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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 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.
  • 3: 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 .
    4: 26.18 Counting Techniques
    With the notation of §26.15, the number of placements of n nonattacking rooks on an n × n chessboard that avoid the squares in a specified subset B is … The number of ways of placing n labeled objects into k labeled boxes so that at least one object is in each box is …
    5: Bibliography P
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • 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.
  • 6: 1.11 Zeros of Polynomials
    Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term. A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. … The square roots are chosen so that … Resolvent cubic is z 3 + 12 z 2 + 20 z + 9 = 0 with roots θ 1 = 1 , θ 2 = 1 2 ( 11 + 85 ) , θ 3 = 1 2 ( 11 85 ) , and θ 1 = 1 , θ 2 = 1 2 ( 17 + 5 ) , θ 3 = 1 2 ( 17 5 ) . …
    7: 26.10 Integer Partitions: Other Restrictions
    p ( 𝒟 k , n ) denotes the number of partitions of n into parts with difference at least k . p ( 𝒟 3 , n ) denotes the number of partitions of n into parts with difference at least 3, except that multiples of 3 must differ by at least 6. …
    Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from A j , k .
    p ( 𝒟 , n ) p ( 𝒟 2 , n ) p ( 𝒟 2 , T , n ) p ( 𝒟 3 , n )
    20 64 31 20 18
    8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    §1.18(ii) L 2 spaces on intervals in
    For a Lebesgue–Stieltjes measure d α on X let L 2 ( X , d α ) be the space of all Lebesgue–Stieltjes measurable complex-valued functions on X which are square integrable with respect to d α , … Eigenfunctions corresponding to the continuous spectrum are non- L 2 functions. … What then is the condition on q ( x ) to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if q ( x ) 0 then there is only a continuous spectrum. Surprisingly, if q ( x ) < 0 on any interval on the real line, even if positive elsewhere, as long as X q ( x ) d x 0 , see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding L 2 ( X ) eigenfunction. …
    9: 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.
  • 10: 31.18 Methods of Computation
    Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …