Article

An improved method for the computation of the Moore–Penrose inverse matrix

Applied Mathematics and Computation (Impact Factor: 1.55). 08/2011; DOI: 10.1016/j.amc.2011.04.080
Source: arXiv

ABSTRACT

In this article we provide a fast computational method in order to calculate the Moore–
Penrose inverse of singular square matrices and of rectangular matrices. The proposed
method proves to be much faster and has significantly better accuracy than the already
proposed methods, while works for full and sparse matrices.

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Available from: Vasilios N Katsikis
    • "The singular value decomposition (SVD) algorithm is the most known between the direct methods [1]. Also, other types of matrix factorizations have been exploited in computation of generalized inverses, such as the QR decomposition [5] [6], LU factorization [7]. Methods based on the application of the Gauss-Jordan elimination process to an appropriate augmented matrix were investigated [8] [9]. "
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    ABSTRACT: We propose two continuous-time neural networks for computing generalized inverses of complex-valued matrices with rank-deficient cases.The first of them is applicable in the pseudoinverse computation and the second one is applicable in construction of outer inverses. The proposed continuous-time neural networks have a low complexity of implementation and they are proved to be globally convergent withut any condition. Compared with the existing algorithms for computing the pseudoinverse and outer inverses of matrices, the global convergence of the proposed continuous-time neural networks is analyzed in the complex domain. Effectiveness of the proposed continuous-time neural networks is evaluated numerically via examples.
    No preview · Article · Jan 2016 · Applied Mathematics and Computation
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    • "where ψ † k is the pseudoinverse or the Moore–Penrose inverse [40] of ψ k . Equation (12) can also be solved by the recursive least squares method [41]. "
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    ABSTRACT: The emergence of smart grids has posed great challenges to traditional power system control given the multitude of new risk factors. This paper proposes an online supplementary learning controller (OSLC) design method to compensate the traditional power system controllers for coping with the dynamic power grid. The proposed OSLC is a supplementary controller based on approximate dynamic programming, which works alongside an existing power system controller. By introducing an action-dependent cost function as the optimization objective, the proposed OSLC is a nonidentifier-based method to provide an online optimal control adaptively as measurement data become available. The online learning of the OSLC enjoys the policy-search efficiency during policy iteration and the data efficiency of the least squares method. For the proposed OSLC, the stability of the controlled system during learning, the monotonic nature of the performance measure of the iterative supplementary controller, and the convergence of the iterative supplementary controller are proved. Furthermore, the efficacy of the proposed OSLC is demonstrated in a challenging power system frequency control problem in the presence of high penetration of wind generation.
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    • "A large number of different methods for computing generalized inverses are available in the literature. Direct methods are usually based on SVD (Singular Value Decomposition), QR factorization [4], Gaussian elimination [11] [13], etc. On the other hand, there are certain iterative methods, mainly based on the appropriate generalizations of the wellknown hyper-power method and the Schultz method as its particular case. "
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    ABSTRACT: In this paper, we propose new iterative schemes for the computation of outer inverse which reduce the total number of matrix multiplications per iteration. In particular, we consider how the hyper-power method of orders 5 and 9 can be accelerated such that they require 4 and 5 matrix multiplications per iteration, respectively. These improvements are tested against quadratically convergent Schultz’ method and fastest Horner scheme hyper-power method of order three. Numerical results show the superiority and practical applicability of the proposed methods. Finally, it is shown that a possibly more efficient method should have the order at least , making it useless for practical applications.
    Full-text · Article · Apr 2015 · Journal of Computational and Applied Mathematics
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