Showing papers in "Linear Algebra and its Applications in 1980"

TL;DR: It is shown that the method of Arnoldi can be successfully used for solvinglarge unsymmetric eigenproblems and bounds for the rates of convergence similar to those for the symmetric Lanczos algorithm are obtained.

TL;DR: A block biconjugate gradient algorithm for general matrices is developed, and block conjugate gradient, minimum residual, and minimum error algorithms for symmetric semidefinite matrices are developed.

TL;DR: All the results are derived and stated in terms of the recent concept of displacement rank, and this is used to extend the scope of the algorithm to include a wider class of matrices than just Toeplitz and also to include block ToePlitz matrices.

TL;DR: Three generalizations of conjugate-gradient acceleration are described which are designed to speed up the convergence of basic iterative methods which are not necessarily symmetrizable.

TL;DR: In this paper, the modulus, real part, and the imaginary part of a linear combination of the ordered eigenvalues of a square complex matrix are obtained for the trace of a matrix and its square.

TL;DR: Eigenvalues and eigenvectors of a large sparse symmetric matrix A can be found accurately and often very quickly using the Lanczos algorithm without reorthogonalization, although it does not necessarily give the correct multiplicity of multiple, or even single, eigenvalues.

TL;DR: In this paper, the combinational properties of all m × n matrices of 0's and 1's having r i 1's in row i and s i 1s in column j were studied.

TL;DR: In this article, a modified minimal residual (least square) method was proposed for unsymmetric and inconsistent systems of equations, which converges for systems with matrices that have a positive definite symmetric part.

TL;DR: This approach allows full exploitation of sparsity, and permits the use of a fixed (static) data structure during the numerical computation, allowing for the convenient use of auxiliary storage and updating operations.

TL;DR: In this article, the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs, and an application to the computation of vertex-transitive graphs is described.

TL;DR: A variant of Lanczos is proposed which maintains robust linear independence of the Lanczos vectors by keeping them in secondary storage and occasionally making use of them.

TL;DR: In this paper, it was shown that each square centrohermitian matrix is similar to a real (pure imaginary) matrix with real entries, and that skew-centro-hermitians are similar to real-valued centro-symmetric matrix.

TL;DR: In this article, the definition of the Drazin inverse of a square matrix with complex elements is extended to rectangular matrices by showing that for any B and W,m by n and n by m, respectively, there exists a unique matrix, X, such that (B) k =(W) k+1 XW for some positive integer k, XWBWX = X, and BWX =XWB.

TL;DR: In this article, a necessary and sufficient condition for the matrix equation AXB + CYD = E to be consistent is established, and if it is, a representation of the general solution is given, thus generalizing earlier results concerning the equation AX + YB = C.

TL;DR: In this paper, the concept of unique exchange on a sequence of bases of a matroid M is defined, where the unique element of a base may be exchanged for another base such that (Xi − {x}) ∪ {y} and (Xj − {y}) ∫ {x} are both bases.

TL;DR: In this article, a new vector iterate is formed from a linear combination of current iterates and correction vectors that are derived from either gradient or conjugate-gradient techniques.

TL;DR: A class of methods for accelerating the convergence of iterative methods for solving linear systems by replacing the given linear system with a derived one of smaller size, the aggregated system is studied.

TL;DR: In this paper, the authors present an iterative algorithm for solving the discrete maximum entropy problem with linear equality constraints, which has applications in image reconstruction from projections, transportation planning, and matrix scaling.

TL;DR: In this article, the extrapolation principle was used to improve and extend most of the theoretical results concerning the accelerated overrelaxation (AOR) method, as well as to discover some new ones.

TL;DR: In this article, the authors give simple geometric proofs of the known results that, for n ⩽4, n × n nonnegative positive semidefinite matrices can be factored into n×n nonnegative factors and for n ≥ 5, these conditions are not sufficient to guarantee the existence of such a factorization.

TL;DR: A direct method based on an LU decomposition of the rectangular coefficient matrix for the solution of sparse linear least squares problems and a general updating scheme for modifying the solution when extra equations are added is described.

TL;DR: An algorithm for the quadratic programming problem of determining a local minimum of ƒ(x) such that ATx⩾b where Q ymmetric matrix which may not be positive definite is presented.

TL;DR: By using a Perron-Frobenius theory for Boolean matrices, contraction is defined for finite cellular automata as discussed by the authors, which can be seen as a link between results of numerical algebra for systems of equations and the domain of finite iterations.

TL;DR: In this paper, the authors partially synthesize and extend work on finite-vector approximative solutions obtained from nxn northwest corner truncations (n)P of P, from the standpoints of (pointwise convergence) algorithms as n→∞.

TL;DR: In this paper, it was shown that the automorphisms of an n × n matrix algebra over a commutative ring can fail to be inner, but the extent of this failure is under control.

TL;DR: For the n × n complex matrix A with complex eigenvalues λ 1,λ 2,λ n, λ n, this article, upper and lower bounds for Σ(Reλ i ) 2 are obtained, extending similar bounds obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974).

TL;DR: The cyclic-shift tensor-factorization interpolation method recently described by de Boor can be used in particular for least-squares fitting of multivariate data on a rectangular grid and for evaluation of the resulting Tensor-product splines, taking advantage of existing linear algebra and univariate spline software.

TL;DR: In this paper, the authors consider the problem of finding a generalized inverse of a real nonnegative definite matrices A and B such that A > B iff B - > A- > A - ‘.

TL;DR: In this paper, a graph-theoretic approach is used to characterize (0, 1)-matrices which are inverses of M -matrices, and the main result is that a 0, 1 -matrix is an inverse of an M-matrix if and only if its graph induces a partial order on its set of vertices and does not contain a certain specific subgraph.

TL;DR: New bounds are found using matrix canonical forms and nonlinear techniques from commutative algebra to obtain upper bounds on the ranks of third order tensors.