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Biconjugate Gradient Method
In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations :A x= b.\, Unlike the conjugate gradient method, this algorithm does not require the matrix A to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose . The algorithm # Choose initial guess x_0\,, two other vectors x_0^* and b^*\, and a preconditioner M\, # r_0 \leftarrow b-A\, x_0\, # r_0^* \leftarrow b^*-x_0^*\, A # p_0 \leftarrow M^ r_0\, # p_0^* \leftarrow r_0^*M^\, # for k=0, 1, \ldots do ## \alpha_k \leftarrow \, ## x_ \leftarrow x_k + \alpha_k \cdot p_k\, ## x_^* \leftarrow x_k^* + \overline\cdot p_k^*\, ## r_ \leftarrow r_k - \alpha_k \cdot A p_k\, ## r_^* \leftarrow r_k^*- \overline \cdot p_k^*\, A ## \beta_k \leftarrow \, ## p_ \leftarrow M^ r_ + \beta_k \cdot p_k\, ## p_^* \leftarrow r_^*M^ + \overline\cdot p_k^*\, In the above formulation, the computed r_k\, and r_k^* satisfy :r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of the common task ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Gradient Method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems. The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it. The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear optimization problems. Description of the problem addressed by co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biconjugate Gradient Stabilized Method
In numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared method (CGS). It is a Krylov subspace method. Unlike the original BiCG method, it doesn't require multiplication by the transpose of the system matrix. Algorithmic steps Unpreconditioned BiCGSTAB To solve a linear system , BiCGSTAB starts with an initial guess and proceeds as follows: # # Choose an arbitrary vector such that , e.g., . denotes the dot product of vectors # # # For ## ## ## ## ## ## ## If is accurate enough, then set and quit ## ## ## ## ## If is accurate enough, then quit ## Preconditioned BiCGSTAB Preconditioners are usually used to accelerat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krylov Subspace
In linear algebra, the order-''r'' Krylov subspace generated by an ''n''-by-''n'' matrix ''A'' and a vector ''b'' of dimension ''n'' is the linear subspace spanned by the images of ''b'' under the first ''r'' powers of ''A'' (starting from A^0=I), that is, :\mathcal_r(A,b) = \operatorname \, \. Background The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about it in 1931. Properties * \mathcal_r(A,b),A\mathcal_r(A,b)\subset \mathcal_(A,b). * Vectors \ are linearly independent until r, where p(A) is the minimal polynomial of A. Furthermore, there exists a b such that r_0 = \deg (A)/math>. * \mathcal_r(A,b) is a cyclic submodule generated by b of the torsion k /math>-module (k^n)^A, where k^n is the linear space on k. * k^n can be decomposed as the direct sum of Krylov subspaces. Use Krylov subspaces are used in algorithms for finding approximate solutions to high-dimensional linear algebra problems. Many linear dyn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biorthogonal System
In mathematics, a biorthogonal system is a pair of indexed families of vectors \tilde v_i \text E \text \tilde u_i \text F such that \left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_, where E and F form a pair of topological vector spaces that are in duality, \langle \,\cdot, \cdot\, \rangle is a bilinear mapping and \delta_ is the Kronecker delta. An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct. A biorthogonal system in which E = F and \tilde v_i = \tilde u_i is an orthonormal system. Projection Related to a biorthogonal system is the projection P := \sum_ \tilde u_i \otimes \tilde v_i, where (u \otimes v) (x) := u \langle v, x \rangle; its image is the linear span of \left\, and the kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute ke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). It is often denoted as \boldsymbol^\mathrm or \boldsymbol^* or \boldsymbol'. H. W. Turnbull, A. C. Aitken, "An Introduction to the Theory of Canonical Matrices," 1932. For real matrices, the conjugate transpose is just the transpose, \boldsymbol^\mathrm = \boldsymbol^\mathsf. Definition The conjugate transpose of an m \times n matrix \boldsymbol is formally defined by where the subscript ij denotes the (i,j)-th entry, for 1 \le i \le n and 1 \le j \le m, and the overbar denotes a scalar complex conjugate. This definition can also be written as :\boldsymbol^\mathrm = \left(\overline\right)^\mathsf = \overline where \boldsymbol^\mathsf denotes the transpose and \overline denotes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Method
Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. The "full" Newton's method requires the Jacobian in order to search for zeros, or the Hessian for finding extrema. Search for zeros: root finding Newton's method to find zeroes of a function g of multiple variables is given by x_ = x_n - _g(x_n) g(x_n), where _g(x_n) is the left inverse of the Jacobian matrix J_g(x_n) of g evaluated for x_n. Strictly speaking, any method that replaces the exact Jacobian J_g(x_n) with an approximation is a quasi-Newton method. For instance, the chord method (where J_g(x_n) is replaced by J_g(x_0) for all iterations) is a simple example. The methods given below for optimization refer to an important subclass of quasi-Newton methods, secant methods. Using methods developed to find extrema in order to fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (linear Algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P : V \to V such that P^2 = P. When V has an inner product and is complete (i.e. when V is a Hilbert space) the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projection on a Hilbert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biconjugate Gradient Stabilized Method
In numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared method (CGS). It is a Krylov subspace method. Unlike the original BiCG method, it doesn't require multiplication by the transpose of the system matrix. Algorithmic steps Unpreconditioned BiCGSTAB To solve a linear system , BiCGSTAB starts with an initial guess and proceeds as follows: # # Choose an arbitrary vector such that , e.g., . denotes the dot product of vectors # # # For ## ## ## ## ## ## ## If is accurate enough, then set and quit ## ## ## ## ## If is accurate enough, then quit ## Preconditioned BiCGSTAB Preconditioners are usually used to accelerat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate tra ... [...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] |
