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1: 3.11 Approximation Techniques
§3.11(v) Least Squares Approximations
For further information on least squares approximations, including examples, see Gautschi (1997a, Chapter 2) and Björck (1996, Chapters 1 and 2). …
2: Bibliography P
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • R. Piessens (1984a) Chebyshev series approximations for the zeros of the Bessel functions. J. Comput. Phys. 53 (1), pp. 188–192.
  • R. Piessens and S. Ahmed (1986) Approximation for the turning points of Bessel functions. J. Comput. Phys. 64 (1), pp. 253–257.
  • M. J. D. Powell (1967) On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9 (4), pp. 404–407.
  • P. J. Prince (1975) Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 1 (4), pp. 372–379.
  • 3: 20 Theta Functions
    Chapter 20 Theta Functions
    4: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • 5: Guide to Searching the DLMF
    Table 1: Query Examples
    Query Matching records contain
    Fourier or series at least one of the words “Fourier” or “series”.
    Fourier (transform or series) at least one of “Fourier transform” or “Fourier series”.
    J_n@(x or z)= at least one of the math fragments J n ( x ) = or J n ( z ) , emphasizing that J n is a function.
    sin x and (J_nu(z) or I_nu(z)) both sin x and at least one of the two functions J ν ( z ) or I ν ( z ) .
    trigonometric^2 + trig$^2 any sum of the squares of two trigonometric functions such as sin 2 z + cos 2 z .
    6: 3.8 Nonlinear Equations
    Bisection of this interval is used to decide where at least one zero is located. … Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … Initial approximations to the zeros can often be found from asymptotic or other approximations to f ( z ) , or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … Consider x = 20 and j = 19 . We have p ( 20 ) = 19 ! and a 19 = 1 + 2 + + 20 = 210 . …
    7: 10.75 Tables
  • Achenbach (1986) tabulates J 0 ( x ) , J 1 ( x ) , Y 0 ( x ) , Y 1 ( x ) , x = 0 ( .1 ) 8 , 20D or 18–20S.

  • Makinouchi (1966) tabulates all values of j ν , m and y ν , m in the interval ( 0 , 100 ) , with at least 29S. These are for ν = 0 ( 1 ) 5 , 10, 20; ν = 3 2 , 5 2 ; ν = m / n with m = 1 ( 1 ) n 1 and n = 3 ( 1 ) 8 , except for ν = 1 2 .

  • Bickley et al. (1952) tabulates x n I n ( x ) or e x I n ( x ) , x n K n ( x ) or e x K n ( x ) , n = 2 ( 1 ) 20 , x = 0 (.01 or .1) 10(.1) 20, 8S; I n ( x ) , K n ( x ) , n = 0 ( 1 ) 20 , x = 0 or 0.1 ( .1 ) 20 , 10S.

  • Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of K n ( z ) and K n ( z ) , for n = 2 ( 1 ) 20 , 9S.

  • Zhang and Jin (1996, p. 322) tabulates ber x , ber x , bei x , bei x , ker x , ker x , kei x , kei x , x = 0 ( 1 ) 20 , 7S.

  • 8: Bibliography Y
  • H. A. Yamani and W. P. Reinhardt (1975) L -squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.
  • K. Yang and M. de Llano (1989) Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.
  • 9: 26.13 Permutations: Cycle Notation
    See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations. … Given a permutation σ 𝔖 n , the inversion number of σ , denoted inv ( σ ) , is the least number of adjacent transpositions required to represent σ . …
    10: 11.13 Methods of Computation
    For simple and effective approximations to 𝐇 0 ( z ) and 𝐇 1 ( z ) see Aarts and Janssen (2016). … For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for 𝐊 ν ( z ) and 𝐌 ν ( z ) . … To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. …