Jacobi-Davidson methods and preconditioning with applications in pole-zero analysis Joost Rommes Utrecht University
[email protected]
Pole-zero analysis is used to obtain information about the characteristics and the stability of an electric circuit. A central role is played by the transfer function of the circuit, which describes the response of circuit variable o to the excitation of source variable i: Xo(s) Hoi (s) = . Ei (s)
Numerically, the elementary response Hoi (s) can be written as
Therefore, one focuses on iterative methods.The Arnoldi method is characterized by slow convergence to all eigenvalues, while the JacobiDavidson methods have fast convergence to a few selected eigenvalues with a certain property. Two Jacobi-Davidson methods are important: The JDQR method for the ordinary eigenproblem, and the JDQZ method for the generalized eigenproblem. The Jacobi-Davidson method constructs a searchspace V for the wanted eigenvector x by executing the following steps iteratively: • Given an eigenpair approximation (θk , uk ), search for a correction v such that
Hoi (s) = eoT (sC + G)−1ei , where e j is the j -th unit n-vector and the matrices C, G ∈ Rn×n are sparse. One is especially interested in the poles and zeroes of the transfer function. The poles pk are given by the generalized eigenvalues λk = − pk of the generalized eigenvalue problem Gx = λk Cx,
x 6= 0.
The zeroes can be found in a similar way.
2 Strategy A general, but dangerous strategy to solve a generalized eigenproblem is to transform the problem to an ordinary eigenproblem by inverting matrix G or C: G −1Cx = λ˜ k x,
x 6= 0.
Drawbacks of this strategy are, besides the computational costs for the LU -decomposition of G, the loss of accuracy and possible numerical instabilities. A striking example is given in Figure 1, where in a Bode-plot the exact solution is compared with the solutions computed by the Q Rmethod and the Q Z -method. −140
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A(uk + v) = λ(uk + v) • Solve v from the Correction Equation, with rk = Auk − θk uk :
• Orthogonally expand V with v The correction equation can be solved exactly or with a linear iterative solver like GMRES. The more accurate the correction equation is solved, the faster the Jacobi-Davidson process converges. A great feature of the Jacobi-Davidson methods is that one can use preconditioning techniques to solve the correction equation, and thereby improve the speed of convergence. In general, a preconditioner K ≈ A − θk I is used. Three problems can be identified beforehand: • K is projected afterwards • K must approximate the ill-conditioned operator A − θk I
• A − θk I changes every Jacobi-Davidson iteration
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Numerical results
To demonstrate the effect of preconditioning, a convergence history for the JDQZ process with ILUT preconditioned GMRES is given in Figure 2.
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exact Arnoldi JDQR JDQZ 2
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Figure 3: Bode-plots computed by Arnoldi, JDQZ and JDQR, and the exact solution.
Application of Jacobi-Davidson methods in stability analysis is even more promising. The stability of a circuit can be determined by computing the positive pole, if present. The imaginary part of this pole is the frequency the circuit oscillates at, and the corresponding eigenvector can be used for the estimation of the periodic steady-state. Such a positive pole clearly can be categorized as an eigenvalue with a certain property, the class of problems for which Jacobi-Davidson methods are designed.
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exact QR QZ 2
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Figure 1: Bode-plots showing the exact solution and the solutions computed by Q R and Q Z .
3 Jacobi-Davidson method The full-space methods Q R and Q Z become less applicable for large problems (n 103).
Correction equation solved with gmres.
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The test subspace is computed as Bv.
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(I − uk u∗k )(A − θk I )(I − uk u∗k )v = −rk
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If one realizes that using GMRES without preconditioning does not lead to convergence at all, the results are good. However, solving the correction equation exactly does lead to quadratical convergence, and this is not the case for preconditioned GMRES, even not if the drop-tolerance is decreased. Nevertheless, JDQR with preconditioned GMRES does show nearly quadratical convergence. The explanation for this lies in the operators of both processes: the operators of the JDQZ process are more ill-conditioned than the operators of the JDQR process. The accuracy of the computed solution is comparable with the solution computed by the Arnoldi method, as can be seen in Figure 3.
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1 Introduction
5 Future research
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• Easy-to-update preconditioners
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• Preconditioners for ill-conditioned operators
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• Adaptive Jacobi-Davidson methods
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• Problem reduction techniques
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50 100 150 200 250 300 350 JDQZ with jmin=10, jmax=20, residual tolerance 1e−08. 2− 5−2002, 17:49:39
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Figure 2: Convergence history for JDQZ with GMRES and ILUT-preconditioning with a droptolerance of 10−5. Each drop below the acceptance level means an accepted eigenvalue.
6 Upcoming event A presentation on this subject at the ECMI-2002: • ECMI-2002, Sept. Latvia
10-14, 2002, Jurmala,
