R. Byers, C. He, and V. Mehrmann, "The matrix sign function method and the computation of invariant subspaces", Tech. Rep., Fakultat f. Mathematik, TU Chemnitz-Zwickau, PSF 964, D-09009 Chemnitz, FRG, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....example, in a recent paper [34] to compute rightmost eigenvalues, Meerbergen and Roose use the Arnoldi process and an approximation M c of (I #A) 1 to compute the largest eigenvalues of A. Other algorithms to compute continuous time invariant subspace include the sign function method (see [35], 8] and their references for details) Its applicability to large sparse systems is still open. Algorithms to stabilize continuous time system are discussed in Chapter 8. 4.8 Summary In this chapter, we have discussed the family of projection methods to stabilize large sparse systems with a ....

C. Y. He P. Byers and V. Mehrmann. The matrix sign function method and the computation of invariant subspace. SIAM J. Matrix Anal. Appl., 18(3), 1997.

....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 ....

R. Byers, C. He, and V. Mehrmann. The matrix sign function method and the computation of invariant subspaces. SIAM J. Matrix Anal. Appl., 18(3):615--632, 1997.

....sign function of A at X given by Roberts [30] #F(sign(A) X)# # l # 2# # max # on # #(#I A) 1 # 2 # #X#, 210 ZHAOJUN BAI AND JAMES DEMMEL where l # is the length of the closed contour #. Recently, an asymptotic perturbation bound of sign(A) was given by Byers, He, and Mehrmann [13]. They show that to first order in #A #sign(A #A) sign(A)# # 4 # # 1 #A 12 # # # 2 ##A#, 9) where A is assumed to have the form of (1) ##A# is su#ciently small, and # = sep(A 11 , A 22 ) # min (I# A 11 A T 22# I) 10) the separation of the matrices A 11 and A ....

.... again we have #E 21 # #A# # O( # u)#A# # # 1 # 2 # , 23) where # = sep(A 11 , A 22 ) is the separation of the matrices A 11 and A 22 , if A is assumed to have the form (11) We note that the error bound (23) is essentially the same as the error bound given by Byers, He, and Mehrmann [13], although we use a di#erent approach. In [13] it is assumed that #F 21 # # O(u)#S# in (19) where F 21 is the (2,1) block of the matrix F . Therefore, the O( # u) term in (23) is replaced by O(u) The bounds (22) and (23) reveal two important features of the matrix signfunction based ....

[Article contains additional citation context not shown here]

<F3.759e+05> R. Byers, C. He, and V.<F3.851e+05> Mehrmann,<F3.469e+05> The Matrix Sign Function Method and the Computation of Invariant<F3.851e+05> Subspaces, Technical report preprint SPC 94-25, Fakultat fur Mathematik, TU Chemnitz-Zwickau, Germany, 1994.

....Newton iteration with global convergence need to compute the inverse of a matrix explicitly in one form or another. Dealing with ill conditioned matrices and instability in the Newton iteration for computing the matrix sign function and the subsequent spectral decomposition have been studied in [3, 8]. Recently, 1450 BAI, DEMMEL, DONGARRA, PETITET, ROBINSON, AND STANLEY an inverse free method for achieving better numerical stability has been proposed in [30, 4] The advantage of the inverse free approach is obtained at the cost of more storage and arithmetic. Since 1) any SDC algorithm could ....

R. BYERS, C. HE, AND V. MEHRMANN, The Matrix Sign Function Method and the Computation of Invariant Subspaces, Tech. report SPC 94-25, Fakultat fur Mathematik, TU Chemnitz-Zwickau, Germany, 1994.

....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 ....

R. Byers, C. He, and V. Mehrmann. The matrix sign function method and the computation of invariant subspaces. SIAM J. Matrix Anal. Appl., 18(3):615--632, 1997.

....cost will be given in Section 4. The following remark does not replace a thorough error analysis but provides an explanation why our method usually provides Lyapunov solutions with an accuracy as predicted by the condition number of the corresponding Lyapunov operator. Remark 2. 2 In [9] it is shown that invariant subspaces computed via the sign function iteration are essentially as accurate as those computed by the QR algorithm, provided the sign function of a matrix can be computed with sufficient accuracy. The same arguments also show that a deflating subspace of a matrix ....

R. Byers, C. He, and V. Mehrmann. The matrix sign function method and the computation of invariant subspaces. SIAM J. Matrix Anal. Appl., 18(3):615--632, 1997.

....Newton iteration with global convergence need to compute the inverse of a matrix explicitly in one form or another. Dealing with ill conditioned matrices and instability in the Newton iteration for computing the matrix sign function and the subsequent spectral decomposition have been studied in [3, 8]. Recently, an inverse free method for achieving better numerical stability has been proposed in [30, 4] The advantage of the inverse free approach is obtained at the cost of more storage and arithmetic. Since 1) any SDC algorithm could suffer numerical instability when some eigenvalues are very ....

R. Byers, C. He, and V. Mehrmann. The matrix sign function method and the computation of invariant subspaces. Technical Report Preprint SPC 94-25, Fakultat fur Mathematik, TU Chemnitz-Zwickau, Germany, 1994.

....matrix sign function of A at X given by Roberts [27] kF(sign(A) X)k l Gamma 2 max i on Gamma k(iI Gamma A) Gamma1 k 2 kXk where l Gamma is the length of the closed contour Gamma. Recently, an asymptotic perturbation bound of sign(A) was given by Byers, He and Mehrmann [11]. They show that to first order in ffi A, ksign(A ffiA) Gamma sign(A)k 4 ffi 1 kA 12 k ffi 2 kffiAk; 2.9) where A is assumed to have the form of (1.1) kffiAk is sufficiently small, and ffi = sep(A 11 ; A 22 ) oe min (I Omega A 11 Gamma A T 22 Omega I) 2.10) the ....

....we have kE 21 k kAk O( p u)kAk ffi q 1 oe 2 l ; 4.23) where ffi = sep(A 11 ; A 22 ) is the separation of the matrices A 11 and A 22 , if A is assumed to have the form (3.11) We note that the error bound (4. 23) is essentially the same as the error bound given by Byers, He and Mehrmann [11], although we use a different approach. In [11] it is assumed that kEk O(u)kSk in (4.19) Therefore, O( p u) term in (4.23) is replaced by O(u) We consider the O(u)kSk accuracy for the computed matrix sign function to be an unrealistic assumption. The bounds (4.22) and (4.23) reveal two ....

[Article contains additional citation context not shown here]

R. Byers, C. He, and V. Mehrmann. The matrix sign function method and the computation of invariant subspaces. Technical Report Preprint SPC 94-25, Fakultat fur Mathematik, TU Chemnitz-Zwickau, Germany, 1994.

....matrix sign function of A at X given by Roberts [30] kF(sign(A) X)k l Gamma 2 max i on Gamma k(iI Gamma A) Gamma1 k 2 kXk where l Gamma is the length of the closed contour Gamma. Recently, an asymptotic perturbation bound of sign(A) was given by Byers, He and Mehrmann [13]. They show that to first order in ffi A, ksign(A ffi A) Gamma sign(A)k 4 ffi 1 kA 12 k ffi 2 kffiAk; 9) 6 Z. BAI AND J. DEMMEL where A is assumed to have the form of (1) kffiAk is sufficiently small and ffi = sep(A 11 ; A 22 ) oe min (I Omega A 11 Gamma A T 22 Omega ....

.... again, we have kE 21 k kAk O( p u)kAk ffi p 1 oe 2 ; 23) where ffi = sep(A 11 ; A 22 ) is the separation of the matrices A 11 and A 22 , if A is assumed to have the form (11) We note that the error bound (23) is essentially the same as the error bound given by Byers, He and Mehrmann [13], although we use a different approach. In [13] it is assumed that kF 21 k O(u)kSk in (19) where F 21 is the (2,1) block of the matrix F . Therefore, O( p u) term in (23) is replaced by O(u) 12 Z. BAI AND J. DEMMEL The bounds (22) and (23) reveal two important features of the matrix ....

[Article contains additional citation context not shown here]

R. Byers, C. He, and V. Mehrmann. The matrix sign function method and the computation of invariant subspaces. Technical Report Preprint SPC 94-25, Fakultat fur Mathematik, TU Chemnitz-Zwickau, Germany, 1994.

....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 ....

R. Byers, C. He, and V. Mehrmann. The matrix sign function method and the computation of invariant subspaces. SIAM J. Matrix Anal. Appl., 18(3):615--632, 1997.

No context found.

R. Byers, C. He, and V. Mehrmann, "The matrix sign function method and the computation of invariant subspaces", Tech. Rep., Fakultat f. Mathematik, TU Chemnitz-Zwickau, PSF 964, D-09009 Chemnitz, FRG, 1994.

....deflating subspaces, solving (generalized) algebraic Riccati equations and (generalized) Lyapunov equations. Matrix sign function algorithms have attracted much recent attention; the survey [23] lists over 100 references. The rounding error analysis and perturbation theory are becoming understood [2, 12, 13, 14, 19, 32]. Because they are rich in matrix matrix operations, matrix sign function algorithms are well suited to computers with advanced architectures [3, 1, 2, 9, 20, 21] Its presence in the ScaLAPACK library [10] is an indication of its maturity and acceptance. The first matrix sign function algorithm ....

R. Byers, C. He, and V. Mehrmann, The matrix sign function method and the computation of invariant subspaces, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 615--632.

....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 computers with advanced architectures [1, 2, 3, 15, 23] It has been demonstrated on matrices of order greater than 10,000 [1] A sign of its maturity and acceptance is its presence in the ScaLAPACK library ....

R. Byers, C. He, and V. Mehrmann, The matrix sign function method and the computation of invariant subspaces, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 615--632.

....k A k ) Gamma1 Delta : Under reasonable assumptions on the scale factors fi k , sign(A) lim k 1 A k . The scale factor fi k is chosen to accelerate convergence and promote numerical stability. Typical choices are fi k = j det(A k )j Gamma1=n and fi k q kA Gamma1 k k=kA k k [10, 11, 25]. Computing sign( A Gamma E) Gamma1 (A E) requires an inverse of A Gamma E followed by several other matrix inverses. The assumption that A Gamma E has no eigenvalue on the unit circle implies that A Gamma E is nonsingular. Nevertheless, A Gamma E may be ill conditioned for inversion. ....

....other matrix inverses. The assumption that A Gamma E has no eigenvalue on the unit circle implies that A Gamma E is nonsingular. Nevertheless, A Gamma E may be ill conditioned for inversion. Moreover, in rare cases, even the wellstudied scaled Newton iteration exhibits numerical instabilities [11, 9, 27]. An alternative way to compute disk functions is the inverse free, implicit squaring algorithm [3, 8, 20, 33, 34, 35] which is related to the AB algorithm [28, 30, 31] UR Gamma VR = lim k 1 (U k V k ) Gamma1 (U k Gamma V k ) where U 0 = A V 0 = E U k 1 = Q H k;12 U k V k 1 = Q H ....

R. Byers, C. He, and V. Mehrmann. The matrix sign function method and the computation of invariant subspaces. Technical report, TU Chemnitz, Germany, 1994. To appear in SIAM J. Matrix Anal. Appl.

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