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Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: A survey of computational methods in linear algebra can be found in this article, where the authors discuss the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, and more traditional questions such as algebraic eigenvalue problems and systems with a square matrix.
Abstract: The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).
667 citations
TL;DR: A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described and a probabilistic algorithm is shown to exist for finding the determinant of a square matrix.
Abstract: A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described. The algorithms discussed all require O(n_{1}(\omega + n_{1})\log^{k}n_{1}) field operations, where n_{1} is the maximum dimension of the coefficient matrix, \omega is approximately the number of field operations required to apply the matrix to a test vector, and the value of k depends on the algorithm. A probabilistic algorithm is shown to exist for finding the determinant of a square matrix. Also, probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error.
617 citations
TL;DR: In the present paper, a class of problems where the dispersion matrix has a known structure is considered and the appropriate statistical methods are discussed.
Abstract: In an earlier paper (Rao, 1959), the author discussed the method of least squares when the observations are dependent and the dispersion matrix is unknown but an independent estimate is available. The unknown dispersion matrix was, however, considered as an arbitrary positive definite matrix. In the present paper we shall consider a class of problems where the dispersion matrix has a known structure and discuss the appropriate statistical methods. More specifically the structure of the dispersion matrix results from considering the parameters in the well-known Gauss-Markoff linear model as random variables. Let Y be a vector random variable with the structure
609 citations
TL;DR: This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required.
Abstract: A method for the inversion of a nonsymmetric matrix has been in use at ORNL and has proved to be highly stable numerically but to require a rather large number of arithmetic operations, including a total of N(N-1)/2 square roots. This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required. For N > 4, this is a savings of (N-4)(N-1)/4 square roots. (T.B.A.)
577 citations
TL;DR: A new backward error analysis of the method is given that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency and new rounding error analysis shows the computed Pade approximant of the scaled matrix to be highly accurate.
Abstract: The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in MATLAB's {\tt expm} function. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Pade approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency. We also give new rounding error analysis that shows the computed Pade approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Pade approximant turns out to be 13, rather than the 6 or 8 used by previous authors. Our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than {\tt expm} when the matrix norm exceeds 1, which can amount to a 37% saving in the number of multiplications, and it is typically more accurate, owing to the fewer required squarings. We also investigate a different scaling and squaring algorithm proposed by Najfeld and Havel that employs a Pade approximation to the function $x \coth(x)$. This method is found to be essentially a variation of the standard one with weaker supporting error analysis.
513 citations