About the Project

on%20intervals

AdvancedHelp

(0.001 seconds)

1—10 of 30 matching pages

1: 6.19 Tables
  • Zhang and Jin (1996, pp. 652, 689) includes Si ( x ) , Ci ( x ) , x = 0 ( .5 ) 20 ( 2 ) 30 , 8D; Ei ( x ) , E 1 ( x ) , x = [ 0 , 100 ] , 8S.

  • 2: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • 3: 10.75 Tables
  • Abramowitz and Stegun (1964, Chapter 9) tabulates j n , m , J n ( j n , m ) , j n , m , J n ( j n , m ) , n = 0 ( 1 ) 8 , m = 1 ( 1 ) 20 , 5D (10D for n = 0 ), y n , m , Y n ( y n , m ) , y n , m , Y n ( y n , m ) , n = 0 ( 1 ) 8 , m = 1 ( 1 ) 20 , 5D (8D for n = 0 ), J 0 ( j 0 , m x ) , m = 1 ( 1 ) 5 , x = 0 ( .02 ) 1 , 5D. Also included are the first 5 zeros of the functions x J 1 ( x ) λ J 0 ( x ) , J 1 ( x ) λ x J 0 ( x ) , J 0 ( x ) Y 0 ( λ x ) Y 0 ( x ) J 0 ( λ x ) , J 1 ( x ) Y 1 ( λ x ) Y 1 ( x ) J 1 ( λ x ) , J 1 ( x ) Y 0 ( λ x ) Y 1 ( x ) J 0 ( λ x ) for various values of λ and λ 1 in the interval [ 0 , 1 ] , 4–8D.

  • 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 .

  • 4: 7.23 Tables
  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf z , x [ 0 , 5 ] , y = 0.5 ( .5 ) 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i t 2 d t , ( 1 / π ) e i ( x 2 + ( π / 4 ) ) x e ± i t 2 d t , x = 0 ( .5 ) 20 ( 1 ) 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

  • 5: 28.35 Tables
  • National Bureau of Standards (1967) includes the eigenvalues a n ( q ) , b n ( q ) for n = 0 ( 1 ) 3 with q = 0 ( .2 ) 20 ( .5 ) 37 ( 1 ) 100 , and n = 4 ( 1 ) 15 with q = 0 ( 2 ) 100 ; Fourier coefficients for ce n ( x , q ) and se n ( x , q ) for n = 0 ( 1 ) 15 , n = 1 ( 1 ) 15 , respectively, and various values of q in the interval [ 0 , 100 ] ; joining factors g e , n ( q ) , f e , n ( q ) for n = 0 ( 1 ) 15 with q = 0 ( .5  to  10 ) 100 (but in a different notation). Also, eigenvalues for large values of q . Precision is generally 8D.

  • 6: 26.6 Other Lattice Path Numbers
    D ( m , n ) is the number of paths from ( 0 , 0 ) to ( m , n ) that are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . … M ( n ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x and are composed of directed line segments of the form ( 2 , 0 ) , ( 0 , 2 ) , or ( 1 , 1 ) . … N ( n , k ) is the number of lattice paths from ( 0 , 0 ) to ( n , n ) that stay on or above the line y = x , are composed of directed line segments of the form ( 1 , 0 ) or ( 0 , 1 ) , and for which there are exactly k occurrences at which a segment of the form ( 0 , 1 ) is followed by a segment of the form ( 1 , 0 ) . …
    Table 26.6.3: Narayana numbers N ( n , k ) .
    n k
    5 0 1 10 20 10 1
    r ( n ) is the number of paths from ( 0 , 0 ) to ( n , n ) that stay on or above the diagonal y = x and are composed of directed line segments of the form ( 1 , 0 ) , ( 0 , 1 ) , or ( 1 , 1 ) . …
    7: 18.40 Methods of Computation
    Let x ( a , b ) . …Results of low ( 2 to 3 decimal digits) precision for w ( x ) are easily obtained for N 10 to 20 . … Here x ( t , N ) is an interpolation of the abscissas x i , N , i = 1 , 2 , , N , that is, x ( i , N ) = x i , N , allowing differentiation by i . … This is a challenging case as the desired w RCP ( x ) on [ 1 , 1 ] has an essential singularity at x = 1 . … Further, exponential convergence in N , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate w ( x ) for these OP systems on x [ 1 , 1 ] and ( , ) respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …
    8: 12.10 Uniform Asymptotic Expansions for Large Parameter
    With the upper sign in (12.10.2), expansions can be constructed for large μ in terms of elementary functions that are uniform for t ( , ) 2.8(ii)). … uniformly for t [ 1 + δ , ) , where … uniformly for t [ 1 + δ , 1 δ ] . … The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when μ uniformly with respect to t [ 1 + δ , ) . … The following expansions hold for large positive real values of μ , uniformly for t [ 1 + δ , ) . …
    9: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals < x 2 and 2 x < , with s = 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 4 , 10 8 , 10 12 .

  • 10: Bibliography G
  • A. Gil, J. Segura, and N. M. Temme (2006a) Computing the real parabolic cylinder functions U ( a , x ) , V ( a , x ) . ACM Trans. Math. Software 32 (1), pp. 70–101.
  • A. Gil, J. Segura, and N. M. Temme (2006b) Algorithm 850: Real parabolic cylinder functions U ( a , x ) , V ( a , x ) . ACM Trans. Math. Software 32 (1), pp. 102–112.
  • A. Gil, J. Segura, and N. M. Temme (2011a) Algorithm 914: parabolic cylinder function W ( a , x ) and its derivative. ACM Trans. Math. Software 38 (1), pp. Art. 6, 5.
  • A. Gil, J. Segura, and N. M. Temme (2011b) Fast and accurate computation of the Weber parabolic cylinder function W ( a , x ) . IMA J. Numer. Anal. 31 (3), pp. 1194–1216.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.