On computing Schur functions and series thereof

Springer Science and Business Media LLC - Tập 50 - Trang 127-141 - 2018
Cy Chan1, Vesselin Drensky2, Alan Edelman3, Raymond Kan4, Plamen Koev5
1Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, USA
2Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
3Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA
4Joseph L. Rotman School of Management, University of Toronto, Toronto, Canada
5Department of Mathematics, San Jose State University, San Jose, USA

Tóm tắt

In this paper, we present two new algorithms for computing all Schur functions $$s_\kappa (x_1,\ldots ,x_n)$$ for partitions $$\kappa $$ such that $$|\kappa |\le N$$ . For nonnegative arguments, $$x_1,\ldots ,x_n$$ , both algorithms are subtraction-free and thus each Schur function is computed to high relative accuracy in floating point arithmetic. The cost of each algorithm per Schur function is $$\mathscr {O}(n^2)$$ .

Tài liệu tham khảo

Biedenharn, L.C., Louck, J.D.: A new class of symmetric polynomials defined in terms of tableaux. Adv. in Appl. Math. 10(4), 396–438 (1989) Butler, R.W., Wood, A.T.A.: Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30(4), 1155–1177 (2002) Byers, G., Takawira, F.: Spatially and temporally correlated MIMO channels: modeling and capacity analysis. IEEE Trans. Veh. Technol. 53, 634–643 (2004) Chen, W.R.: Table for upper percentage points of the largest root of a determinantal equation with five roots. Interstat (2003). http://interstat.statjournals.net/YEAR/2003/articles/0302005.pdf Chen, W.R.: Some new tables of the largest root of a matrix in multivariate analysis: a computer approach from 2 to 6. In: Presented at the 2002 American Statistical Association (2002) Chen, W.Y.C., Louck, J.D.: The factorial Schur function. J. Math. Phys. 34(9), 4144–4160 (1993) Demmel, J., Gu, M., Eisenstat, S., Slapničar, I., Veselić, K., Drmač, Z.: Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl. 299(1–3), 21–80 (1999) Demmel, J., Koev, P.: Accurate and efficient evaluation of Schur and Jack functions. Math. Comp. 75(253), 223–239 (2006) Dumitriu, I., Edelman, A.: Matrix models for beta-ensembles. J. Math. Phys. 43, 5830–5847 (2002) Edelman, A., Koev, P.: Eigenvalue distributions of Beta–Wishart matrices. Random Matrices Theory Appl. 03, 1450,009 (2014) Forrester, P.J.: Log-Gases and Random Matrices, London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010). https://doi.org/10.1515/9781400835416 Gander, W.: Change of basis in polynomial interpolation. Numer. Linear Algebra Appl. 12, 769–778 (2005) Gantmacher, F., Krein, M.: Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, revised edn. AMS Chelsea, Providence (2002) Gao, H., Smith, P., Clark, M.: Theoretical reliability of MMSE linear diversity combining in Rayleigh-fading additive interference channels. IEEE Trans. Commun. 46, 666–672 (1998). https://doi.org/10.1109/26.668742 Gautschi, W., Inglese, G.: Lower bounds for the condition number of Vandermonde matrices. Numer. Math. 52(3), 241–250 (1988) Golub, G., Van Loan, C.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996) Grant, A.: Performance analysis of transmit beamforming. IEEE Trans. Commun. 53, 738–744 (2005) Gross, K.I., Richards, D.S.P.: Total positivity, spherical series, and hypergeometric functions of matrix argument. J. Approx. Theory 59(2), 224–246 (1989) Gutiérrez, R., Rodriguez, J., Sáez, A.J.: Approximation of hypergeometric functions with matricial argument through their development in series of zonal polynomials. Electron. Trans. Numer. Anal. 11, 121–130 (2000) Harding, M.: Explaining the single factor bias of arbitrage pricing models in finite samples. Econom. Lett. 9, 85–88 (2008) James, A.T.: Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Stat. 35, 475–501 (1964) Jeffris, M.: Robust 3D ATR techniques based on geometric invariants. In: Sadjadi, F.A. (ed.) Automatic Target Recognition XV, Proceedings of SPIE, vol. 5807, pp. 190–200. SPIE, Bellingham, WA (2005) Kan, R., Koev, P.: Densities of the extreme eigenvalues of Beta–MANOVA matrices. Preprint (2017) Kaneko, J.: Selberg integrals and hypergeometric functions associated with Jack polynomials. SIAM J. Math. Anal. 24(4), 1086–1110 (1993) Kang, M., Alouini, M.S.: Largest eigenvalue of complex Wishart matrices and performance analysis of MIMO MRC systems. IEEE J. Sel. Areas Commun. 21(3), 418–431 (2003) Koev, P.: http://www.math.sjsu.edu/~koev Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27(1), 1–23 (2005) Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007) Koev, P., Edelman, A.: The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp. 75(254), 833–846 (2006) Macdonald, I.G.: Schur functions: theme and variations. In: Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), Publ. Inst. Rech. Math. Av., vol. 498, pp. 5–39. Univ. Louis Pasteur, Strasbourg (1992) Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995) McKay, M., Collings, I.: Capacity bounds for correlated rician MIMO channels. In: 2005 IEEE International Conference on Communications. ICC 2005., vol. 2, pp. 772–776 (2005) Méndez, M.A.: The umbral calculus of symmetric functions. Adv. Math. 124(2), 207–271 (1996) Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982) Neuman, E.: On complete symmetric functions. SIAM J. Math. Anal. 19(3), 736–750 (1988) Ozyildirim, A., Tanik, Y.: SIR statistics in antenna arrays in the presence of correlated Rayleigh fading. In: IEEE VTS 50th Vehicular Technology Conference, 1999. VTC 1999 - Fall, vol. 1, pp. 67–71 (1999). https://doi.org/10.1109/VETECF.1999.797042 Patterson, N., Price, A., Reich, D.: Population structure and eigenanalysis. PLoS Genet. 2, 2074–2093 (2006) Smidl, V., Quinn, A.: Fast variational PCA for functional analysis of dynamic image sequences. In: Proceedings of the 3rd International Symposium on Image and Signal Processing and Analysis, 2003. ISPA 2003, vol. 1, pp. 555–560 (2003). https://doi.org/10.1109/ISPA.2003.1296958 Stanley, R.P.: Some combinatorial properties of Jack symmetric functions. Adv. Math. 77(1), 76–115 (1989) Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999) Van de Vel, H.: Numerical treatment of a generalized Vandermonde system of equations. Linear Algebra Appl. 17, 149–179 (1977) Verde-Star, L.: Biorthogonal polynomial bases and Vandermonde-like matrices. Stud. Appl. Math. 95(3), 269–295 (1995) Verde-Star, L.: Representation of symmetric functions as Gram determinants. Adv. Math. 140(1), 128–143 (1998)