Package jline.api.pfqn.nc

Class Pfqn_aghq

java.lang.Object
jline.api.pfqn.nc.Pfqn_aghq
public final class Pfqn_aghq extends Object
Adaptive Gauss-Hermite quadrature of the simplex factor of the McKenna-Mitra integral. Rescaling by the logistic-expansion mode and curvature, w = w* + A^(-1/2)*z, and applying the q-node probabilists' Gauss-Hermite rule in each of the M-1 directions gives a convergent rule whose q = 1 member is Pfqn_le itself: a single node at the mode with weight sqrt(2*pi). So LE is the first term of a convergent quadrature rather than an approximation of unknown accuracy. The cost is q^(M-1) evaluations, which is what confines the method to small M. A tensor rule is NOT invariant to the choice of square root of A: any B with B*B' = inv(A) is admissible and they place the nodes differently. The principal-axis frame from the eigendecomposition is used here, as in the reference results; where two curvatures are close to equal the frame is close to arbitrary and two valid rules can part company well above their own error, converging back together as q grows. Do not compare this across codebases node by node. With Z > 0 the radius is integrated numerically and the rule is applied to the M-1 simplex directions, so every node costs one radial quadrature. Because the radius is integrated rather than Laplaced, q = 1 there is the logistic expansion with an exact radius, which is NOT Pfqn_le's own Z > 0 branch. References: J. McKenna, D. Mitra, "Integral Representations and Asymptotic Expansions for Closed Markovian Queueing Networks: Normal Usage", Bell Syst. Tech. J. 61(5), 1982. Q. Liu, D. A. Pierce, "A Note on Gauss-Hermite Quadrature", Biometrika 81(3), 1994.
Since:
LINE 3.0