least%20squares%20approximations
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1: 3.11 Approximation Techniques
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§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
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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Chebyshev series approximations for the zeros of the Bessel functions.
J. Comput. Phys. 53 (1), pp. 188–192.
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Approximation for the turning points of Bessel functions.
J. Comput. Phys. 64 (1), pp. 253–257.
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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria.
Comput. J. 9 (4), pp. 404–407.
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Algorithm 498: Airy functions using Chebyshev series approximations.
ACM Trans. Math. Software 1 (4), pp. 372–379.
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3: 20 Theta Functions
Chapter 20 Theta Functions
…4: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Numerical Methods for Least Squares Problems.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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5: Guide to Searching the DLMF
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Table 1: Query Examples
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Query | Matching records contain |
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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”. |
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J_n@(x or z)= |
at least one of the math fragments or , emphasizing that is a function. |
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sin x and (J_nu(z) or I_nu(z)) |
both and at least one of the two functions or . |
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trigonometric^2 + trig$^2 |
any sum of the squares of two trigonometric functions such as . |
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6: 3.8 Nonlinear Equations
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►Bisection of this interval is used to decide where at least one zero is located.
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►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation:
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►Initial approximations to the zeros can often be found from asymptotic or other approximations to , or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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►Consider and .
We have and .
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7: 10.75 Tables
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Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Makinouchi (1966) tabulates all values of and in the interval , with at least 29S. These are for , 10, 20; , ; with and , except for .
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
Zhang and Jin (1996, p. 322) tabulates , , , , , , , , , 7S.
8: Bibliography Y
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-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian.
Phys. Rev. A 11 (4), pp. 1144–1156.
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Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State.
American Journal of Physics 57 (1), pp. 85–86.
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9: 26.13 Permutations: Cycle Notation
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►See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►Given a permutation , the inversion number of , denoted , is the least number of adjacent transpositions required to represent .
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10: 11.13 Methods of Computation
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►For simple and effective approximations to and see Aarts and Janssen (2016).
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►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 and .
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►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.
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