K. Gallivan, E. Grimme, and P. V. Dooren, A rational Lanczos algorithm for model reduction, to appear, (1995).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....s 1 ) m 2 (s s 1 ) m 3 (s s 1 ) H(s) m 0 m 1 (s s 1 ) m 2 (s s 1 ) m 3 (s s 1 ) such that m i = m i for i = 0; 2k 1: One shows that there always exists a kth order model that matches 2k moments of such an expansion. Di erent Interpolation Points ( 9] [6]) A more elaborate approach is to match the rst moments of H(s) with those of H(s) in several expansion points. For two points s 1 and s 2 this would amount to : H(s) m 0;s1 m 1;s1 (s s 1 ) m 2;s1 (s s 1 ) m 0;s2 m 1;s2 (s s 2 ) m 2;s2 (s s 2 ) H(s) m 0;s1 m 1;s1 (s s 1 ) ....

K. Gallivan, E. Grimme and P. Van Dooren, A Rational Lanczos Algorithm for Model Reduction, Numerical Algorithms, 12:33-63, 1996.

....r #sI ,A r # B r ; Y r #s#=G r #s#U r #s# (10) is close to the original. 3.1 Moment Matching Methods Up to now, model order reduction of linear systems has usually gone in one of two directions. One is moment matching, which includes Pade, partial realization, and their shifted versions [7, 9, 11]. These methods usually utilize the Arnoldi or Lanczos method to find an orthonormal basis for some combination of Krylov subspaces, K J #A;B#, K J #A #, B#,orK J ##A #,where K J #A;B#=spanfB;AB;A B;### ;A #J,1# Bg: 11) The result is that moments, when K J ##A ; B# is ....

K. Gallivan, E. J. Grimme, and P. Van Dooren. A Rational Lanczos Algorithm for Model Reduction. Numer. Algorithms, 1996

....[ i=1 K Psiae i (C H (G s i C) GammaH ; d) 13) One sided and two sided methods use the same number of basis vectors to match the same number of moments. An advantage with two sided methods is that, by using a pair of biorthogonal bases, short term recurrences can be developed; see [18] and [9] One disadvantage with the biorthogonalisation process is that it is not as stable as the orthogonalisation process. The biorthogonalisation process can break down in a way that the orthogonalisation process cannot do; this can be remedied with look ahead [16, 15, 37] For an orthogonal ....

.... Freund described how to generate a stable and passive reduced order model via 6 the PVL algorithm [1] Bai and Ye develop an error estimate for the transfer function computed by Pad# via Lanczos [2] The rational Lanczos algorithm discussed by Gallivan, Grimme and Van Dooren in their later work [18] is a multipoint Pad# approximation; the method builds up a biorthogonal pair of bases. Multipoint Pad# and Pad# type methods are also discussed by Grimme in his PhD thesis [20] Nguyen and Li discuss a block rational Lanczos algorithm [22] Silveria, Kamon, Elfadel and White create a passive ....

[Article contains additional citation context not shown here]

K. Gallivan, E. J. Grimme, and P. Van Dooren. A rational Lanczos algorithm for model reduction. Numerical Algorithms, 12:3363, 1996.

....a consequence the matrix H k 2;k 1 will be real. 4. 3 Related Work The biorthogonal rational Lanczos algorithm is similar in behaviour, regarding the nonzero structure for general matrices, to the rational Krylov algorithm for Hermitian matrices; see the paper by Gallivan, Grimme and Van Dooren [3]. The nonzero structure discussed here is also related to the case where the DIOM ( FOM with incomplete orthogonalisation) generates orthonormal basis vectors; see page 186 in the book by Saad [11] 9 5 Implementation 5.1 Introduction The parallel programs described in this section are ....

K. Gallivan, E. Grimme, and P. Van Dooren. A rational Lanczos algorithm for model reduction. Numerical Algorithms, 12:3364, 1996.

....(1) of a dynamical system provided that x(0) 0. The basic principle of the Pad e approximation based model reduction algorithms (Pad e algorithms) is an expansion of G(s) about a point s 0 2 C [ f1g. A frequent choice is s 0 = 1 although other or even multiple expansion points (e.g. [13]) can be used. For example, the expansion about in nity delivers G(s) 1 X i=0 1 s i 1 M i with M i = CA i B; 12) which can be interpreted as a Taylor series, that converges if jsj is suciently large. The matrices M i , which are called Markov parameters, are system invariants. The ....

K. Gallivan, E. Grimme, and P. Van Dooren. A rational Lanczos algorithm for model reduction. Numer. Alg., 12:33-63, 1996.

....spaces, computed by Lanczos or Arnoldi algorithms applied to a shifted and inverted matrix pencil (8) have caught on, especially in Electrical Engineering, Freund and Feldmann at Bell Laboratories [4, 5] Elfadel and Ling et al. at IBM [2] but also in Mechanical Engineering (E. Grimme et al. [6]) and even in Quantum Chemistry, the tridiagonal matrix from Lanczos has been used to predict photoionization cross sections ( H. Karlsson [7] A Rational Krylov approach in the spirit we are going to discuss today has already been tried at several places, see [6] and [1] 2 Rational Krylov A ....

....Engineering (E. Grimme et al. [6] and even in Quantum Chemistry, the tridiagonal matrix from Lanczos has been used to predict photoionization cross sections ( H. Karlsson [7] A Rational Krylov approach in the spirit we are going to discuss today has already been tried at several places, see [6] and [1] 2 Rational Krylov A standard practice to find eigenvalues of large matrix pencils (1) is to apply a Krylov space algorithm, Lanczos or Arnoldi, to a shift invert spectral transformation, C = A Gamma B) Gamma1 B : 8) It will compute eigenvalues close to the shift point in the ....

K. Gallivan, E. Grimme, and P. Van Dooren, A rational Lanczos algorithm for model reduction, Numerical Algorithms, 12 (1996), pp. 33--64.

....resulting vectors are used to span up a subspace where approximations are found. This way we get a jth degree polynomial after just one, albeit big, step. Rational functions are also natural in model reduction problems in Control theory, see the recent report by Gallivan, Grimme and Van Dooren [5]. The outline of this contribution is as follows: In x2 we formulate the RKS algorithm and derive its basic recursion, much like in our earlier contributions. Then in x3, we will study different ways to get approximate eigenvalues and eigenvectors. We give a general framework that covers all ....

K. Gallivan, E. Grimme, and P. Van Dooren, A rational Lanczos algorithm for model reduction, Numerical Algorithms, 12 (1996), pp. 33--64.

....exposition, but note that the second form indicates how a parallel variant, where all the shifts 1 : j are applied at the same time, can be implemented. Rational functions are also natural in model reduction problems in Control theory, see the recent report by Gallivan, Grimme and Van Dooren [6]. After formulating the algorithm in x2.1 and deriving its basic recursion in x2.2, we will study different ways to get approximate eigenvalues and vectors in x2.3 and x2.4. It appears that the theory of harmonic Ritz values, expounded by Paige, Parlett and Van der Vorst [11] gives a good ....

K. Gallivan, E. Grimme, and P. V. Dooren, A rational Lanczos algorithm for model reduction, to appear, (1995).

....A r ) Gamma1 B r ; Y r (s) G r (s)U r (s) 10) is close to the original. 3.1 Moment Matching Methods Up to now, model order reduction of linear systems has usually gone in one of two directions. One is moment matching, which includes Pad e, partial realization, and their shifted versions [7, 9, 11]. These methods usually utilize the Arnoldi or Lanczos method to find an orthonormal basis for some combination of Krylov subspaces, K J (A; B) K J (A T ; C T ) K J ( A Gamma pI) Gamma1 ; B) or K J ( A T Gamma pI) Gamma1 ; C T ) where K J (A; B) spanfB; AB;A 2 B; Delta ....

K. Gallivan, E. J. Grimme, and P. Van Dooren. A Rational Lanczos Algorithm for Model Reduction. Numer. Algorithms, 1996

.... a large region within the complex plane) gives motivation to the development of the Rational Krylov Sequence (RKS) method of Ruhe [66, 68, 67] The RKS method can also be used as a tool to construct a reduced order model for a large scale single input single output (SISO) linear (control) system [25]. Given a sequence of shifts oe i , the RKS method mimics the shift invert Arnoldi iteration by generating an orthonormal basis of a Rational Krylov Subspace S = fw 1 ; w 2 ; w k g; where w j 1 = K Gamma oe j M) Gamma1 Mw j : 4.3) Each new basis vector is generated by applying (K ....

K. Gallivan and E. Grimme. A rational Lanczos algorithm for model reduction. Numerical Algorithms, 13:631--644, 1995.

....one expansion point, s 0 , so called multi point Pad e approximants (see, e.g. 11] match moments at several, say M , expansion points, s 0 ; s 1 ; s M Gamma1 . A multi point moment matching algorithm based on AWE type computations was proposed in [23] Gallivan, Grimme, and Van Dooren [60], for the single input single output case, advocated the use of multi point Pad e approximation for reduced order modeling and showed how such approximants can be computed by means of the so called rational Lanczos method [103, 104] Nguyen and Li [86] developed a multi point Pad e approach for ....

K. Gallivan, E. J. Grimme, and P. Van Dooren. A rational Lanczos algorithm for model reduction. Numer. Algorithms, 12:33--63, 1996.

....system by a small system with similar behavior is called model reduction, which is very useful in engineering applications. For linear systems, this reduction can be achieved by using the traditional Krylov subspace approximation, and a perturbed system can be analyzed in this approximate subspace [4]. For eigenvalue problems, Krylov subspace will not be appropriate for model reduction. In practice, one often employs the first order perturbation theory for this kind of analysis. We think that this work, which allows high order perturbation, will lead to useful insight into this problem. To ....

K. Gallivan. E. Grimme and P. Van Dooren. "A rational lanczos algorithm for model reduction. ", Numer. Algorithms, 12 (1996), pp. 33--63.

....system described by # sEx(s) Ax(s) Bu(s) y(s) Cx(s) Du(s) K. Gallivan and P. Van Dooren Pre filtering rational approximations 13 in the Laplace domain. The transfer function R(s) D C(sE A) 1 B can be expanded around a finite point s = # (which is not a pole of R(s) as follows [2], 3] R(s) D C[ s #)E (A #E) 1 B = D # # i=0 C[ A #E) 1 E] i (#E A) 1 B(s #) i . It is clear that in this context the role of the matrices A g , b g and c g , is now replaced by (A #E) 1 E, #E A) 1 B and C, respectively. For more details on these connections, we ....

....R(s) D C[ s #)E (A #E) 1 B = D # # i=0 C[ A #E) 1 E] i (#E A) 1 B(s #) i . It is clear that in this context the role of the matrices A g , b g and c g , is now replaced by (A #E) 1 E, #E A) 1 B and C, respectively. For more details on these connections, we refer to [2], 3] A further extension is the application of the above ideas to the stabilization of an unstable system via pole zero dislocations (see [11] Assume we have a k th order model A, b, c with transfer function g(s) c(sI k A) 1 b, but this model has unstable and hence ....

K. Gallivan, E. Grimme, P. Van Dooren, "A Rational Lanczos Algorithm for Model Reduction ", Numerical Algorithms, Vol. 12, pp. 33-63, 1995.

....and rational Lanczos algorithms lies in step (A2.4) of Algorithm 1. In rational Lanczos the matrix, A Gamma sE) Gamma1 E, multiplying the previous v vector changes with the expansion frequency. By making this matrix a function of s, the following sequences of Krylov spaces are computed (see [14] for a proof) Theorem 1 If V k and W k are the results of the first k steps of the rational Lanczos algorithm with 1 k then colsp(V k ) 2 ae K k Gamma (i Gamma1) Gamma (A Gamma s i E) Gamma1 E; A Gamma s i E) Gamma1 b Delta S i Gamma1 l=1 K Gamma (A Gamma s l ....

....x to be K Gamma1 k;k W T k x. Comparing (9.1) and (9.4) indicates that the prescribed choices for A, E, b and c are quite logical. However, 9.4) was obtained assuming K k;k to be invertible. This assumption is in fact not necessary for our purposes. The following result (see [14] for a proof) states that the reduced order model corresponding to (9.3) matches the desired moments of the original system without placing any restrictions on the invertibility of K k;k or E. Theorem 2 Let the j th moments of the original and reduced order systems about the expansion frequency ....

[Article contains additional citation context not shown here]

K. Gallivan, E. Grimme and P. Van Dooren, "A rational Lanczos algorithm for model reduction", Numer. Algorithms, to appear.

No context found.

K. Gallivan, E. Grimme, P. Van Dooren, "A Rational Lanczos Algorithm for Model Reduction", Numerical Algorithms, Vol. 12, pp. 33-63, 1995.

No context found.

K. Gallivan, E. Grimme, and P. V. Dooren, A rational Lanczos algorithm for model reduction, to appear, (1995).

No context found.

K. Gallivan, E. Grimme, and P. Van Dooren. A Rational Lanczos Algorithm for Model Reduction. Numerical Algorithms, 12:33--63, 1995.

No context found.

K. Gallivan, E.J. Grimme, and P. Van Dooren, A rational Lanczos algorithm for model reduction, Numer. Algorithm, vol. 12, pp. 33-63, 1996.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC