On the Foundations of Computational Mathematics

Handbook of Numerical Analysis

Volumn 11, Issue , 2003, Pages 3-34

  Baxter, B J C ^a   Iserles, A ^b  

^a UNIVERSITY OF LONDON   (United Kingdom)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2942687316     PISSN: 15708659     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1570-8659(02)11001-5     Document Type: Review

References (78)

1. Allgower E., and Georg K. Numerical Continuation Methods. An Introduction (1990), Springer-Verlag, Berlin
2. Arnold V. Mathematical Methods of Classical Mechanics (1989), Springer-Verlag, Berlin
3. Baxter, B. (2002). Positive definite functions on Hilbert space, Tech. rep., Numerical Analysis Group, University of Cambridge
4. Bellman R. Stability Theory of Differential Equations (1969), Dover, New York
5. Berg C., Christensen J.P.R., and Ressel P. Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Graduate Texts in Math. 100 (1984), Springer, New York
6. Blum L., Cucker F., Shub M., and Smale S. Complexity and Real Computation (1998), Springer, New York
7. Brouder C. Runge-Kutta methods and renormalization. Eur. Phys. J. 12 (1999) 521-534
8. Buchberger B. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes Math. 4 (1970) 374-383
9. In: Buchberger B., and Winkler F. (Eds). Gröbner Bases and Applications. London Math. Soc. Lecture Note Ser. 251 (1998), Cambridge University Press, Cambridge
10. Budd C., and Iserles A. Geometric integration: Numerical solution of differential equations on manifolds. Philos. Trans. Royal Soc. A 357 (1999) 945-956
11. Budd C., and Piggott M. Geometric integration and its applications. In: Ciarlet P.G., and Cucker F. (Eds). Foundations of Computational Mathematics. Handbook of Numerical Analysis 11 (2003), North-Holland, Amsterdam 35-139
12. Buhmann M. Radial basis functions. Acta Numerica 9 (2000) 1-38
13. Butcher J. An algebraic theory of integration methods. Math. Comp. 26 (1972) 79-106
14. Chard J., and Shapiro V. A multivector data structure for differential forms and equations. Math. Comput. Simulation 54 (2000) 33-64
15. Cheung D., Cucker F., and Ye Y. Linear programming and condition numbers under the real number computation model. In: Ciarlet P.G., and Cucker F. (Eds). Foundations of Computational Mathematics. Handbook of Numerical Analysis 11 (2003), North-Holland, Amsterdam 141-207
16. Ciarlet P.G. The Finite Element Method for Elliptic Problems (1978), North-Holland, Amsterdam
17. Cohen A., Dahmen W., Daubechies I., and DeVore R. Tree approximation and optimal encoding. Appl. Comput. Harmon. Anal. 11 (2001) 192-226
18. Connes A., and Kreimer D. Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199 (1998) 203-242
19. Cox D., Little J., and O'Shea D. Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra (1997), Springer, New York
20. Curry H., and Schoenberg I. Pólya frequency functions IV. The fundamental spline functions and their limits. J. Anal. Math. 17 (1966) 71-107
21. Dahmen W. Wavelet and multiscale methods for operator equations. Acta Numerica 6 (1997) 55-228
22. Daubechies I. Ten Lectures on Wavelets (1992), SIAM, Philadelphia
23. de Boor C., Höllig K., and Riemenschneider S. Box Splines (1993), Springer, New York
24. DeVore R. Nonlinear approximation. Acta Numerica 7 (1998) 51-150
25. Duchon J. Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. RAIRO Anal. Numer. 10 (1976) 5-12
26. Edelman, A. (20xx). Random matrices, Unpublished course notes for MIT's class 18.325
27. Franke R. Scattered data interpolation: Tests of some methods. Math. Comp. 38 (1982) 181-200
28. Gardner R. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. 39 (2002) 355-405
29. Golub G., and Van Loan C. Matrix Computations. 3rd edn. (1996), Johns Hopkins University Press, Baltimore
30. Hairer E., Lubich C., and Wanner G. Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations (2002), Springer-Verlag, Berlin
31. Hammel S., Yorke J., and Grebogi C. Do numerical orbits of chaotic dynamical processes represent true orbits?. J. Complexity 3 (1987) 136-145
32. Harder R., and Desmarais R. Interpolation using surface splines. J. Aircraft 9 (1972) 189-191
33. Hardy G.H. A Mathematician's Apology (1992), Cambridge University Press, Cambridge
34. Hardy R. Multiquadric equations of tomography and other irregular surfaces. J. Geophys. Res. 76 (1971) 1905-1915
35. Helgason S. Topics in Harmonic Analysis on Homogeneous Spaces (1981), Birkhäuser, Boston
36. Hestenes D. Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics (1984), D. Reidel, Dordrecht
37. Hiptmair R. Finite elements in computational electromagnetism. Acta Numerica 11 (2002) 237-339
38. Hodges A. Alan Turing, The Enigma (1983), Burnet Books, London
39. Hubert E. Factorization-free decomposition algorithms in differential algebra. J. Symbolic Comput. 29 (2000) 641-662
40. Iserles A., Munthe-Kaas H.Z., Nørsett S.P., and Zanna A. Lie-group methods. Acta Numerica 9 (2000) 215-365
41. Keller H. Lectures on Numerical Methods in Bifurcation Problems. Tata Institute of Fundamental Research Lectures on Mathematics and Physics 79 (1987), Springer-Verlag, Berlin
42. Körner T. Fourier Analysis (1988), Cambridge University Press, Cambridge
43. In: Levy S. (Ed). Flavors of Modern Geometry (1997), Cambridge University Press, Cambridge
44. Li T. Numerical solution of polynomial systems by homotopy continuation methods. In: Ciarlet P.G., and Cucker F. (Eds). Foundations of Computational Mathematics. Handbook of Numerical Analysis 11 (2003), North-Holland, Amsterdam 209-303
45. Marsden J., and West M. Discrete mechanics and variational integrators. Acta Numerica 10 (2001) 357-514
46. McLachlan R., and Quispel G. Six lectures on the geometric integration of ODEs. In: DeVore R., Iserles A., and Sűli E. (Eds). Foundations of Computational Mathematics, Oxford 1999 (2001), Cambridge University Press, Cambridge 155-210
47. Micchelli C. Interpolation of scattered data: distance matrices and conditionally positive functions. Constr. Approx. 2 (1986) 11-22
48. Micchelli C. Mathematical Aspects of Geometric Modeling (1995), SIAM, Philadelphia
49. Milman V., and Schechtman G. Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Math. 1200 (1986), Springer-Verlag, Berlin
50. Natterer F. Numerical methods in tomography. Acta Numerica 8 (1999) 107-142
51. Parlett B. Some basic information on information-based complexity theory. Bull. Amer. Math. Soc. 26 (1992) 3-27
52. Sapiro G. Introduction to partial differential equations and variational formulations in image processing. In: Ciarlet P.G., and Cucker F. (Eds). Foundations of Computational Mathematics. Handbook of Numerical Analysis 11 (2003), North-Holland, Amsterdam 383-461
53. Schoenberg I. Remarks to Maurice Fréchet's article: 'Sur la definition d'une classe d'espace distanciés vectoriellement applicable sur l'espace d'Hilbert. Ann. of Math. 36 (1935) 724-732
54. Schoenberg I. On certain metric spaces arising from Euclidean space by a change of metric and their embedding in Hilbert space. Ann. of Math. 38 (1937) 787-793
55. Schoenberg I. Metric spaces and completely monotone functions. Ann. of Math. 39 (1938) 811-841
56. Smale S. Some remarks on the foundations of numerical analysis. SIAM Rev. 32 (1990) 211-220
57. Smale S. Complexity theory and numerical analysis. Acta Numerica 6 (1997) 523-551
58. Stewart G. The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM J. Numer. Anal. 17 (1980) 403-409
59. Strang G., and Fix G. An Analysis of the Finite Element Method (1973), Prentice-Hall, Englewood Cliffs, NJ
60. Stuart A., and Humphries A. Numerical Analysis of Dynamical Systems (1996), Cambridge University Press, Cambridge
61. Temlyakov V. Nonlinear methods of approximation. Found. Comp. Math. 3 (2002) 33-107
62. Terras A. Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts 4 (1999), Cambridge University Press, Cambridge
63. Tonti E. Sulla struttura formale delle teorie fisiche. Rend. Sem. Mat. Fis. Milano 46 (1976) 163-257 (in Italian)
64. Traub J., and Woźniakowski H. Perspectives on information-based complexity. Bull. Amer. Math. Soc. 26 (1992) 29-52
65. Trotter H. Eigenvalue distributions of large Hermitian matrices; Wigner's semi-circle law and a theorem of Kac, Murdock and Szegő. Adv. Math. 54 (1984) 67-82
66. von Goethe J. Erotic Poems (1997), Oxford University Press (transl. by D. Luke)
67. von Neumann J., and Schoenberg I. Fourier integrals and metric geometry. Trans. Amer. Math. Soc. 50 (1941) 226-251
68. von zur Gathen J., and Gerhard J. Modern Computer Algebra (1999), Cambridge University Press, New York
69. Watson G. Choice of norms for data fitting and function approximation. Acta Numerica 7 (1998) 337-377
70. Werschulz A. The Computational Complexity of Differential and Integral Equations. An Information-Based Approach (1991), Oxford University Press, New York
71. Wilkinson J. Error analysis of floating-point computation. Numer. Math. 2 (1960) 319-340
72. Williams D. Weighing the Odds (2001), Cambridge University Press, Cambridge
73. Wilson K. Grand challenges to computational science. Future Gen. Comput. Sys. 5 (1989) 171-189
74. Winograd S. On the multiplicative complexity of the discrete Fourier transform. Adv. in Math. 32 (1979) 83-117
75. Yamaguti M., and Maeda Y. Chaos in finite difference schemes. In: Ciarlet P.G., and Cucker F. (Eds). Foundations of Computational Mathematics. Handbook of Numerical Analysis 11 (2003), North-Holland, Amsterdam 305-381
76. Young N. An Introduction to Hilbert Space (1988), Cambridge University Press, Cambridge
77. In: Zaremba S. (Ed). Applications of Number Theory to Numerical Analysis (1972), Academic Press, New York
78. Zeilberger D. A fast algorithm for proving terminating hypergeometric series identities. Discrete Math. 80 (1990) 207-211