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1: 3.8 Nonlinear Equations
Bisection Method
Bisection of this interval is used to decide where at least one zero is located. …
Secant Method
Eigenvalue Methods
2: Bibliography N
  • National Physical Laboratory (1961) Modern Computing Methods. 2nd edition, Notes on Applied Science, No. 16, Her Majesty’s Stationery Office, London.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.
  • G. Nemes (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.
  • Y. Nievergelt (1995) Bisection hardly ever converges linearly. Numer. Math. 70 (1), pp. 111–118.
  • 3: Bibliography K
  • D. K. Kahaner, C. Moler, and S. Nash (1989) Numerical Methods and Software. Prentice Hall, Englewood Cliffs, N.J..
  • E. H. Kaufman and T. D. Lenker (1986) Linear convergence and the bisection algorithm. Amer. Math. Monthly 93 (1), pp. 48–51.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • M. K. Kerimov (1999) The Rayleigh function: Theory and computational methods. Zh. Vychisl. Mat. Mat. Fiz. 39 (12), pp. 1962–2006.
  • A. D. Kerr (1978) An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.
  • 4: 20.11 Generalizations and Analogs
    Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …
    5: 27.15 Chinese Remainder Theorem
    Their product m has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result ( mod m ) , which is correct to 20 digits. … Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …
    6: 34.9 Graphical Method
    §34.9 Graphical Method
    The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For an account of this method see Brink and Satchler (1993, Chapter VII). For specific examples of the graphical method of representing sums involving the 3 j , 6 j , and 9 j symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).
    7: 20 Theta Functions
    Chapter 20 Theta Functions
    8: Bibliography I
  • L. Infeld and T. E. Hull (1951) The factorization method. Rev. Modern Phys. 23 (1), pp. 21–68.
  • K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
  • M. E. H. Ismail and E. Koelink (2011) The J -matrix method. Adv. in Appl. Math. 46 (1-4), pp. 379–395.
  • A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.
  • A. R. Its and V. Yu. Novokshënov (1986) The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, Vol. 1191, Springer-Verlag, Berlin.
  • 9: Publications
  • A. Youssef (2007) Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007. PDF
  • B. Saunders and Q. Wang (2010) Tensor Product B-Spline Mesh Generation for Accurate Surface Visualizations in the NIST Digital Library of Mathematical Functions, in Mathematical Methods for Curves and Surfaces, Proceedings of the 2008 International Conference on Mathematical Methods for Curves and Surfaces (MMCS 2008), Lecture Notes in Computer Science, Vol. 5862, (M. Dæhlen, M. Floater., T. Lyche, J. L. Merrien, K. Mørken, L. L. Schumaker, eds), Springer, Berlin, Heidelberg (2010) pp. 385–393. PDF
  • B. I. Schneider, B. R. Miller and B. V. Saunders (2018) NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53. PDF
  • 10: 17.18 Methods of Computation
    §17.18 Methods of Computation
    Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. Shanks (1955) applies such methods in several q -series problems; see Andrews et al. (1986).