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[3] ai.viXra.org:2611.0005 [pdf] submitted on 2026-11-02 10:25:30
Authors: Kenneth C. Johnson
Comments: 15 Pages.
The Magnus exponent is formulated over an integration interval [−t, t] that is centered on the origin, so that the solution at t is obtained from the solution at −t. In this centered formulation the propagator satisfies U(−t) = U(t)−1, the Magnus exponent is an odd function of t, and the Magnus theorem takes a form in which the even and odd parts of the coefficient matrix appear separately. The Magnus exponent is then expressed in Legendre moments of the coefficient matrix over the centered interval, and quadrature formulas with residual errors of order t3, t5, t7 and t9 are given, using 0, 1, 3 and 6 commutators. The moments are evaluated by Gauss-Legendre, Newton-Cotes or Gauss-Lobatto quadrature, and Gauss-Kronrod rules are shown to give embedded error estimates. The methodology and results are compared with those of Blanes, Casas, Oteo and Ros.
Category: General Mathematics
[2] ai.viXra.org:2611.0004 [pdf] submitted on 2026-11-02 10:28:01
Authors: Kenneth C. Johnson
Comments: 16 Pages.
This paper extends centered Magnus quadrature formulas of orders 2 to 8 to order 10 (residual error O(t11) over the interval [−t, t]). The formula uses the centered Legendre moments g0, …, g4 of the coefficient matrix and nine commutators; no formula with eight commutators was found. Of its 81 nonzero coefficients, 25 are small rationals and 56 are irrational algebraic numbers, defined as the isolated real solution of 70 polynomial equations with rational coefficients. They were refined by Newton's method to more than 190 digits, and the conditioning of the refinement is reported. The moments can be computed with five Gauss-Legendre nodes, with six Gauss-Lobatto nodes (five new samples per step), or with nine nodes that include the order-8 Newton-Cotes nodes. An embedded pair on seven Gauss-Kronrod nodes gives an order-6 estimate with three commutators and an order-10 estimate with seven more, whose difference serves as an error estimate.
Category: General Mathematics
[1] ai.viXra.org:2611.0003 [pdf] submitted on 2026-11-01 04:15:36
Authors: Xuda Ye, Yuliang Wang
Comments: 49 Pages.
We build a class of stochastic gradient integrators with second order accuracy via noise correction. The integrators multiply the injected Gaussian noise of the stochastic gradient Leimkuhler--Matthews integrator by a noise factor, obtained by applying a scalar function, the noise corrector, to the eigenvalues of an estimate of the covariance of the batch gradient from the same batch. For nonconvex potentials that are strongly convex outside a ball, every noise corrector satisfying a polynomial bound and every batch size, we prove that the numerical bias is of second order in the step size $h$ and that the mean square error of the time average of $N$ samples is bounded by a constant times $(Nh)^{-1} + h^4$. Under Gaussian gradient noise and an additional polynomial bound on the square of the noise corrector, we separate the coefficient of $h^2$ in the numerical bias into explicit terms from discretization and noise correction.
Category: General Mathematics