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1: 3.8 Nonlinear Equations
Bisection Method
Secant Method
Steffensen’s Method
Eigenvalue Methods
2: 29.20 Methods of Computation
A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …
3: 3.9 Acceleration of Convergence
§3.9(v) Levin’s and Weniger’s Transformations
4: Bibliography T
  • J. F. Traub (1964) Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..
  • 5: 3.11 Approximation Techniques
    to f ( x ) , then the coefficients a k can be computed iteratively. … The iterative process converges locally and quadratically (§3.8(i)). A method for obtaining a sufficiently accurate first approximation is described in the next subsection. … For details and examples of these methods, see Clenshaw (1957, 1962) and Miller (1966). … The method of the fast Fourier transform (FFT) exploits the structure of the matrix Ω with elements ω n j k , j , k = 0 , 1 , , n - 1 . …
    6: Bibliography S
  • D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.
  • M. J. Seaton (1984) The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.
  • J. Segura (1998) A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.
  • J. Segura (2001) Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.
  • A. Sidi (2003) Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.
  • 7: 19.36 Methods of Computation
    §19.36 Methods of Computation
    §19.36(i) Duplication Method
    When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …
    §19.36(iv) Other Methods
    Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. …
    8: Bibliography G
  • B. Gabutti and G. Allasia (2008) Evaluation of q -gamma function and q -analogues by iterative algorithms. Numer. Algorithms 49 (1-4), pp. 159–168.
  • B. Gabutti (1979) On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.
  • B. Gabutti (1980) On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.
  • M. L. Glasser (1979) A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.
  • B. Guo (1998) Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.
  • 9: 22.20 Methods of Computation
    §22.20 Methods of Computation
    Four iterations of (22.20.1) lead to c 4 = 6.5×10⁻¹² . … If k = k = 1 / 2 , then three iterations of (22.20.1) give M = 0.84721 30848 , and from (22.20.6) K = π / ( 2 M ) = 1.85407 46773 — in agreement with the value of ( Γ ( 1 4 ) ) 2 / ( 4 π ) ; compare (23.17.3) and (23.22.2). … If k = 1 - i , then four iterations of (22.20.1) give K = 1.23969 74481 + i 0.56499 30988 . … For additional information on methods of computation for the Jacobi and related functions, see the introductory sections in the following books: Lawden (1989), Curtis (1964b), Milne-Thomson (1950), and Spenceley and Spenceley (1947). …
    10: Bibliography O
  • A. M. Odlyzko (1995) Asymptotic Enumeration Methods. In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.), pp. 1063–1229.
  • T. Oliveira e Silva (2006) Computing π ( x ) : The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • F. W. J. Olver (1950) A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations. Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.
  • J. M. Ortega and W. C. Rheinboldt (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.