A Divide-and-Conquer Algorithm for Functions of Triangular Matrices
نویسنده
چکیده
We propose a divide-and-conquer algorithm for computing arbitrary functions of upper triangular matrices, which requires approximately the same number of arithmetic operations as Parlett’s algorithm. However, the new algorithm has better performance on computers with two levels of memory due to its block structure and thus, less memory-cache traffic requirements. Like Parlett’s algorithm, the new algorithm also requires that the eigenvalues (main diagonal elements) of the input matrix be distinct, and computes the matrix function nearly as accurately.
منابع مشابه
Free Vibration Analysis of Repetitive Structures using Decomposition, and Divide-Conquer Methods
This paper consists of three sections. In the first section an efficient method is used for decomposition of the canonical matrices associated with repetitive structures. to this end, cylindrical coordinate system, as well as a special numbering scheme were employed. In the second section, divide and conquer method have been used for eigensolution of these structures, where the matrices are in ...
متن کاملSuperfast solution of linear convolutional Volterra equations using QTT approximation
We address a linear fractional differential equation and develop effective solution methods using algorithms for inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini’s algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula f...
متن کاملAnalysis of a QR Algorithm for Computing Singular Values
We extend the Golub-Kahan algorithm for computing the singular value decomposition of bidiagonal matrices to triangular matrices R. Our algorithm avoids the explicit formation of R T R or RRT. We derive a relation between left and right singular vectors of triangular matrices and use it to prove monotonic convergence of singular values and singular vectors. The convergence rate for singular val...
متن کاملComputational method based on triangular operational matrices for solving nonlinear stochastic differential equations
In this article, a new numerical method based on triangular functions for solving nonlinear stochastic differential equations is presented. For this, the stochastic operational matrix of triangular functions for It^{o} integral are determined. Computation of presented method is very simple and attractive. In addition, convergence analysis and numerical examples that illustrate accuracy and eff...
متن کاملEfficient parallel and incremental parsing of practical context-free languages
We present a divide-and-conquer algorithm for parsing context-free languages efficiently. Our algorithm is an instance of Valiant’s (1975), who reduced the problem of parsing to matrix multiplications. We show that, while the conquer step of Valiant’s is O(n), it improves to O(log n) under certain conditions satisfied by many useful inputs, and if one uses a sparse representation of matrices. T...
متن کامل