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A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices

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A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices Appl. Math. Inf. Sci. 9, No. 2L, 319-324 (2015) 319 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/092L06 A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices Waldemar Hołubowski1, Dariusz Kurzyk1,∗ and Tomasz Trawi´nski2 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, Gliwice 44–100, Poland 2 Mechatronics Division, Silesian University of Technology, Akademicka 10a, Gliwice 44–100, Poland Received: 6 Jul. 2014, Revised: 7 Oct. 2014, Accepted: 8 Oct. 2014 Published online: 1 Apr. 2015 Abstract: We propose an effective method to find the inverse of symmetric block arrowhead matrices which often appear in areas of applied science and engineering such as head-positioning systems of hard disk drives or kinematic chains of industrial robots. Block arrowhead matrices can be considered as generalisation of arrowhead matrices occurring in physical problems and engineering. The proposed method is based on LDLT decomposition and we show that the inversion of the large block arrowhead matrices can be more effective when one uses our method. Numerical results are presented in the examples. Keywords: matrix inversion, block arrowhead matrices, LDLT decomposition, mechatronic systems 1 Introduction thermal and many others. Considered subsystems are highly differentiated, hence formulation of uniform and simple mathematical model describing their static and A square matrix which has entries equal zero except for dynamic states becomes problematic. The process of its main diagonal, a one row and a column, is called the preparing a proper mathematical model is often based on arrowhead matrix. Wide area of applications causes that the formulation of the equations associated with this type of matrices is popular subject of research related Lagrangian formalism [9], which is a convenient way to with mathematics, physics or engineering, such as describe the equations of mechanical, electromechanical computing spectral decomposition [1], solving inverse and other components. As a result of application of eigenvalue problems [2], solving symmetric arrowhead Lagrange formalism, we get the mathematical model systems [3], computing the inverse of arrowhead matrices describing the dynamic of the system. The obtained [4], modelling of radiationless transitions in isolated model is given by second order differential equation, molecules [5,6], oscillators vibrationally coupled with a which can be expressed as matrix equation. In matrix Fermi liquid [6], modelling of wireless communication notation of equation of the modelled system, it is possible systems [7,8]. to distinguish matrix of inertia, whose structure In this paper we propose a simple and effective corresponds to the structure of the real object – the method to find the inverse of the symmetric block mechatronic system. The inertia matrices usually are arrowhead matrices, which have wide applications in symmetric. Additionally, in many cases these matrices mechatronics. In order to illustrate the significance of can be expressed as symmetric arrowhead matrices or arrowhead matrices in process of designing mechatronic symmetric block arrowhead matrices. It is typical of the systems, we should look at this in more details. mathematical models describing the following devices: Colloquially, the mechatronic system is a combination of different elements. It means that devices are made in different technologies which are strongly coupled to each –electromechanical transducers [10,11,12,13]. This other. The systems are built from following main component is included for instance in squirrel-cage subsystems such as: mechanical, electromagnetic, induction motors, where the inertia matrix can be electronic and informatics. Such systems can also have represented as a block arrowhead matrix. The number subsystems associated with pneumatics, hydraulics, of blocks in the matrix depends on the number of ∗ Corresponding author e-mail: [email protected] c 2015 NSP Natural Sciences Publishing Cor. 320 W. Hołubowski et. al. : A Fast Method for Computing the Inverse of... harmonics of the magnetic field in the airgap and the matrix P, the Aˆ can be transformed to number of rotor bars in a cage [12,13]. T –electromechanicaltransducers with a double stator [14, A1 0 0 ··· B1 T 15]. This type of transducers can be successfully used 0 A2 0 ··· B2 as generators in production of wind energy. ˆ T 0 0 A ··· BT A = PAP = 3 3 , (2) –kinematic chains of industrial robots [16]. Depending . . .. on the configuration of the open kinematic chain of an industrial robotic manipulator, the inertia matrix can B1 B2 B3 ··· Ak be expressed as block arrowhead matrix, which represents kinematic relationships between actuators where A1,A2,...Ak are n1,n2,...,nk dimensional matrices, and the elements of the Stewart platform [17]. respectively. In particular if Pˆ is expressed as –head-positioning systems of hard disk drives (HDD) [18,19]. Drivetrain of a head-positioning control 0 0 ··· Ik system can be analysed as a special case of branched . .. Pˆ = . , (3) robotic manipulator [18,19,20]. 0 I2 ··· 0 I 0 ··· 0 The block arrowhead inertia matrices are widely used 1 in modeling of mechatronic systems. Its inverses have where I ,I ,...,I are identity matrices with dimensions important meaning e.g. in: 1 2 k n1,n2,...,nk then –reduction the time of designing and development of ˆ ··· ˆT mentioned device models, A1 0 0 B1 ˆ ˆT –increasing efficiency of simulation of the modelled 0 A2 0 ··· B2 ˆ ˆ ˆT 0 0 Aˆ ··· BˆT systems, A = PAP = 3 3 . (4) –eliminating torsional vibrations in the drive systems. . . .. ˆ ˆ ˆ ˆ Considered inertia matrices depending on the systems can B1 B2 B3 ··· Ak have large sizes, hence there is a need for improvement of methods for block arrowhead matrices inversion. Hence, the A consists of the same blocks Aˆ1,Aˆ2,...,Aˆk and ˆ ˆ ˆ ˆ Generally, matrix inversion is not harder than matrix B1,B2,...,Bk as matrix A. If we consider permutation ˜ multiplication [21,22]. Computational complexity of matrix P expressed as matrix inversion based on Gauss-Jordan elimination is O(n3) [23]. Strassen algorithm, which can be used to 00 ··· 1 inverse of matrix with complexity O(n2.8074)[21], is more . .. P˜ = . , (5) efficient. Coppersmith and Winograd [24,25] show that 01 ··· 0 O 2.3755 matrix multiplication can be obtained in (n ). Now, 10 ··· 0 the fastest algorithm of matrix multiplication running in O(n2.3727) time was performed by Williams [26]. Two then entries of A˜ = P˜AˆP˜T are given by A˜ = Aˆ , last mentioned algorithms are rarely used in practice. i, j n−i+1,n− j+1 where n is a size of Aˆ. The form of a matrix given by (1) occurs many times in different cases e.g. during designing the mechatronic 2 Inverse of block arrowhead matrix system. Blocks of the matrix are related with the structure of a modelled object. Hence the blocks should be invariant under transformation of the matrix. Expressed in 2.1 Arrowhead matrix (3) changes the structure of the matrix, but the blocks remain unchanged. Let Aˆ be a square block matrix given in the following way ∗ Aˆ1 Bˆ1 Bˆ2 ··· Bˆk−1 2.2 LDL decomposition ˆT ˆ B1 A2 0 ··· 0 ˆ BˆT 0 Aˆ ··· 0 Let A be a Hermitian positive definite matrix. A = 2 3 , (1) . .. Decomposition the A into the product of a lower . triangular matrix and its conjugate transpose is called ˆT ˆ Bk−1 0 0 ··· Ak Cholesky decomposition. Thus, every Hermitian matrix can be expressed as where n1,n2,...,nk are dimensions of matrices ˆ ˆ ˆ ∑k ∗ A1,A2,...,Ak and i=1 ni = n. By use of permutation A = LL , (6) c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 9, No. 2L, 319-324 (2015) / www.naturalspublishing.com/Journals.asp 321 where L is a lower triangular matrix. If A is a symmetric where for 1 ≤ i ≤ n − 1 positive definite matrix, then A can be factorized into A = T n−1 LL . If we consider (6) in following way bi d = a , l = and d = a − ∑ l2d . (12) i i i d n n k k ∗ ′ 1 ′ 1 ′ 1 1 ∗ ′ ∗ ′ ′ ∗ i k=1 A=LL =L D 2 (L D 2 )=L D 2 (D 2 ) (L ) =L D(L ) , (7) O ′ The computationalcomplexity of the factorization is (n). where L is a lower triangular matrix and D is a diagonal n×n ∗ Let A = [Ai j] ∈ R be a symmetric block arrowhead matrix, then we get LDL decomposition. This variant of matrix expressed as matrix in (2). The A can be Cholesky decomposition is also useful for the Hermitian decomposed into a products of matrices L, D and LT , nonpositive matrix A. For real symmetric matrices, the where factorization has the form A = LDLT . The computational 3 complexity of Cholesky decomposition is O(n ) and the D1 0 0 ... 0 F1 0 0 ... 0 most efficient algorithms used for the factorization 0 D2 0 ... 0 0 F2 0 ... 0 require 1 n3 operations. The complexity of LDL∗ 6 D = 0 0 D3 ... 0 L = 0 0 F3 ... 0 . (13) decomposition is the same as Cholesky decomposition. . . n×n . .. . .. Let A = [ai j] ∈ C be the Hermitian matrix. The LDL∗ decomposition factorizes A into a lower triangular 0 0 0 ... Dn L1 L2 L3 ... Fk ∈ n×n matrix L = [li j] C , a diagonal matrix T ∈ n×n Matrices Fi and Di are obtained as results of LDL D = [di j] C and conjugate transpose of L expressed −1 T −1 as A = LDL∗, where decomposition of a matrix Ai and Li = Bi(Fi ) Di , where 1 ≤ i ≤ k − 1. Next, Fk and Dk are results of the i−1 ˜ ∑k−1 T factorization of a matrix Ak = Ak − i=1 LiDiLi .
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