Numerical Linear Algebra
Canadian mathematical bulletin, Tome 9 (1966) no. 5, pp. 757-801

Voir la notice de l'article provenant de la source Cambridge University Press

The primordial problems of linear algebra are the solution of a system of linear equations and the solution of the eigenvalue problem for the eigenvalues λk, and corresponding eigenvectors of a given matrix A.
Kahan, W. Numerical Linear Algebra. Canadian mathematical bulletin, Tome 9 (1966) no. 5, pp. 757-801. doi: 10.4153/CMB-1966-083-2
@article{10_4153_CMB_1966_083_2,
     author = {Kahan, W.},
     title = {Numerical {Linear} {Algebra}},
     journal = {Canadian mathematical bulletin},
     pages = {757--801},
     year = {1966},
     volume = {9},
     number = {5},
     doi = {10.4153/CMB-1966-083-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-083-2/}
}
TY  - JOUR
AU  - Kahan, W.
TI  - Numerical Linear Algebra
JO  - Canadian mathematical bulletin
PY  - 1966
SP  - 757
EP  - 801
VL  - 9
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-083-2/
DO  - 10.4153/CMB-1966-083-2
ID  - 10_4153_CMB_1966_083_2
ER  - 
%0 Journal Article
%A Kahan, W.
%T Numerical Linear Algebra
%J Canadian mathematical bulletin
%D 1966
%P 757-801
%V 9
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-083-2/
%R 10.4153/CMB-1966-083-2
%F 10_4153_CMB_1966_083_2

[1] 1. Barron, D.W. and Swinnerton-Dyer, H.P.F., Solution of Simultaneous Linear Equations Using a Magnetic-Tape Store. The Computer Journal, Vol. 3, (1960) pp. 28-33. Google Scholar

[2] 2. Bauer, F.L., Optimally Scaled Matrices. Numerische Math., Vol. 5, (1963) pp. 73-87. Google Scholar

[3] 3. Birkhoff, G., Varga, R.S., and Young, D., Alternating Direction Implicit Methods. Advances in Computers, Vol. 3, Academic Press (1962). Google Scholar

[4] 4. Bodewig, E., Matrix Calculus, 2nd ed. North Holland (1959). (A catalogue of methods and tricks, with historical asides.) Google Scholar

[5] 5. Cayless, M.A., Solution of Systems of Ordinary and Partial Differential Equations by Quasi- Diagonal Matrices. The Computer Journal, Vol. 4, (1961) pp. 54-61. Google Scholar

[6] 6. Day, M.M., Normed Linear Spaces. Springer (1962). Google Scholar

[7] 7. Douglas, J. and Gunn, J.E., A General Formulation of Alternating Direction Methods, part I. Numerische Math. Vol. 6, (1965) pp. 428-453. Google Scholar

[8] 8. Dunford, and Schwartz, , Linear Operators, part I: General Theory. Interscience (1958). Google Scholar

[9] 9. Engeli, M. et. al., Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems. Mitteilung Nr. 8 aus dem Inst, für angew. Math, an dur E.T.H., Zurich: Birkhauser (1959). Google Scholar

[10] 10. Faddeev, D.K. and Faddeeva, V.N., Computational Methods of Linear Algebra, translated from the Russian by Williams, R.C.. Freeman, W.H. (1964). (This text is a useful catalogue, but weak on error-analysis. A new augmented Russian edition has appeared.) Google Scholar

[11] 11. Forsythe, G.E. and Wasow, W.R., Finite Difference Methods for Partial Differential Equations. Wiley (1960). (A detailed text.) Google Scholar

[12] 12. Fox, L., The Numerical Solution of Two-Point Boundary Problems in Ordinary Differential Equations. Oxford Univ. Press (1957). Google Scholar

[13] 13. Fox, L., Numerical Solution of Ordinary and Partial Differential Equations. Pergamon Press (1962) (Ed.). (Based on a Summer School held in Oxford, Aug.-Sept. 1961.) Google Scholar

[14] 14. Fox, L., An Introduction to Numerical Linear Algebra. Oxford Univ. Press (1964). (This is an excellent introduction.) Google Scholar

[15] 15. Gauss, C.F., Letter to Gerling, 26 Dec. 1823. Werke Vol. 9, (1823) pp. 278-281. A translation by Forsythe, G.E. appears in MTAC Vol. 5 (1950), pp. 255–258. Google Scholar

[16] 16. Gauss, C.F., Supplementum …. Werke, GÖttingen, Vol. 4, (1826) pp. 55-93. Google Scholar

[17] 17. Golub, G. and Kahan, W., Calculating the Singular Values and Pseudo-Inverse of a Matrix. J. SIAM Numer. Anal. (B), Vol. 2, (1965) pp. 205-224. Google Scholar

[18] 18. Gunn, J.E., The Solution of Elliptic Difference Equations by Semi-Explicit Iterative Techniques. J. SIAM Numer. Anal. Ser. B, Vol. 2, (1965) pp. 24-45. Google Scholar

[19] 19. Householder, A.S., The Theory of Matrices in Numerical Analysis. Blaisdell (1964). (An elegant but terse treatment, including material on matrix norms which is otherwise hard to find in Numerical Analysis texts.) Google Scholar

[20] 20. IFIP: Proceedings of the Congress of the International Federation for Information Processing, held in New York City, May 24–29, 1965. Spartan Books (1965). Google Scholar

[21] 21. Jacobi, C.G.J., Űber eine neue AuflÖsumgsart …. Astr. Nachr. Vol. 22, No. 523, (1845) pp. 297-306. (Reprinted in his Werke Vol. 3, p. 467.) Google Scholar

[22] 22. Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, translated from the Russian by Brown, D.E.. Pergamon (1964). Google Scholar

[23] 23. Kellog, R.B. and Spanier, J., On Optimal Alternating Direction Parameters for Singular Matrices. Math, of Comp., Vol. 19, (1965) pp. 448-451. Google Scholar

[24] 24. Klyuyev, V.V. and Kokovkin-Shcherbak, N.I., On the Minimization of the Number of Arithmetic Operations for the Solution of Linear Algebraic Systems of Equations. Journal of Computational Math, and Math. Phys., Vol. 5, (1965) pp. 21–33 (Russian). A translation, by Tee, G.J., is available as Tech. Rep't CS24 from the Computer Sci. Dep't of Stanford University. (My copy has mistakes in it which I have not yet sorted out.) Google Scholar

[25] 25. Liouville, J., Sur le développement des fonctions en series …. II, J. Math, pures appl. (1), Vol. 2, (1837) pp. 16-37. Google Scholar

[26] 26. Martin, D.W. and Tee, G.J., Iterative Methods for Linear Equations with Symmetric Positive Definite Matrix. The Computer Journal Vol. 4, (1961) pp. 242-254. (An excellent survey.) Google Scholar

[27] 27. Murray, W.A. and Lynn, M.S., A Computer-Oriented Description of the Peaceman-Rachford ADI Method. The Computer Journal, Vol. 8, (1965) pp. 166-175. Google Scholar

[28] 28. von Neumann, J. and Goldstine, H.H., Numerical Inverting of Matrices of High Order. Bull. Amer. Math. Soc, Vol. 53, (1947) pp. 1021-1099. Google Scholar

[29] 29. von Neumann, J. and Goldstine, H.H., "…. part II" Proc. Amer. Math. Soc, Vol. 2, (1951) pp. 188-202. Google Scholar

[30] 30. Rall, L.B., Error in-Digital Computation. Two volumes (1965) Wiley. (Contains a valuable bibliography.) Google Scholar

[31] 31. Smith, G.D., Numerical Solution of Partial Differential Equations. Oxford Univ. Press (1965). (This is an introductory text.) Google Scholar

[32] 32. Stiefel, E.L., Some Special Methods of Relaxation Technique appearing in Simultaneous Linear Equations and the Determination of Eigenvalues. National Bureau of Standards Applied Math. Series No. 29 (1953). (A subsequent article by Rosser, J.B. in this same book contains more details about conjugate gradient methods.) Google Scholar

[33] 33. Stiefel, E.L., Kernel Polynomials in Linear Algebra and their Applications. in Further Contributions …, National Bureau of Standards Applied Math., Series No. 49 (1958). Google Scholar

[34] 34. Turing, A.M., Rounding-off Errors in Matrix Processes. Quart. J. Mech. Appl. Math. 1, (1948) pp. 287-308. Google Scholar

[35] 35. Varga, R.S., Matrix Iterative Analysis. Prentice Hall (1962). (An important treatise on those iterative methods most widely used to solve large boundary-value problems.) Google Scholar

[36] 36. Wilkinson, J.H., Rounding Errors in Algebraic Processes, in Information Processing, (1960) pp. 44-53. Proceedings of a UNESCO conference held in Paris in 1959. Google Scholar

[37] 37. Wilkinson, J.H., Error Analysis of Direct Methods of Matrix Inversion. J. Assoc. Computing Machinery, Vol. 8 (1961) pp. 281-330. Google Scholar

[38] 38. Wilkinson, J.H., Rounding Errors in Algebraic Processes. National Physical Lab. Note on Applied Science No. 32(1963), Her Majesty's Stationery Office. Google Scholar

Cité par Sources :