Optimal Rational Functions for the Generalized Zolotarev Problem in the Complex Plane

Abstract

Abstract. It has been long recognized that the determination of optimal parameters for the classical alternating direction implicit (ADI) method leads to the Zolotarev problemwhere Rnn is the collection of rational functions of order n. In the case where E and F are real intervals, it was more recently pointed out that if they have different lengths, it is of interest to generalize the foregoing problem to the set Rmn with unequal numerator degree m and denominator degree n. The object of the paper is to investigate the generalized Zolotarev problem in the complex plane. A method is proposed to construct the optimal rational function, which is then applied to the particular example of two line segments-E on the real axis, F parallel to the imaginary axis. Numerical experiments show that the improvement on the classical solution may be amazingly great. Explicit expressions are also provided for special values of data.

Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts

“…The results of [6] and Theorem 4.3 indicate that Theorem 4.2 gives an optimal or asymptotically optimal choice of s j for the RKSR approximation of (A + sI) −1 ϕ on iR without information about distribution of the spectral measure Downloaded 12/30/12 to 129.173.72.87. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php of A within [λ min , λ max ].…”

“…Augmenting U n with the initial vector ϕ (such subspaces are considered, for example, in [7]), one can obtain a sharper bound (without c) than (4.3) in terms of the residual: Looking at (4.5), one can easily recognize the third Zolotaryov problem in the extended complex plane [50,20]. It was discovered in [6] that this problem has a solution with real parameters satisfying at least necessary optimality conditions. This solution can be obtained by restricting (4.6) s j = λ j , j = 1, .…”

“…In [31] (for a slightly different problem of the computation of the resolvent of an unbounded operator on a bounded interval of the imaginary axis) we obtained asymptotically optimal (in the Cauchy-Hadamard sense) pure imaginary s j in a closed form via elliptic integrals. Surprisingly, for the problem considered here, i.e., the optimal rational approximation of the resolvent of a bounded A on the entire imaginary axis, a classical Zolotaryov solution with real s j gives an asymptotically optimal solution that also according to [6] satisfies the necessary optimality conditions. It is important because real shifts have a clear advantage over complex ones for the solution of linear systems with matrices A + s j I (especially with the preconditioned Krylov methods).…”

“…As was already mentioned, the parameter set providingσ n satisfies the necessary optimality conditions[6]. It is not clear if it is the true global minimum of the approximation error function max λ∈[λmin,λmax] |r(λ)| min s∈iR∪{∞} |r(s)| .…”

Zolotarev Iterations for the Matrix Square Root

“…This paper builds upon a stream of research that, in recent years, has sparked renewed interest in the applications of Zolotarev's work on rational approximation to numerical linear algebra. These applications include algorithms for the SVD, the symmetric eigendecomposition, and the polar decomposition [22]; algorithms for the CS decomposition [8]; bounds on the singular values of matrices with displacement structure [4]; computation of spectral projectors [18,9,20]; and the selection of optimal parameters for the alternating direction implicit (ADI) method [26,19]. Zolotarev's functions have even been used to compute the matrix square root [10], however, there is an important distinction between that work and ours: In [10], Zolotarev's functions are not used as the basis of an iterative method.…”