Accurate and efficient evaluation of Schur and Jack functions

Mathematics of Computation

Volumn 74, Issue 253, 2006, Pages 223-239

  Demmel, James ^a   Koev, Plamen ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

High relative accuracy; Jack function; Schur function; Zonal polynomial

Indexed keywords

EID: 32944461288     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-05-01780-1     Document Type: Article

References (30)

1. P.-A. Absil, A. Edelman, and P. Koev, On the largest principal angle between random subspaces, http : //www.csit.fsu.edu/~absil/Publi/ RandomSubspaces.htm, Submitted to Linear Algebra Appl..
2. ANSI/IEEE, New York, IEEE Standard for Binary Floating Point Arithmetic, Std 754-1985 ed., 1985.
3. Á. Björck and V. Pereyra, Solution of Vandermonde systems of equations, Math. Comp. 24 (1970), 893-903. MR0290541 (44:7721)
4. R. W. Butler and A. T. A. Wood, Laplace approximations for hypergeometric functions with matrix argument, Ann. Statist. 30 (2002), no. 4, 1155-1177. MR1926172 (2003h:62076)
5. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to algorithms, second ed., MIT Press, Cambridge, MA, 2001. MR1848805 (2002e:68001)
6. J. Demmel, Accurate SVDs of structured matrices, SIAM J. Matrix Anal. Appl. 21 (1999), no. 2, 562-580. MR1742810 (2001h:65036)
7. J. Demmel and P. Koev, Necessary and sufficient conditions for accurate and efficient rational function evaluation and factorizations of rational matrices. Structured matrices in mathematics, computer science, and engineering. II (Boulder, CO, 1999), Amer. Math. Soc., Providence, RI, 2001, pp. 117-143. MR1855508 (2002h:65055)
8. _, The accurate, and efficient solution of a totally positive generalized Vandermonde linear system. SIAM J. Matrix Anal. Appl., 27 (2005), no. 1, 142-152.
9. I. Dumitriu, A. Edelman, and F. Shuman, A Maple Package for Computing Multivariate Orthogonal Polynomials, http://www.math.berkeley.edu/~dumitriu.
10. A. Edelman and B. Sutton. Tails of condition number distributions, submitted to SIAM J. Matrix Anal. Appl.
11. W. Gautschi, How (un)stable are Vandermonde systems?, Asymptotic and computational analysis (Winnipeg, MB, 1989), Lecture Notes in Pure and Appl. Math., vol. 124, Dekker, New York, 1990, pp. 193-210. MR1052434 (91f:65080)
12. R. Gutiérrez, J. Rodriguez, and A. J. Sáez, Approximation of hypergeometric functions with matricial argument through their development in series of zonal polynomials, Electron. Trans. Numer. Anal. 11 (2000), 121-130. MR1799027 (2002b:33004)
13. G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work., AMS Chelsea Publishing Company, New York, 1999. MR0106147 (21:4881)
14. N. J. Higham, Stability analysis of algorithms for solving confluent Vandermonde-like systems, SIAM J. Mat. Anal. Appl. 11 (1990), no. 1, 23-41. MR1032215 (91d:65062)
15. _, Accuracy and stability of numerical algorithms, second ed., Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. MR1927606 (2003g:65064)
16. G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981, With a foreword by P. M. Cohn, With an introduction by Gilbert de B. Robinson. MR0644144 (83k:20003)
17. F. Knop and S. Sahi. A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 9-22. MR1437493 (98k:33040)
18. P. Koev, http://www-math.mit.edu/~plamen.
19. P. Koev, Accurate eigenvalues and SVDs of totally nonnegative matrices, SIAM J. Matrix Anal. Appl. 27 (2005), no. 1, 1-23.
20. P. Koev and A. Edelman, The efficient, evaluation of the hypergeometric function of matrix argument, Math. Comp., to appear.
21. I. G. Macdonald, Schur functions: theme and variations, Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), Publ. Inst. Rech. Math. Av., vol. 498, Univ. Louis Pasteur, Strasbourg, 1992, pp. 5-39. MR1308728 (95m:05245)
22. _, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, 1995. MR1354144 (96h:05207)
23. The MathWorks, Inc., Natick, MA, MATLAB reference guide, 1992.
24. M. Monagan, K. Geddes, K. Heal, G. Labahn, and S. Vorkoetter, Maple V Programming guide for release 5, Springer-Verlag, 1997.
25. R. J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons Inc., New York, 1982. Wiley Series in Probability and Mathematical Statistics. MR0652932 (84c:62073)
26. D. Priest, Algorithms for arbitrary precision floating point arithmetic, Proceedings of the 10th Symposium on Computer Arithmetic (Grenoble, France) (P. Kornerup and D. Matula, eds.), IEEE Computer Society Press, June 26-28 1991, pp. 132-145.
27. A. Schönhage and V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing (Arch. Elektron. Rechnen) 7 (1971), 281-292. MR0292344 (45:1431)
28. R. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989). no. 1, 76-115. MR1014073 (90g:05020)
29. _, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. MR1676282 (2000k:05026)
30. H. Van de Vel, Numerical treatment of a generalized Vandermonde system of equations, Linear Algebra Appl. 17 (1977), 149-179. MR0474757 (57:14390)