, S_ \\ S'_ &= s^2 \,S_ + 2 s c\, S_ + c^2 \, S_ \\ S'_ &= S'_ = (c^2 - s^2 ) \, S_ + s c \, (S_ - S_ ) \\ S'_ &= S'_ = c \, S_ - s \, S_ & k \ne i,j \\ S'_ &= S'_ = s \, S_ + c \, S_ & k \ne i,j \\ S'_ &= S_ &k,l \ne i,j \end where s=\sin(\theta) and c=\cos(\theta). Since G is orthogonal, S and S^\prime have the same Frobenius norm , , \cdot, , _F (the square-root sum of squares ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 sciences ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m rows and n columns and entries in the field K. A matrix norm is a norm on K^. This article will always write such norms with double vertical bars (like so: \, A\, ). Thus, the matrix norm is a function \, \cdot\, : K^ \to \R that must satisfy the following properties: For all scalars \alpha \in K and matrices A, B \in K^, *\, A\, \ge 0 (''positive-valued'') *\, A\, = 0 \iff A=0_ (''definite'') *\left\, \alpha A\right\, =\left, \alpha\ \left\, A\right\, (''absolutely homogeneous'') *\, A+B\, \le \, A\, +\, B\, (''sub-additive'' or satisfying the ''triangle inequality'') The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative: *\left\, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Round-off Error
A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error. When using approximation equations or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of numerical analysis is to estimate computation errors. Computation errors, also called numerical errors, include both truncation errors and roundoff errors. When a sequence of calculations with an input involving any roundoff error are made, errors may accumulate, sometimes dominating the calculation. In ill-conditioned problems, significant error may accumulate. In short, there are two major facets of roundoff errors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Method For Complex Hermitian Matrices
In mathematics, the Jacobi method for complex Hermitian matrices is a generalization of the Jacobi iteration method. The Jacobi iteration method is also explained in "Introduction to Linear Algebra" by . Derivation The complex unitary rotation matrices ''R''''pq'' can be used for Jacobi iteration of complex Hermitian matrices in order to find a numerical estimation of their eigenvectors and eigenvalues simultaneously. Similar to the Givens rotation matrices, ''R''''pq'' are defined as: : \begin (R_)_ & = \delta_ & \qquad m,n \ne p,q, \\0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^ \end Each rotation matrix, ''R''''pq'', will modify only the ''p''th and ''q''th rows or columns of a matrix ''M'' if it is applied from left or right, respectively: : \begin (R_ M)_ & = \begin M_ & m \ne p,q \\ pt\frac (M_ e^ - M_ e^) & m = p \\ pt\frac (M_ e^ + M_ e^) & m = q \end \\ pt(MR_^\dagger)_ & = \begin M_ & n \ne p,q \\ \frac (M_ e^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Matrix
In linear algebra, a Hilbert matrix, introduced by , is a square matrix with entries being the unit fractions : H_ = \frac. For example, this is the 5 × 5 Hilbert matrix: : H = \begin 1 & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \\ \frac & \frac & \frac & \frac & \frac \end. The Hilbert matrix can be regarded as derived from the integral : H_ = \int_0^1 x^ \, dx, that is, as a Gramian matrix for powers of ''x''. It arises in the least squares approximation of arbitrary functions by polynomials. The Hilbert matrices are canonical examples of ill-conditioned matrices, being notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8. Historical note introduced the Hilbert matrix to study the following question in approximation theory: "Assume that , is a real interval. Is it then po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Schönhage
Arnold Schönhage (born 1 December 1934 in Lockhausen, now Bad Salzuflen) is a German mathematician and computer scientist. Schönhage was professor at the Rheinische Friedrich-Wilhelms-Universität, Bonn, and also in Tübingen and Konstanz. He now lives near Bonn. Together with Volker Strassen he developed the Schönhage–Strassen algorithm for fast integer multiplication that has a run-time of '' O''(''N'' log ''N'' log log ''N''). Schönhage designed and implemented together with Andreas F. W. Grotefeld and Ekkehart Vetter a multitape Turing machine, called TP, in software. The machine is programmed in TPAL, an assembler language In computer programming, assembly language (or assembler language, or symbolic machine code), often referred to simply as Assembly and commonly abbreviated as ASM or asm, is any low-level programming language with a very strong correspondence be .... They implemented numerous numerical algorithms including the Sch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Rotation
In numerical linear algebra, a Jacobi rotation is a rotation, ''Q''''k''ℓ, of a 2-dimensional linear subspace of an ''n-''dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an ''n''×''n'' real symmetric matrix, ''A'', when applied as a similarity transformation: : A \mapsto Q_^T A Q_ = A' . \,\! : \begin & & & \cdots & & & * \\ & \ddots & & & & & \\ & & a_ & \cdots & a_ & & \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ & & a_ & \cdots & a_ & & \\ & & & & & \ddots & \\ & & & \cdots & & & * \end \to \begin & & & \cdots & & & * \\ & \ddots & & & & & \\ & & a'_ & \cdots & 0 & & \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ & & 0 & \cdots & a'_ & & \\ & & & & & \ddots & \\ & & & \cdots & & & * \end. It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Off-diagonal Element
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek διαγώνιος ''diagonios'', "from angle to angle" (from διά- ''dia-'', "through", "across" and γωνία ''gonia'', "angle", related to ''gony'' "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as ''diagonus'' ("slanting line"). In matrix algebra, the diagonal of a square matrix consists of the entries on the line from the top left corner to the bottom right corner. There are also other, non-mathematical uses. Non-mathematical uses In engineering, a diagonal brace is a beam used to brace a rectangular structure (such as scaffolding) to withstand strong forces pushing into it; although called a diagonal, due to practical considerations d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similar (linear Algebra)
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two (possibly) different bases, with being the change of basis matrix. A transformation is called a similarity transformation or conjugation of the matrix . In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that be chosen to lie in . Motivating example When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in when the axis of rotation is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the previous ones. A specific implementation of an iterative method, including the Algorithm#Termination, termination criteria, is an algorithm of the iterative method. An iterative method is called convergent if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations A\mathbf=\mathbf by Gaussian elimination). Iterative methods are often the only cho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Givens Rotation
In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory. Matrix representation A Givens rotation is represented by a matrix of the form :G(i, j, \theta) = \begin 1 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & & \vdots & & \vdots \\ 0 & \cdots & c & \cdots & -s & \cdots & 0 \\ \vdots & & \vdots & \ddots & \vdots & & \vdots \\ 0 & \cdots & s & \cdots & c & \cdots & 0 \\ \vdots & & \vdots & & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & \cdots & 0 & \cdots & 1 \end, where and appear at the int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
