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A384589 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384587.

%I A384589 #11 Jun 26 2025 08:00:09
%S A384589 0,0,0,5,3,9,2,9,4,7,0,5,5,6,1,3,2,7,4,5,0,1,0,3,7,9,0,5,6,7,6,2,0,5,
%T A384589 9,3,2,1,2,2,7,7,2,5,6,9,6,6,4,3,3,2,4,4,0,8,5,4,6,6,4,9,9,4,7,7,9,0,
%U A384589 1,0,9,1,7,5,6,9,3,7,2,3,0,2,7,8,5,7,9,1,1,6
%N A384589 Decimal expansion of the weight factor for Laguerre-Gauss quadrature corresponding to abscissa A384587.
%H A384589 Paolo Xausa, <a href="/A384589/b384589.txt">Table of n, a(n) for n = 0..10000</a>
%H A384589 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Table 25.9, n = 4.
%H A384589 A.H.M. Smeets, <a href="/A384277/a384277.txt">Abscissas and weight factors for Laguerre integration for some larger degrees</a>.
%H A384589 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Laguerre-GaussQuadrature.html">Laguerre-Gauss Quadrature</a>.
%H A384589 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A384589 Smallest root of 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9 = 0.
%e A384589 0.00053929470556132745010379056762059321227725696643324...
%t A384589 First[RealDigits[Root[1990656*#^4 - 1990656*#^3 + 504576*#^2 - 16960*# + 9 &, 1], 10, 100, -1]] (* _Paolo Xausa_, Jun 26 2025 *)
%o A384589 (PARI) solve(x = 0.0, 0.01, 1990656*x^4 - 1990656*x^3 + 504576*x^2 - 16960*x + 9)
%Y A384589 Cf. A384590.
%Y A384589 There are k positive real zeros of the Laguerre polynomial of degree k:
%Y A384589    k | zeros                                    | corresponding weights for Laguerre-Gauss quadrature
%Y A384589   ---+------------------------------------------+-----------------------------------------------------
%Y A384589    2 | A101465, 1+A014176                       | A201488, A100954-3
%Y A384589    3 | A384277, A384278, A384279                | A384463, A384464, A384465
%Y A384589    4 | A384280, A384281, A384586, A384587       | A384466, A384467, A384588, this sequence
%K A384589 nonn,cons
%O A384589 0,4
%A A384589 _A.H.M. Smeets_, Jun 14 2025