R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The scaled Newton iteration for sign(B)hastheform # k X k # 1 X 1 , X 0=B , 3.1) which corresponds to replacing X k by # k X k , for some scaling parameter # k ,atthe start of each iteration. We consider just one of the various scaling schemes: the determinantal scaling of Byers [6], which takes det(X k ) 1 n . This scaling parameter is easily formed during the computation of X 1 k . To translate the scaling to the DB iteration (1.3) for the square root, we use the connection between these two iterations established in section 2. From (2.5) we have det(X k ) ....

R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Linear Algebra Appl. 85 (1987) 267--279.

....which will play a dominant role in our algorithms presented in next sections. 3 LEFT INVARIANT SUBSPACE COMPUTATION VIA MATRIX SIGN ITERATIONS In the past, the matrix sign function has been used to solve some important equations in control theory, such as Riccati or Lyapunov equations [15] 45] [9], 20] employing left invariant subspace computations. In this paper, we will use the same concept in another field, namely applied probability, in the context of numerical solutions of the nonlinear matrix equations of the form (4) and (7) We first provide some introductory material regarding ....

.... Gamma ff k )Z k ; 18) 1 2 (fl k Z k fl k Z k ) 19) where ff k and fl k are appropriately chosen scalars. Two simple scaling schemes based on the determinant of the intermediate matrices which have proven to be useful are Balzer [6] ff k = j det Z k j 1=m 1) 20) Byers [9] : fl k = j det Z k j Gamma1=m : 21) A comparison of different scaling schemes for different ff k is presented by Balzer [6] and for different fl k by Kenney and Laub [31] In the implementations of the scaling schemes (20) and (21) we note that the determinant 11 det Z k can be calculated ....

R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987. 35

....we then devise algorithms for M G 1 and G M 1 type Markov chains based on the matrix sign function iteration approach. Although several methods for finding the sign of a matrix exist in the literature, we focus on the generally accepted Newton s iteration with a particular scaling proposed in [5]. These iterations are globally quadratically convergent and the computational load at each iteration consists of only one matrix inversion and one matrix addition. We note that there are also variants of matrix sign function iteration algorithms that have a convergence rate of arbitrary order ....

R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987.

....k = 0; 1; 2; 19) Under the given assumptions for the spectrum of Z, the sequence fZ k g 1 k=0 converges to sign(Z) 40] Although the convergence of the Newton iteration is globally quadratic, the initial convergence may be slow. Acceleration is possible, e.g. via determinantal scaling [15], Z k c k Z k ; c k = j det (Z k )j 1 n ; where det (Z k ) denotes the determinant of Z k . Other acceleration schemes can be employed; see [4] for a comparison of these schemes. Roberts [40] was the rst to use the matrix sign function for solving Lyapunov (and Riccati) equations. In the ....

R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl., 85:267-279, 1987.

....e = W fl fl Note that all the eigenvalues of W and W e are the same except W has one eigenvalue at the origin which has moved to one for W e . 4. Find matrix sign of W e using the matrix sign iterations, i.e. S = sgn(W e ) The basic sign function algorithm for an m Theta m matrix M is (see [13] and [14] Z 0 = M; Z k 1 = 1 2ck (Z k c k Z Gamma1 k ) c k = j det(Z k )j 1=m Then lim k 1 Z k = Z = sgn(M) and the convergence is quadratic. The stopping criterion is jjZ k 1 Gamma Z k jj 1 ffljjZ k jj 1 with a small, user specified error bound ffl. 5. Find a rank revealing ....

R. Byers, "Solving the algebraic Riccati equation with the matrix sign function," Lin. Alg. Appl., vol. 85, pp. 267--279, 1987.

....T 1 spans and write X = U 12 U Gamma1 11 . Since the major tool is the computation of an invariant subspace, all these methods are referred to invariant subspace methods. These methods include Schur methods [28, 29] iterative refinement techniques [21] and the matrix sign function algorithms [8, 10, 20, 24]. It is the invariant subspace approach that has proved to be the most suitable approach for solving the ARE among the three basic approaches mentioned above [29] due to its numerical efficiency and reliability features. As shown above, the solution of algebraic Riccati equations and QBD chains ....

....the eigenvectors of the original matrix (see Figure 2) This feature of the matrix sign function makes it computationally attractive for studying the eigenstructures of matrices without explicitly computing the eigenvalues. The Newton iteration below [29] with the determinantal scaling proposed by [10] S 0 = M S k 1 = 1 2c k (S k c 2 k S Gamma1 k ) c k = j det(S k )j 1=n is known to yield lim k 1 S k = S = sgn(M) It is known that the above iteration is ultimately quadratically convergent [8] 10] For a review of existing algorithms for the matrix sign, we refer the reader ....

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R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987.

.... of the stable invariant subspace of H in (2) These methods include the Schur vector method of Laub [15] the Hamiltonian QR algorithm of Byers [8] the SR algorithm of Bunse Gerstner and Mehrmann [5] the HHDR algorithm of Bunse Gerstner and Mehrmann [6] and the matrix sign function method [9]. Unlike earlier attempts based on the computation of eigenvectors of the Hamiltonian matrix H , Laub s method uses the numerically reliable QR algorithm to compute the desired invariant subspace. This Schur vector method is numerically backwards stable and of complexity O(n 3 ) However, this ....

R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Lin. Alg. Appl. 85 (1987), pp. 267--279.

.... its inverse: G 0 = H G k 1 = G k (1=2) G k ) Gamma1 Gamma G k ] k = 0; 1; 4:7) To speed up convergence, Gardiner and Laub (1986) suggest using the recursion G 0 = H; G k 1 = 1=2ffl k ) G k ffl k 2 G k Gamma1 ) where ffl k = j det G k j 1=n : 4:8) Bierman (1984) and Byers (1987) propose a further refinement, which exploits the fact that the matrix G k is a Hamiltonian matrix for each k. Recall that if H is a Hamiltonian matrix, then JH is symmetric where J = 0 GammaI I 0 : Hence JG k 1 = 1 2ffl k (JG k ffl k 2 JJG k Gamma1 J) 4:9) where ffl k is ....

Byers, R. (1987). `Solving the Algebraic Riccati Equation with the Matrix Sign Function'. Linear Algebra and its Applications, Vol. 85, pp. 267--279.

....Lyapunov equation (B = A ) in the common case where A is positive stable, that is, Re i (A) 0 for all i. The matrix sign function was the subject of a steady stream of papers throughout the 1970s and 1980s. It has been widely used to solve the algebraic Riccati equation; see, for example, [8], and see [36] for a survey. It has also been applied to eigensystem computations (see, for example, 5] 11] 28] though some of the proposed algorithms are of dubious computational merit. Recently there has been a resurgence of interest in the matrix sign function because of its suitability ....

Ralph Byers. Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra and Appl., 85:267--279, 1987.

.... G k 1 = G k (1=2) G k ) Gamma1 Gamma G k ] k = 0; 1; sign ] 4:7) To speed up convergence, Gardiner and Laub (1986) suggest using the recursion G 0 = H; G k 1 = 1=2ffl k ) G k ffl k 2 G k Gamma1 ) where ffl k = j det G k j 1=n : lars16 ] 4:8) Bierman (1984) and Byers (1987) propose a further refinement, which exploits the fact that the matrix G k is a Hamiltonian matrix for each k. Recall that if H is a Hamiltonian matrix, then JH is symmetric where J = 0 GammaI I 0 : Hence JG k 1 = 1 2ffl k (JG k ffl k 2 JJG k Gamma1 J) lars17 ] 4:9) where ffl ....

Byers, R. (1987). `Solving the Algebraic Riccati Equation with the Matrix Sign Function'. Linear Algebra and its Applications, Vol. 85, pp. 267--279.

....Road, Oxford, OX2 8DR. nagjdc vax.oxford.ac.uk) z Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK. mbbgsnh cms.mcc.ac.uk) 1 Nevertheless, there are some applications that genuinely require computation of a matrix inverse see [1, sec. 7.5] 14, p. 342ff] and [4, 10] for example. LAPACK [3] like LINPACK before it, will include routines for matrix inversion. LAPACK will support inversion of triangular matrices and of general, symmetric indefinite, and symmetric positive definite matrices via an LU (or related) factorization. Each of these matrix inversions ....

R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Linear Algebra and Appl., 85 (1987), pp. 267--279.

....more choices of the parameters. Underlined values indicate the default values. For each example, we provide norms and condition numbers of the stabilizing solution X and the Hamiltonian matrix H. A large condition number of H may signal that one can expect problems using the sign function method [17, 24, 45] since the underlying Newton iteration depends on inversion of H. On the other hand, a large condition number may also be due to a large norm of H rather than to ill conditioning with respect to inversion as in Example 4.4. If no analytical solution is available, we computed approximations by the ....

....despite the scaling of the output matrix. Without the scaling corresponding to WC , these values are even larger by about 10 orders of magnitude. This is due to the large entries in A, i.e. the large values j . The reference solution was computed by the sign function method as proposed in [17] where the defect correction was performed using Newton s method combined with exact line search [9] Due to the bad scaling of this example, it was necessary to scale the Lyapunov equation (5) by jjAjj 1 when computing KU . Acknowledgments We would like to express our gratitude to the co authors ....

R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl., 85:267--279, 1987.

....the convergence of the Newton iteration is globally quadratic, the initial convergence may be slow. Several quasi optimal acceleration schemes have been proposed to speed up the initial convergence. Optimal schemes require complete knowledge of (A) Among these, the determinantal scaling [11] is usually preferred because of its efficiency and simplicity. Specifically, when using the determinantal scaling, iteration (3) becomes A k 1 = 1 2 i A k =fl k fl k A Gamma1 k j ; A 0 = A; Q k 1 = 1 2 i Q k =fl k fl k A GammaT k Q k A Gamma1 k j ; Q 0 = Q; 4) 4 with ....

R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl., 85:267--279, 1987.

....[13] Such error bounds are implemented in the Fortran 77 solvers ricc and ricm used in the comparison done in this report. The solver ricc implements the Schur method [16, 17] with block scaling [21] which enhances the numerical stability while ricm implements the matrix sign function method [22, 5, 15, 9, 23]. Both solvers make use of LAPACK [1] Release 3.0, and are implemented as mex files which makes them easily accessible from MATLAB. Note that the Riccati solvers implemented in MATLAB do not produce condition estimates and forward error estimates. 3 Numerical experiments In this section we ....

R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl., 85:267--279, 1987.

....invariant subspace, deflating subspaces AMS subject classifications. 65F15, 65F35, 65F30, 15A18 PII. S0895479896297719 1. Introduction. Since the matrix sign function was introduced in the early 1970s, it has been the subject of numerous studies and used in many applications. For example, see [30, 31, 11, 26, 23] and references therein. Our main interest here is to use the matrix sign function to build parallel algorithms for computing invariant subspaces of nonsymmetric matrices, as well as their associated eigenvalues. It is a challenge to design a parallel algorithm for the nonsymmetric eigenproblem ....

<F3.759e+05> R.<F3.851e+05> Byers,<F3.469e+05> Solving the algebraic Riccati equation with the matrix sign<F3.851e+05> function, Linear Algebra Appl., 85 (1987), pp. 267--279.

....G M 1 type Markov chains, one based on the matrix sign function iteration approach, and the other based on the Schur decomposition. Although numerous approaches for finding the sign of a matrix exist in the literature, we focus on the generally accepted Newton iteration with scaling proposed by [11]. The significance of this iteration is that it is globally quadratically convergent and the computational load at each iteration consists of only one matrix inversion and one matrix N. Akar and K. Sohraby, On Computational Aspects of the Invariant Subspace Approach 8 addition (of size mf Theta ....

....and Z k 1 = 1 2 (fl k Z k fl Gamma1 k Z Gamma1 k ) where ff k and fl k are appropriately chosen scalars. Two simple scaling schemes based on the determinant of the intermediate matrices which have proven to be useful are Balzer [9] ff k = j detZ k j 1=m 1) Gamma1 ; 20) Byers [11] : fl k = j det Z k j Gamma1=m : 21) A comparison of different scaling schemes for different ff k is presented by Balzer [9] and for different fl k by Kenney and Laub [28] Unless otherwise stated, we will focus on the numerical experimentation of the scaling scheme (21) since it is widely ....

R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987.

....in [3] for general M G 1 and G M 1 paradigms, we further show that the two matrix geometric factors R 1 and R 2 in the expression (11) are related to the left and right invariant subspaces of one square real matrix of size 2m, respectively. We also provide a particular matrix sign iteration [6] with quadratic convergence rates to compute a basis for these two subspaces, which in turn gives the two matrix geometric factors. The core of the algorithm lies in the matrix sign iterations for finding the associated subspaces, each iteration of which has a time complexity of O(m 3 ) and ....

....or LAPACK [4] The above definition for the matrix sign does not lend itself to an efficient computation but there are several ways of evaluating the matrix sign function. A numerically efficient iteration scheme is an extension of Newton s method applied to sgn(M) 2 = I with scaling given in [6]: Set Z 0 = M and Z k 1 = 1 2 (fl k Z k fl Gamma1 k Z Gamma1 k ) fl k = j det Z k j Gamma1=m : 43) Then lim k 1Z k = Z = sgn(M) and the convergence is quadratic [6] A comparison of different scaling schemes for matrix sign function iterations is beyond the scope of this paper. ....

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R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987.

....once an arbitrary basis for each of the two subspaces is found. As the engine for simultaneous computation of these subspaces, which turns out to be the core of the algorithm, we propose to use either Schur decomposition [19] or a matrix sign iteration algorithm with a quadratic convergence rate [13]. The advantages of the proposed method can be summarized as follows. i) The method is simple to implement, efficient, and does not require much storage: it is feasible to implement and use it on small computers even for large scale QBD processes. ii) The number of levels, K 1, does not have ....

.... stability [28] 19] the former one is promising especially for large scale problems as it is amenable to parallel implementation [35] Both methods have also been successfully used in the past to solve important equations in control theory, such as matrix Riccati and Lyapunov equations [37] [13], 17] 28] Here, given the matrix E T m , we aim at simultaneous computation of the matrix geometric factors R 1 and R 2 through finding either the matrix sign or the Schur decomposition of E T m . The two methods are introduced next in that spirit. Akar, Oguz, and Sohraby, A Novel ....

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R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987.

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R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987.

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R. Byers, "Solving the algebraic Riccati equation with the matrix sign function", Linear Algebra Appl., vol. 85, pp. 267--279, 1987.

....function as Z 0 Z; FOR j = 0; 1; 2; Z j 1 1 2 (c j Z j (c j Z j ) Gamma1 ) 27) where c j 0 is a scalar acceleration parameter. This iteration converges globally and quadratically to sign (Z) lim j 1 Z j [42] For our implementations we choose the determinantal scaling [17] given by c j = j det (Z j )j Gamma 1 n . Many other iterative methods to compute the matrix sign function are known [32] Among these, the Halley iteration Z 0 Z; FOR j = 0; 1; 2; Z j 1 Z j (3I n Z 2 j ) I n 3Z 2 j ) Gamma1 ; 28) and the Newton Schulz iteration, Z ....

R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Linear Algebra Appl., 85:267--279, 1987.

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R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267--279, 1987.

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R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Lin. Alg. Appl. 85 (1987), pp. 267--279.

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R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Lin. Alg. Appl. 85 (1987), 267--279.

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IL. #7# R. Byers. Solving the algebraic Riccati equation with the matrix sign function. Lin. Alg. Appl., 85:267#279, 1987.

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