## 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

Zbl 1390.41015

### Software:

FEAST; NLEIGS; RKToolbox; Matlab; AAA
Full Text:

### References:

  Berljafa, M.; Elsworth, S.; Güttel, S., A Rational Krylov Toolbox for MATLAB, (2017), Manchester Institute for Mathematical Sciences, The University of Manchester, UK, available for download at  Ruhe, A., Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. Matrix pairs, Linear Algebra Appl., 198, 283-295, (1994) · Zbl 0810.65031  Berljafa, M.; Güttel, S., The RKFIT algorithm for nonlinear rational approximation, SIAM J. Sci. Comput., 39, 5, A2049-A2071, (2017) · Zbl 1373.65037  Berljafa, M.; Güttel, S., Generalized rational Krylov decompositions with an application to rational approximation, SIAM J. Matrix Anal. Appl., 36, 2, 894-916, (2015) · Zbl 1319.65028  Berljafa, M., Rational Krylov Decompositions: Theory and Applications, (2017), The University of Manchester: The University of Manchester Manchester, UK, available as MIMS EPrint 2017.6 at  Nakatsukasa, Y.; Sète, O.; Trefethen, L. N., The AAA algorithm for rational approximation, SIAM J. Sci. Comput., 40, 3, A1494-A1522, (2018) · Zbl 1390.41015  Schneider, C.; Werner, W., Some new aspects of rational interpolation, Math. Comp., 47, 175, 285-299, (1986) · Zbl 0612.65005  Lawrence, P. W.; Corless, R. M., Stability of rootfinding for barycentric Lagrange interpolants, Numer. Algorithms, 65, 3, 447-464, (2014) · Zbl 1297.65051  Klein, G., Applications of Linear Barycentric Rational Interpolation, (2012), Université de Fribourg, Ph.D. thesis  Van Beeumen, R.; Meerbergen, K.; Michiels, W., Compact rational Krylov methods for nonlinear eigenvalue problems, SIAM J. Matrix Anal. Appl., 36, 2, 820-838, (2015) · Zbl 1319.65042  Mehrmann, V.; Voss, H., Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods, GAMM-Mitt., 27, 121-152, (2004) · Zbl 1071.65074  Güttel, S.; Tisseur, F., The nonlinear eigenvalue problem, Acta Numer., 26, 1-94, (2017) · Zbl 1377.65061  Güttel, S.; Van Beeumen, R.; Meerbergen, K.; Michiels, W., NLEIGS: a class of fully rational Krylov methods for nonlinear eigenvalue problems, SIAM J. Sci. Comput., 36, 6, A2842-A2864, (2014) · Zbl 1321.47128  Bagby, T., On interpolation by rational functions, Duke Math. J., 36, 1, 95-104, (1969) · Zbl 0223.30049  Sakurai, T.; Sugiura, H., A projection method for generalized eigenvalue problems using numerical integration, J. Comput. Appl. Math., 159, 1, 119-128, (2003) · Zbl 1037.65040  Beyn, W.-J., An integral method for solving nonlinear eigenvalue problems, Linear Algebra Appl., 436, 10, 3839-3863, (2012) · Zbl 1237.65035  Güttel, S.; Polizzi, E.; Tang, P. T.P.; Viaud, G., Zolotarev quadrature rules and load balancing for the FEAST eigensolver, SIAM J. Sci. Comput., 37, 4, A2100-A2122, (2015) · Zbl 1321.65055  Austin, A. P.; Trefethen, L. N., Computing eigenvalues of real symmetric matrices with rational filters in real arithmetic, SIAM J. Sci. Comput., 37, 3, A1365-A1387, (2015) · Zbl 1328.15016  Van Barel, M., Designing rational filter functions for solving eigenvalue problems by contour integration, Linear Algebra Appl., 502, 346-365, (2016) · Zbl 1386.65115  Lietaert, P.; Pérez, J.; Vandereycken, B.; Meerbergen, K., Automatic rational approximation and linearization of nonlinear eigenvalue problems, (2018), preprint  Vandebril, R.; Van Barel, M.; Mastronardi, N., Matrix Computations and Semiseparable Matrices, Volume 2: Eigenvalue and Singular Value Methods, (2008), Johns Hopkins University Press: Johns Hopkins University Press Baltimore, MD, USA · Zbl 1175.65045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.