numerical-methods

We develop an exact and parameterized theory of iterated complex roots under explicit branch conventions. The principal negative-root recursion z n+1 = F r (z n) := Root pr r (−z n), r ≥ 2, separates completely into radial and angular dynamics. The radius obeys the exact law |z n | = |z 0 | 1/r n, while the principal argument is governed by a two-branch contraction. This yields a closed formula f…

Fred James (1939 – 2026) — Fred James, CERN’s renowned expert on data analysis and statistics, pioneer of numerical methods, and co-author of the popular book Statistical Methods in Experimental Physics, sadly passed away on 31 July.  Fred was born in Detroit, USA, on 28 May 1939, and graduated in 1960 from Princeton University with a degree in physics. […]

We proved that linear partial differential operator with constant coefficients has locally integrable fundamental solutions, elementarily.

For decades, the mathematical backbone of high-precision Global Navigation Satellite System (GNSS) positioning has rested on a Gaussian assumption that rarely holds in the real world. Multipath reflections, atmospheric interference, and signal blockages produce erratic, heavy-tailed errors that classical models systematically underestimate--leading to incorrect integer fixes when accuracy matters…

Problem-solving ability are essential competencies that students need to develop from elementary education to higher education. However, in developing countries such as Indonesia, the assessment of process-based problem-solving skills remains limited. Educational practices in Indonesia still focus more on outcome-based assessment rather than evaluating the problem-solving process itself. Therefor…

This is an application of functional analysis to the existence and smoothness of the Navier–Stokes equations using elementary weak solutions in Sobolev spaces. We solve the problem in mathematics. The problems are not in physics, so we do not use any physics or assumptions-falsified mathematics. We use mathematics only. We can solve the problem by using an exactly and completely FALSIFIED resolut…

The previous post gave several examples of three-term recurrence relations for special functions. These relations can be computationally useful, but they have to be applied carefully. Several years ago I wrote a post on stable and unstable recurrences. In that post I show that the stability of the recurrence relation for Bessel functions produces depends […] The post Numerical (in)stability of re…

If the domains of functions is R^N then embedding theorem follows even if on bounded domains without extension operators.

For decades, numerical weather prediction (NWP) models have leaned on physics-based equations to simulate the atmosphere. These models pack a […]

Given an appropriate situation, for example, a case where there are 2 grounded conductors (infinite sheet charges), one at $y=0$ and one at $y=a$ , and a 3rd conductor (at $x=0$ ) perpendicular to both, maintained at a potential $V_{0}(y) $ , (This is the example situation given in DJ Griffiths 4th ed. Example 3, chapter 3) we can readily assume that the potential within the bounds takes the form…

Say that I have a representation $V_\ell$ of $\rm SO(3)$ , with angular momentum number $\ell$ . In Dirac notation, states in $V_\ell$ can be represented in the form $$ |v⟩ = \sum_{m=-\ell}^\ell c_m |\ell,m⟩, $$ where, if the representation is real-valued, we require $c_{-m} = (-1)^m c_m^*$ . I would like to randomly sample from this space , i.e., the vector $|v⟩$ and/or its coefficients $c_m$ , …

A GPU-friendly algorithm for tight certified singular-value endpoint bounds using scaled Gram matrices, SYRKs, Frobenius reductions, and a small scalar moment problem.

A hardware-aware hybrid polar decomposition for ML: one Dynamic Weighted Halley (rational) step to handle the hard early regime, then two Polar Express (polynomial) cleanup steps once the spectrum is easy. The result is exactly two rectangular GEMMs, no eigendecomposition or power iteration, and robust convergence from condition numbers up to 1000.

Geomagnetic field forecasting is critical for mitigating space weather hazards, yet single-station prediction remains a challenge due to the complex, non-linear coupling of vector components. In this work, we propose a Graph Neural Network (GNN) architecture enhanced with Temporal Convolutional Networks (TCN) to forecast the H, D, and Z components. By modeling the observatory’s sensors as nodes i…

If you ever need more precision than what 15 decimal digits of the double format can offer, there is a neat trick: glue two doubles together and treat them as one number. This gives you ~31 decimal digits for roughly 9x the cost of a plain double in a real kernel (4-12x per isolated operation). With no heap allocation and no dependencies, this puts it almost exactly halfway between a double and a…