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Conversions between barycentric, RKFUN, and Newton representations of rational interpolants. (English) Zbl 1429.65075

Let \(r(z)\) be the rational function of type \([m,m]\) interpolating the data \((z_i,f_i)_{i=0}^m\). Different forms to represent \(r(z)\) exist. The barycentric form: \(r(z)=\sum_{j=0}^m f_j r_j(z)\), \(r_j(z)=\frac{w_j/(z-z_j)}{\sum_{i=0}^m w_i/(z-z_i)}\), and the Newton form: \(r(z)=\sum_{j=0}^m d_jb_j(z)\), with Newton basis \(b_j(z)=\frac{z-\sigma_{j-1}}{\beta_j(h_j-k_jz)}b_{j-1}(z)\), \(b_0=1\).
The link between both is given in this paper via the representation used in the Matlab RKFUN (Rational Krylov) toolbox. There both previous representations result in the same shape since they represent the basis functions as solution of an Arnoldi process: \(z[r_0,\ldots,r_m]W_m=[r_0,\ldots,r_m]Z_mW_m\) and \(z[b_0,\ldots,b_m]M_m=[b_0,\ldots,b_m]N_m\), respectively, where \(Z_m=\mathrm{diag}(z_0,\ldots,z_m)\) and \(W_m,M_m,N_m\) are all bidiagonal upper Hessenberg matrices of size \((m+1)\times m\). This allows one to easily switch between the parameters in both forms. This idea is used to get a rational matrix approximant for solving a nonlinear eigenvalue problem. The AAA algorithm of Y. Nakatsukasa et al. [SIAM J. Sci. Comput. 40, No. 3, A1494–A1522 (2018; Zbl 1390.41015)] helps to find a good approximant in barycentric form which is directly converted to Newton form for the linearization which can then be solved using for example a rational Krylov method.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
41A20 Approximation by rational functions
30E10 Approximation in the complex plane
65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems

Citations:

Zbl 1390.41015
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References:

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