P. Benner and E.S. Quintana-Ort. Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms, 20(1):75-100, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....solving the generalized Lyapunov equation (6.10) The numerical solution of the standard Lyapunov equation has been studied in numerous publications (see, e.q. 2, 29] and the references therein) Numerical methods for the generalized Lyapunov equation with nonsingular A have been considered in [3, 20, 21, 45]. However, the case of singular A is more complicated, since the solution of the generalized Lyapunov equation is not unique. We need the special solution Z of (6.10) namely, such that Z = Z Pi. In the next section we present an algorithm for computing the projections P , Pi and the desired ....

....as Sigma Z 11 F F Z 11 Sigma = GammaG; 7.9) where F = B 11 Gamma B 12 B Gamma1 22 B 21 and G = I B 21 B Gamma 22 B Gamma1 22 B 21 . This equation with nonsingular Sigma can be solved using the generalized Bartels Stewart algorithm [20, 21, 45] or the sign function method [3]. Note that for computing the matrix G we have to multiply the matrices B 21 B Gamma 22 and B Gamma1 22 B 21 . This may lead to a larger sensitivity, in the worst case the condition number may be squared. In fact, this multiplication is not necessary. The matrix G can be represented as ....

P. Benner, E.S. Quintana-Ort'i. Solving Stable Generalized Lyapunov Equations with the matrix Sign Function. -- Numerical Algorithms 20, No. 1, 1999, pp. 75-100.

....equations (the latter is also known as the Stein equation) The theoretical analysis, numerical solution and perturbation theory for these equations has been the topic of numerous publications, see [1, 13, 16, 17, 27] and the references therein. The case of nonsingular E has been considered in [3, 28]. However, many applications in singular systems or descriptor systems [9] lead to generalized Lyapunov equations with a singular matrix E, see [2, 21, 24, 32, 31] The solvability of the generalized Lyapunov equations (1.1) and (1.2) can be described in terms of the generalized eigenstructure of ....

P. Benner and E.S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Alg., 20(1):75--100, 1999. 36

....system solvers, and matrix products, which are highly efficient on current high performance parallel distributed architectures. Although these methods cannot be considered as numerically stable, the numerical results obtained in practice are close to those obtained by means of QR type solvers [8]. In this paper we show that by coding these new algorithms with PLAPACK [41] and using the given off the shelf kernels in this library, it is easy to obtain parallel iterative distributed solvers for these matrix equations. Furthermore, we also show that the flexible infrastructure provided by ....

....maximum attainable accuracy. The numerical solution of Lyapunov equations by means of the matrix sign function cannot be considered a numerically stable procedure; in fact, the numerical stability depends on the distance from the eigenspectrum of A to the imaginary axis. However, recent studies [8, 12] show that the matrix sign function approach, with careful shifts and scaling, can obtain numerical results which are close in quality to those obtained by means of the numerically stable QR type algorithms. The Newton iterative scheme has a cost of 6n 3 flops per iteration. In practice, 7 10 ....

P. Benner and E.S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms, 20(1):75--100, 1999.

....products X A i = Z A i Z A i T and X il = Z il Z T il ; 13) respectively. This is always possible because the iterates X A i and X il can be shown recursively to be symmetric and positive semidefinite. Although a similar approach is pursued in several methods for Lyapunov equations, e.g. [AL93, BQO97, Ham82, HR92, JK94, Saa90], this has not been done in combination with ADI or Smith like methods. LR ADI is based on the ADI single sweep (3) Using (13) this formula can be rewritten in terms of the matrices Z A i as Z A i = h (A T Gamma p i I) A T p i I) Gamma1 Z A i Gamma1 p Gamma2p i (A T p i I) ....

P. Benner and E. Quintana-Orti. Solving stable generalized Lyapunov equations with the matrix sign function. Preprint SFB97-23, Technische Universitat Chemnitz, Chemnitz, Germany, 1997.

....algorithm requires 84n 3 flops. 4. Subsequent spectral divisions with the matrix sign function become much cheaper as they operate on smaller matrices. 5. The serial QZ algorithm can still be applied locally when the problem is decoupled into a sufficiently small subproblem. 6. Some applications [4,5,24] do not require all the information provided by the QZ algorithm but only bases for specific deflating subspaces. 3. Accurate Subspace Extraction with the Generalized Newton Iteration The accuracy of the generalized Newton iteration depends, among other factors, on the condition number of A and ....

P. Benner and E. S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Technical Report SFB393/97-23, TU ChemnitzZwickau, 1997.

....di er in several ways from the ones considered in [41, 45] though they are mathematically equivalent. In particular we focus on the implementation on modern computer architectures such as distributed memory parallel computers. The Lyapunov equation (4) is solved using a sign function based method [11, 40] from which we obtain a full rank factorization PG = S T S, i.e. S 2 R nc n where n c = rank (S) rank (PG ) This is usually faster and more reliable than computing the Cholesky factor using Hammarling s method [28] see [12] for details. The same approach is used to compute a full rank ....

....as their condition number can be up to the square of the condition number of the Cholesky factors. In these algorithms equations (4) and (5) are initially solved for the Cholesky factors without ever forming the Gramians explicitly. This can be achieved, e.g. by the algorithms described in [11, 28] applied to (4) and (9) Then, the SVD of the product SR T = U 1 U 2 ] 1 0 0 2 V T 1 V T 2 (10) is computed. Here, the matrices are partitioned at a given dimension r, with 1 = diag ( 1 ; r ) and 2 = diag ( r 1 ; n ) such that 1 2 : ....

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P. Benner and E.S. Quintana-Ort. Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms, 20(1):75-100, 1999.

....THE GRAMIANS All the model reduction methods in the previous section require, as a first stage, the solutions (or their Cholesky factors) of two Lyapunov equations. In this section we describe Lyapunov equation solvers based on the matrix sign function. Details of the algorithms can be found in [4]. The algorithms are specially appropriate for parallel distributed memory computers. Here we describe in particular those modifications needed for computing full rank factors of the Gramians. Roberts [16] was the first to use the matrix sign function for solving Lyapunov (and Riccati) equations. ....

....Q k c 2 k (A Gamma1 k ) T Q k A Gamma1 k Delta = 2c k ) Here, c k is a scaling parameter used to accelerate convergence in the early steps of the iteration. At convergence, W c = P1 =2 and W o = Q1 =2. Efficient convergence criteria for these iterations have been proposed in [4]. As A is a stable matrix, A1 = lim k 1 A k = GammaI n , and a suitable convergence criterion for the iterations is to stop when the relative error in the A iterates drops below a tolerance threshold, i.e. if kA k I n k1 kA k k1 for a user defined tolerance . In [4, 13] iteration (9) was ....

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P. Benner and E.S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms, 20(1):75--100, 1999.

....this stopping criterion is satisfied, usually two additional iterations are enough to reach the attainable accuracy due to the quadratic convergence of Newton s method close to the solution. Moreover, this approach avoids possible stagnation of the method due to an un attainable stopping criterion [8,12]. A more reliable stopping criterion can be based on the following observations. It is well known [41] that the Newton iteration is able to improve the relative 12 accuracy of an approximate solution X j of the CARE as long as kX Gamma X j k K CARE kX k: This may be used as follows: once ....

.... det(M)j) 1 n L j 1 (L j c 2 j ML Gamma1 j M) 2c j ) 33) The iterates satisfy lim j 1 L j = Msign (M Gamma1 L) In case oe ( E; A) ae C Gamma , we can apply iteration (33) to the matrix pencil corresponding to (2) and obtain the following iteration which is derived in [8,12] A 0 A, Q 0 Q, FOR j = 0; 1; 2; A j 1 i A j E A Gamma1 j E j =2, Q j 1 i Q j E T A GammaT j Q j A Gamma1 j E j =2. 34) The cost per iteration is O( 26 3 n 3 ) flops, and the solution Y is obtained from ....

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P. Benner and E.S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms, 20(1):75--100, 1999.

....of the spectrum of (A; B) In case the complete spectrum is desired, the matrix sign function can be used as an initial spectral divide and conquer technique. On the other hand, several applications in linear control systems design only require an orthonormal base for a certain deflating subspace [2,7,10,14, 17,24]. The matrix sign function was first defined by Roberts [24] as sign(A) def = Y GammaI r 0 0 I n Gammar Y Gamma1 ; 2) X. Sun was supported in part by the Advanced Research Projects Agency, under contract P 95006. E. S. Quintana received support from the CICYT project TIC ....

P. Benner and E. S. Quintana-Ort' i, Solving stable generalized Lyapunov equations with the matrix sign function, Tech. Report SFB393/97-23, Faculty of Mathematics, Technische Universitat Chemnitz-Zwickau, 1997. To appear in Numer. Alg.

...., C = C T , and X 2 R n Thetan is the sought after solution. It easily follows that if there exists a unique solution to (1) then this solution has to be symmetric. Stein equations play a fundamental role in linear control and filtering theory for discrete time systems (see references in [3]) Throughout this paper we assume (1) to be Schur stable, that is, if ae(A) denotes the spectral radius of A, then ae(A) 1. It is well known that under this assumption, the Stein equation (1) has a unique solution; see, e.g. 7] Schur stable Stein equations appear in many computational ....

....computational kernels. These are the squared Smith iteration and the sign function method applied to the Lyapunov equation resulting from a Cayley transformation of (1) The Smith iteration will be reviewed in Section 2. For a description of the sign function method for the Lyapunov equation, see [3]. The algorithms considered here are implemented using the ScaLAPACK library [4] to ensure their portability. The computational performance and scalability of the implemented algorithms will be reported in Section 3. Some final remarks are given in Section 4. 2 The Smith Iteration We can rewrite ....

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P. Benner and E.S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms, 20(1):75--100, 1999.

....and matrix products, which are highly efficient on current high performance parallel distributed architectures. Furthermore, although these methods can not be considered as numerically stable, the numerical results obtained in practice are close to those obtained by means of QR type solvers [4]. In this paper we show that by coding in PLAPACK [21] and using the given off theshelf kernels in this library, it is easy to obtain parallel iterative distributed solvers for these matrix equations. Furthermore, we also show that PLAPACK allows us to specialize our parallel kernels to exploit ....

....are performed as the ultimate quadratic convergence of the Newton iteration will then ensure the maximum attainable accuracy. The numerical solution of Lyapunov equations by means of the matrix sign function can not be considered as a numerically stable procedure. However, recent studies [4, 8] show that the matrix sign function approach, with careful shifts and scaling, can obtain 3 numerical results which are close to those obtained by means of the numerically stable QR type algorithms. The Newton iterative scheme has a cost of 6n 3 flops per iteration. In practice, 7 10 ....

P. Benner and E.S. Quintana. Solving stable generalized Lyapunov equations with the matrix sign function. Tech. Rep. SPC 97 23, Fak. f. Mathematik, TU Chemnitz, Germany, 1997.

....in, and ensures the feasibility of our solvers based on the matrix sign function. The property will be assumed throughout this paper and the associated Lyapunov equation will be called stable Lyapunov equation (the anti stable case, i.e. oe (A; E) ae C Gamma , can be treated analogously [4]) Moreover, if Q is positive negative (semi )definite, then the solution X of (1) is also positive negative (semi )definite [20, 23] We say then that the equation is a (semi )definite generalized Lyapunov equation. Numerical solution methods for generalized Lyapunov equations are studied in ....

.... involving medium size matrices (i.e. of order O(10 3 ) and has therefore been chosen as one of the basic algorithms in ScaLAPACK [5] Moreover, it has also shown its efficiency for parallel control relevant computations; see, e.g. 21, 11] In Section 2 we review the algorithms suggested in [4] for solving stable generalized Lyapunov equations with the matrix sign function. The algorithms and a brief study of the computational and communication cost are described in Section 3. In Section 4 we analyze the performance of our parallel solvers on an ibm sp2 parallel distributed ....

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P. Benner and E. Quintana-Ort' i, Solving stable generalized lyapunov equations with the matrix sign function, Tech. Rep. SFB393/97-23, Fak. f. Mathematik, TU Chemnitz, 09107 Chemnitz, FRG, 1997.

....equations of the form (4) with E = I n . Roberts also shows how to solve stable Sylvester and Lyapunov equations via the matrix sign function. The application to generalized algebraic Riccati equations with E 6= I n is investigated in [10] while the application to (3) with E 6= I n is examined in [6]. The computation of the sign function is based on basic numerical linear algebra tools like matrix multiplication, inversion and or solving linear systems. These computations are implemented efficiently on most parallel architectures and in particular, ScaLAPACK [9] provides easy to use and ....

.... takes the form H Gamma K = A 0 Q GammaA T Gamma E 0 0 E T : 7) For stable systems, i.e. oe (A; E) ae C Gamma , H Gamma K is regular and H Gamma K has an n dimensional stable deflating subspace such that the solution of (3) can be obtained analogously to that of (4) In [6] it is observed that applying the generalized Newton iteration (5) to the matrix pencil H Gamma K in (7) and exploiting the block triangular structure of all matrices involved, 5) boils down to A 0 : A; Q 0 : Q; for k = 0; 1; 2; A k 1 : 1 2c k Gamma A k c 2 k EA Gamma1 k E ....

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P. Benner and E.S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Numer. Algorithms, 20(1):75--100, 1999.

.... we can apply iteration (7) to the matrix pencil Z Gamma Y : A 0 Q GammaA T Gamma E 0 0 E T : 9) It can be shown that in this case, Z1 : GammaE 0 2E T XE E T (10) and the solution X of (8) can be obtained from Z1 by solving two linear systems of equations; see [3, 4, 7, 20]. Remark 2.1 In case E = I n , the solution X can be read off directly from the lower left n Theta n block of Z1 and (7) reduces to the standard Newton iteration for the computation of the matrix sign function as introduced in [20] In [3, 4] and in [20] for the case E = I n ) it is observed ....

....solving two linear systems of equations; see [3, 4, 7, 20] Remark 2. 1 In case E = I n , the solution X can be read off directly from the lower left n Theta n block of Z1 and (7) reduces to the standard Newton iteration for the computation of the matrix sign function as introduced in [20] In [3, 4] (and in [20] for the case E = I n ) it is observed that the iteration (7) greatly simplifies when applied to Z Gamma Y from (9) Rather than one iteration with 2n Theta 2n matrices Z k , it is sufficient to consider two iterations with n Theta n matrices: A 0 : A, Q 0 : Q. FOR k = 0; 1; 2; ....

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P. Benner and E. S. Quintana-Ort' i, Solving stable generalized Lyapunov equations with the matrix sign function, Tech. Rep. SFB393/97-23, Fak. f. Mathematik, TU Chemnitz, 09107 Chemnitz, FRG, 1997.

....algebraic Riccati equations of the form (9) with E = I n . Roberts also shows how to solve stable Sylvester and Lyapunov equations via the matrix sign function. The application to CAREs and DAREs with E 6= I n is investigated in [21, 22] while the application to (14) with E 6= I n is examined in [14]. The computation of the sign function requires basic numerical linear algebra tools like matrix multiplication, inversion and or solving linear systems. These computations are implemented efficiently on most parallel architectures and, in particular, ScaLAPACK [16] provides easy to use and ....

.... then takes the form H Gamma K = A 0 E Gamma A T # Gamma C 0 0 C T # : 29) For stable matrix pencils A Gamma C, H Gamma K is regular and has an n dimensional stable deflating subspace such that the solution of (14) can be obtained analogously to that of (9) In [14] it is observed that applying the generalized Newton iteration (27) to the matrix pencil H Gamma K in (29) and exploiting the block triangular structure of all matrices involved, 27) boils down to A 0 : A; A k 1 : 1 2 i A k CA Gamma1 k C j ; E 0 : E; E k 1 : 1 2 i E k ....

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P. Benner and E.S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Numerical Algorithms, to appear.

.... of A 2 R n Thetan corresponding to eigenvalues with negative real part, then V Gamma = range(sign(A) Gamma I) The matrix sign function may be used to find invariant subspaces and to solve related problems like finding solutions of the algebraic Riccati equation and stable Lyapunov equation [2, 6, 10, 15, 19, 23, 26]. The matrix sign function is attracting much attention. The survey [19] lists over 100 references. The rounding error analysis and perturbation theory are becoming understood [2, 9, 10, 11, 13, 27] Because it is rich in matrixmatrix operations, the matrix sign function is well suited to ....

P. Benner and E. S. Quintana-Ort' i, Solving stable generalized Lyapunov equations with the matrix sign function, Tech. Rep. SFB393/97-23, Sonderforschungsbereich 393 -- Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz , 1997.

....and matrix products, which are highly efficient on current high performance parallel distributed architectures. Furthermore, although these methods can not be considered as numerically stable, the numerical results obtained in practice are close to those obtained by means of QR type solvers [7]. In this paper we show that by coding in PLAPACK [31] and using the given off the shelf kernels in this library, it is easy to obtain parallel iterative distributed solvers for these matrix equations. Furthermore, we also show that PLAPACK allows us to specialize our parallel kernels to exploit ....

....maximum attainable accuracy. The numerical solution of Lyapunov equations by means of the matrix sign function can not be considered as a numerically stable procedure; in fact, the numerical stability depends on the distance of the eigenspectrum of A to the imaginary axis. However, recent studies [7, 11] show that the matrix sign function approach, with careful shifts and scaling, can obtain numerical results which are close to those obtained by means of the numerically stable QR type algorithms. A more detailed study of the numerical stability of the iterative solvers based on the matrix sign ....

P. Benner and E.S. Quintana-Ort'i. Solving stable generalized Lyapunov equations with the matrix sign function. Technical Report SPC 97 23, Fak. f. Mathematik, TU Chemnitz, 09107 Chemnitz, FRG, 1997.

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