Fast Algorithms for Verifying Centrosymmetric Solutions of Sylvester Matrix Equations
Based on floating point operations, we study the accuracy of numerically computed centrosymmetric solutions in Sylvester matrix equations. Propose a fast algorithm which outputs the lower bound and upper bound of the exact centrosymmetric solution. Numeri
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College of Economics and Management, Shanghai Maritime University, Shanghai 201306, China College of Mathematics and Statistics, Beihua University, Jilin 132013, China [email protected]
Abstract. Based on floating point operations, we study the accuracy of numerically computed centrosymmetric solutions in Sylvester matrix equations. Propose a fast algorithm which outputs the lower bound and upper bound of the exact centrosymmetric solution. Numerical experiments show the properties of the proposed algorithm.
Keywords: Fast algorithm metric solution · INTLAB
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Sylvester matrix equation
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Introduction
The numerical computation plays a crucial role in scientific calculation. It has the characteristics of fast and it is used widely in our practical work. However, due to the raw data error, calculation error accumulation and the limited precision said real numbers, the numerical calculation cannot guarantee the accuracy of results, which could lead to a major accident in the high risk field [1]. Therefore, reliable computing is widely applied across thees high risk areas such as rocket design, nuclear magnetic resonance (NMR) machine and digital machine theory. And in many important fields such as biomathematics, mechanics, physics and control theory [2–4], some problems can be come down to compute a centrosymmetric solution of the Sylvester matrix equation AX + XB = C.
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In this paper, the accuracy of numerically computed centrosymmetric solutions in the Sylvester equation (1) is concerned. We investigate the methods for computing the lower bound and upper bound of the exact centrosymmetric solution where A ∈ Cm×m , B ∈ Cm×m , X, C ∈ Cm×n . X, While there are well-established methods for enclosing solutions of Sylvester matrix equations [5–8], less attention has been paid to centrosymmetric solutions. It is well known Eq. (1) can be written as a system of linear equations as follow P vec(X) = vec(C), (2) c Springer Nature Singapore Pte Ltd. 2016 M. Gong et al. (Eds.): BIC-TA 2016, Part II, CCIS 682, pp. 524–529, 2016. DOI: 10.1007/978-981-10-3614-9 65
Fast Algorithms for Verifying Centrosymmetric Solutions
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where P = In ⊗ A + B T ⊗ Im , ⊗ represent the Kronecker product [9] and vec is the operation of stacking the columns of a matrix in order to obtain one long vector. The purpose of this paper is to obtain the error bounds X ε ∈ Rm×n satisfying and X denote the numerical solution and exact solution − X| ≤ X ε , where X |X − X) ij | and inequalities in (1), |X − X| denotes the matrix with elements |(X between matrices hold componentwise. This method requires only O(m3 + n3 ) operations if A and B are diagonalizable. This paper is organized as follows. In Sect. 2 we introduce the preliminary definitions and notation we shall use. The main result, the algorithm for verifying the centrosymmetric solution of Sylvester matrix equations, is presented in Sect. 3. In Sect. 4 we provide some examples for demonstrating the performance of our algorithm.
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Notation an
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