G. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. ETNA, Kent State University, 1, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....MIPSpro Fortran 77 compiler with options 64 TARG:platform=ip27 Ofa st=ip27 LNO. The programs called optimized BLAS and LAPACK subroutines from the SGI Cray Scientific Library version 1.2.0.0. Timings were carried out using matrices with pseudorandom entries uniformly distributed in the interval [ 1, 1]. Unblocked code was used for all subproblems with column or row dimension smaller than 65 (NX = 64) Each matrix was stored in an array with leading dimension slightly larger than the number of rows to avoid unnecessary cache conflicts. Table 6.1 shows the result for DGESQB, an implementation of ....

....basically covers the range of practically important one sided orthogonal symplectic factorizations an important class of two sided factorizations, so called PVL reductions [15] is not considered in this work. The PVL reduction of a Hamiltonian matrix is extensively used in the OSMARE algorithm [1]. However, with increasing matrix dimensions, OSMARE DGESUB n n b R U V 128 1 0.53 0.11 0.11 128 8 0 .48 0 .08 0 .09 128 16 0.52 0 .08 0 .09 256 1 6.91 0.87 0.89 256 8 4 .74 0 .50 0 .52 256 16 5.12 0 .50 0.53 256 32 5.82 0.55 0.58 512 1 66.79 13.04 12.90 512 8 42.17 4.82 5.15 512 16 42 .05 ....

G. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electron. Trans. Numer. Anal., 1(Sept.):33--48 (electronic only), 1993.

....This idea, which was previously discussed in [17] and tried for the symmetric case in [13] could lead to a highly parallel algorithm. A multishift QR like algorithm for solving the linear quadratic Gaussian problem of control theory was proposed by Ammar and Mehrmann [2] 3] and extended in [1] and [14] This algorithm, which uses iterations of multiplicity n on matrices of dimension 2n, has problems similar to those of multishift QR; it works well for small multiplicities and poorly for larger ones. Although the analysis of multishift QR does not carry over directly to this algorithm, ....

G. Ammar, P. Benner, and V. Mehrmann, A multishift algorithm for the numerical solution of algebraic Riccati equations, Elec. Trans. Numer. Anal., 1:33-48 (1993).

....was stationary at about 10 ff from the iter th iteration forward, and failed to satisfy the stopping criterion even after 60 iterations. All random matrices used below are with entries independent and normally distributed with mean 0 and variance 1. Example 1. This example is taken from [4, 1]. Let B = 0 B B B 0j 1 0 0 01 0j 0 0 0 0 j 1 0 0 01 j 1 C C C A ; G = R = 0 B B B 1 1 1 1 1 C C C A i 1 1 1 1 j : and A = Q T B R G 0B T Q: where Q is an orthogonal matrix generated from the QR decomposition of a random matrix. As j 0, two pairs of complex conjugate ....

G. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. ETNA, Kent State University, 1, 1993.

....(the theory is not ready yet) We leave these issues open for discussions. 2.6. 3 Symplectic methods to solve Riccati equations Structure preserving algorithms to solve Riccati based on the reduction of Hamiltonian matrices by using orthogonal symplectic transformations has been proposed in [2, 1] and an implementation of this method led to the OSMARE subroutine submitted to SLICOT in October 1994 and also used in [37] A new version of OSMARE will be ready soon and will employ symplectic balancing, the best you can do for relatively modest dimensions (n 50) A new solver based on new ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....in part by NSF grant ASC 9102963 and in part by ARPA grant DM28E04120 via a subcontract from Argonne National Laboratory. y Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506 0027 2 Bai and Qian include the Schur vector method [10] the Hamiltonian QR algorithm [5, 1], the SRalgorithm [4] and the matrix sign function method [13, 6] Among all these existing algorithms, the matrix sign function method is apparently the most suitable for parallel implementation [9] However, in the inner loop iteration of the method, it is required to compute matrix inversion, ....

G. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Elect. Trans. on Numer. Anal., Kent State University, 1:33--48, 1993.

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G. Ammar, P. Benner, and V. Mehrmann, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electr. Trans. Num. Anal., 1 (1993), pp. 33--48.

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G. Ammar, P. Benner, and V. Mehrmann, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electr. Trans. Num. Anal., 1 (1993), pp. 33--48.

....structure preserving numerical method to compute it. It has been argued in [2] that, except in special cases [13, 14] QR like algorithms are impractically expensive because of the lack of a Hamiltonian Hessenberg like form. For this reason other methods such as the multishift method of [1] and the structured implicit product methods of [6, 7, 38] do not follow the QR algorithm paradigm. The implicit product methods [6, 7] do come quite close to optimality. We extend the method of [6] to skew Hamiltonian Hamiltonian matrix pencils in section 4. A third di#culty arises when the ....

G. Ammar, P. Benner, and V. Mehrmann, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electron. Trans. Numer. Anal., 1 (1993), pp. 33--48.

.... 14] Unfortunately, the construction of a method for the computation of this Schur form that has complexity O(n 3 ) and is strongly stable, i.e. computes the Hamiltonian Schur form of a nearby Hamiltonian matrix, is still an open problem, although a lot of progress has been made in recent years [1, 4, 5, 8]. For real skew Hamiltonian matrices N , Van Loan s method presented in [18] can be used to develop a structural QR algorithm as follows. First, a real matrix Q 1 # US 2n is computed such that Q T 1 NQ 1 = F 1 D 1 0 F T 1 , 1.2) where F 1 is upper Hessenberg and D 1 is skew ....

....chosen real. Here, the loss of accuracy of order H 2 # in Van Loan s method is obvious while both ZHAEV and ZGEEV compute all eigenvalues to full accuracy. EXAMPLE 4. We tested our subroutines for randomly generated Hamiltonian matrices with entries distributed uniformly in the interval [ 1, 1 ]. The eigenvalues computed by ZHAEV are as accurate as for ZGEEV. In Figure 5.1 we present the minimum singular value of H #I 2n , denoted by # min (H #I 2n ) for an n = 50 example and all eigenvalues in the closed right half plane (in this case, these are 52, i.e. 4 eigenvalues are ....

G. S. AMMAR, P. BENNER, AND V. MEHRMANN, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electron. Trans. Numer. Anal., 1 (1993), pp. 33--48.

....satisfactory structure preserving, numerical method to compute it. It has been argued in [2] that, except in special cases [13, 14] QR like algorithms are impractically expensive because of the lack of a Hamiltonian Hessenberg like form. For this reason other methods like the multishift method of [1], the structured implicit product methods of [6, 7, 38] do not follow the QR algorithm paradigm. The implicit product methods [6, 7] do come quite close to optimality. We extend the method of [6] to skew Hamiltonian Hamiltonian matrix pencils in Section 4. A third diculty arises when the ....

G. Ammar, P. Benner, and V. Mehrmann, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electr. Trans. Num. Anal., 1 (1993), pp. 33-48.

.... algebraic Riccati equations as well as properties of solutions of the CARE (2) we refer to [25] Knowledge of approximations to the eigenvalues of the corresponding Hamiltonian matrix is crucial for some numerical methods for the CARE (2) e.g. for the multishift QR like methods proposed in [2, 3, 34] and the algorithm presented in [32] As suggested in [35] they can significantly improve the convergence of the SR algorithm [12] when employed as shifts. Van Loan s method uses the square reduced form of a Hamiltonian matrix which will be introduced in Section 2. Square reduced Hamiltonian ....

.... distance to instability problem [16, 37] If, for any reason, the Hamiltonian matrix cannot be overwritten by K, the algorithm requires an additional workspace of size 2n 2 O(n) This is the case when the eigenvalues of the Hamiltonian matrix are used as shifts in CARE solution methods as in [1, 2, 3, 32, 34, 35] since the original Hamiltonian matrix (or a similar Hamiltonian matrix) is needed in following steps of the algorithm. This additional workspace can be avoided using the implicit version of the algorithm as given in [39] That is, the orthogonal symplectic matrix U from Step 2 of Algorithm 4 is ....

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G. Ammar, P. Benner, and V. Mehrmann, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electr. Trans. Num. Anal., 1 (1993), pp. 33--48.

....balancing as implemented in the LAPACK expert driver routine DGEEVX [3] applied to the Hamiltonian matrix and using quadruple precision. Moreover, we tested the effects of balancing when solving algebraic Riccati equations with the structure preserving multishift method presented in [1] for the examples from the benchmark collection [6] We only present some of the most intriguing results. Example 6.1. 6, Example 6] The system data come from an optimal control problem for a J 100 jet engine as a special case of a multivariable servomechanism problem. The resulting ....

....10 15 10 14 10 13 10 12 10 11 10 10 10 9 10 8 eigenvalue number with symplectic balancing, o without balancing Fig. 6.2. symplectic URV periodic QR. Using the balanced matrix in order to solve algebraic Riccati equations by the multishift method as described in [1], we obtain the following results: if the multishift method is applied to the unbalanced data, the computed solution yields a residual r F : kQ A T X XA Gamma XGXkF (6.1) of size 1:5 Theta 10 Gamma6 while using the balanced Hamiltonian matrix we get r F = 8:1 Theta 10 Gamma10 . ....

G. Ammar, P. Benner, and V. Mehrmann, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electr. Trans. Num. Anal., 1 (1993), pp. 33--48.

....exploiting the structure of the symmetric equations (28) 34) and (39) Theorems 3.7 and 4.2 and Corollary 3.9 show that for these symmetric equations the related matrix pencils have certain symmetry structures. Special equivalence transformations may be employed to preserve these structures, see [2, 10, 14, 33]. However, a numerically stable and efficient method for computing the structured decompositions (30) 33) 35) and (41) in general is still an open problem. The numerical methods for linear matrix equations can be simplified by using the block triangular forms of the related matrices and the ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....the structure of the symmetric equations (28) 35) and (44) Theorems 3.8 and 4.2 and Corollary 3.12 show that for these symmetric equations the related matrix pencils have certain symmetry structures. Special equivalence transformations may be employed to preserve these structures, see [2, 11, 12, 18, 19, 44, 45]. However, a numerically stable and efficient method for computing the structured decompositions (31) 34) 36) and (46) in general is still an open problem. For the eigenvalue problem corresponding to (37) there is a simplified QR like method for computing the generalized Schur form if A 21 and ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993. 25

....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 multishift algorithm [1] and the Schur vector method [34] If possible, these approximations were refined by Newton s method [32] possibly combined with an exact line search [8, 9] to achieve the highest possible accuracy. We then chose the approximate solution with smallest residual norm to compute the properties of the ....

....: 0 0 1 Gamma2 1 0 : 0 . 0 1 1 0 : 0 1 Gamma2 3 7 7 7 7 7 7 7 5 ; G = Q = I n : Most eigenvalues of the Hamiltonian matrix have multiplicity 2. For invariant subspace methods that use deflation techniques (e.g. Hamiltonian SR [14, 15, 47] multishift QR [1, 44]) this may cause a lot of deflation steps and hence may slow down convergence. Growth of the problem size n does not influence norms and condition numbers. All the closed loop modes are real and of magnitude O(1) Therefore, this example is perfectly suited to test the behavior of algorithms for ....

[Article contains additional citation context not shown here]

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....is no known structure preserving, numerical method to compute it. It has been argued in [2] that, except in special cases [13, 14] QR like algorithms are impractically expensive because of the lack of a Hamiltonian Hessenberg like form. For this reason other methods like the multishift method of [1], the structured implicit product methods of [4, 5, 6, 46] do not follow the QR algorithm paradigm. The implicit product methods [5, 6] do come quite close to optimality. We extend the method of [5] to skew Hamiltonian Hamiltonian matrix pencils in Section 5. A third difficulty arises when the ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....this problem, see [8, 15, 17] and the references therein, but it has been shown in [2] that a modification of standard QRlike methods to solve this problem is in general hopeless, due to the missing reduction to a Hessenberg like form. For this reason other methods like the multishift method of [1] were developed that do not follow the direct line of a standard QR like method. The structure of the multishift method is at first a computation of the eigenvalues followed by a sequence Fakultat fur Mathematik, TU Chemnitz Zwickau, D 09107 Chemnitz, FRG. e mail: ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

.... suggested in [58] but not every Hamiltonian matrix or skew Hamiltonian Hamiltonian pencil has such a condensed form, see [47, 50, 51] The second difficulty arises from the fact that even if a Hamiltonian Schur form exists, it is still difficult to construct a method with the desired features, see [2, 3, 9, 10, 19, 20]. We dicuss here only the computation of the structured Schur form for Hamiltonian matrices. For skew Hamiltonian Hamiltonian pencils we refer the reader to [9, 50, 51] Necessary and sufficient conditions for the Hamiltonian Schur form are given by the following theorem. Theorem 4 [47] Let H be ....

G.S. Ammar, P. Benner and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....of SLICOT; see Section 5.3. Some of the most recent developments are: balancing free square root methods for model reduction [46] periodic Schur methods for the solution of periodic Lyapunov equations [48] descriptor systems analysis procedures [28] symplectic methods to solve Riccati equations [1, 4, 32], and subspace identification methods [53, 44, 31] 3 Retrospect 3.1 Short history of control subroutine libraries The development of efficient, reliable, and portable numerical software requires joining expertise in theory, in numerical mathematics, in numerical programming, and in numerical ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....only in special cases a satisfactory solution has been obtained [9, 10] Furthermore it has been shown in [1] that a modification of standard QR like methods is in general hopeless, due to the missing reduction to a Hessenberg like form. For this reason other methods like the multishift method of [2] were developed that do not follow the direct line of a standard QRlike method. The multishift method is in principle a satisfactory solution, but unfortunately it sometimes has convergence problems, in particular for large n. Recently the authors have proposed a method to compute the eigenvalues ....

....and in the other case we stopped the algorithm after Step 2 (superscript 2 ) as discussed in Remark 3.5. We compared the results with the result obtained using the Schur vector method as proposed in [19] and implemented in the MATLAB function are [22] and the multishift method as described in [2]. We refrain from reproducing all the data here. In general, Algorithm 1 produces errors of the same order as the other two methods. For the problems of larger dimension (Examples 15, 16, 18, 19) the new method produced the best results while the multishift method suffers from convergence ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....in special cases a satisfactory solution has been obtained [12, 13] Furthermore it has been shown in [1] that a modification of standard QR like methods is in general hopeless, due to the missing reduction to a Hessenberg like form. For this reason other methods like the multishift method of [2] were developed that do not follow the direct line of a standard QRlike method. The multishift method is in principle a satisfactory solution, but unfortunately it sometimes has convergence problems, in particular for large n. Recently the authors have proposed a method to compute the eigenvalues ....

....solver contained in the MATLAB LMI Toolbox [16] This solver is based on the deflating subspace approach [33] as presented in [5] ffl aresolv The Schur vector method [23] implementation from the MATLAB Robust Control Toolbox, Version 2. 0b [27] ffl osmare The multishift method as described in [2] (MATLAB codes as described in [6] Note that all algorithms are implemented without any kind of scaling. Computations were performed either on a PC Pentium s with IEEE standard double precision arithmetic and machine precision ffl 2:22 Theta 10 Gamma16 or on a SunSparc ULTRA 1 under ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....(1) should consist only of orthogonal symplectic similarity transformations. An algorithm with this property was proposed in [11] for the case that rank G = 1 or rank Q = 1. To the best of our knowledge, the only exisiting algorithm for the general case satisfying this demand was proposed in [1]. But for growing dimension n, this method suffers from convergence problems. The Lanczos method proposed here for the large scale problem exploits the structure of the problem by weakening orthogonality to J orthogonality. In exact arithmetic, the method would compute a symplectic (nonorthogonal) ....

G. Ammar, P. Benner, and V. Mehrmann, A multishift algorithm for the numerical solution of algebraic Riccati equations, Electr. Trans. Num. Anal., 1 (1993), pp. 33--48.

....fail as small perturbations can cause eigenvalues with small real part to cross the imaginary axis; see, e.g. 59] For computing the system properties we use the exact stabilizing solution if an analytical solution is available. Otherwise, we computed approximations by the multishift algorithm [2] and the Schur vector method [38] If possible, these approximations were refined by Newton s method [33] possibly combined with an exact line search [9, 10] to achieve the highest possible accuracy. We then chose the approximate solution with smallest residual norm to compute the properties of ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

....to compute it numerically. It has been shown in [1] that a modification of standard QR like methods to solve this problem is (except for special cases [15, 16] in general hopeless, due to the missing reduction to a Hessenberg like form. For this reason other methods like the multishift method of [2] or the structured method of [9] were developed that do not follow the direct line of a standard QR like method. Although these methods still do not fulfill all the requirements to a full extend, they come quite close to the optimal methods. We will review the method of [9] and indicate how it can ....

....achieving full possible accuracy. Unfortunately this new approach is not perfect. We would like to have the Hamiltonian Schur form, since it would give us the eigenvalues and also the deflating subspaces. Currently the only other candidate for an optimal algorithm, the multishift algorithm of [2], sometimes has convergence problems. For the computation of the deflating subspaces we now use another embedding procedure. These ideas are presented in the next section. 5 Invariant subspace computation for Hamiltonian matrices In this section we discuss structure preserving methods to compute ....

G.S. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. Electr. Trans. Num. Anal., 1:33--48, 1993.

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G. Ammar, P. Benner, and V. Mehrmann. A multishift algorithm for the numerical solution of algebraic Riccati equations. ETNA, Kent State University, 1, 1993.

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