numerical-analysis

Here I derive the elimination tree for the (right-looking) sparse Cholesky algorithm for computing A = LL^T for lower triangular L and sparse matrices A . This tree forms the foundation for most sparse factorization software, even when the underlying assumptions of Cholesky (symmetric and definite) do not apply. Ultimately this tree tells us two things: 1. where nonzeros appear in the matrix L ev…

C23 requires (while IEEE-754 'recommends') a few new mathematical functions, and among which are the pown and rootn functions for raising a floating point number to integer power, and taking an ...

I simulated four SDEs and saved their paths at time T1 (these were Xk with K=4) and then I continued the simulation to a later time T2 and from the paths at time T2, I found the value of sum of four SDE joint simulation paths and this was variable to be conditionally predicted and I called it Y. The problem to solve is to fiund the conditional density of sum \(Y\,=\,X_1(T_2)\,+\,X_2(T_2)\,+X_3(T_…

I'm trying to find the first 2 roots of equation g[t,10,t1]==Tan[t] for t>0 which turns into h[t,10,t]==Tanh[t] for t<0. For the interval 0<t<11 I get the result and moreover I get the ...

We propose an axiomatic framework for variational analysis in which a single scalar functional E is fixed independently of the choice of representation. After an admissible repre- sentation is chosen, the field variables admit a decomposition into a retained component and a complementary component. The basic non-exposure axiom states that the complementary component is not itself a retained varia…

Scientific Reports, Published online: 06 May 2026; doi:10.1038/s41598-026-51692-8 An analytical exploration of dark and bright solitary wave solutions of time-fractional (3+1) -dimensional generalized Painlevé-type equation

Let $u$ solve the heat equation$$ u_t = u_{xx} \quad \text{in } (0,\pi)\times(0,\infty),$$ with Dirichlet boundary conditions $$ u(0,t)=u(\pi,t)=0,$$ and initial data $$ u(x,0)=f(x), $$ where $f \in ...

Pseudospectral collocation is a widely used direct transcription method that discretizes continuous systems with spectral accuracy and high fidelity while requiring relatively few collocation points. The special orthogonal group, SO(3), is both a Lie group and a manifold, and thus demands on-manifold optimization techniques to preserve its geometric structure. Riemannian optimization is enabled b…

If you’ve spent any time in the open-source AI community recently, you’ve probably seen someone excitedly announce they are running a 70B parameter model locally, only to follow up an hour later asking why their system crashed with an OOM (Out of Memory) error. Deploying Large Language Models (LLMs) locally—whether for privacy, cost savings, or offline availability—is the new frontier for develop…

Has any one any idea to solve this equation J 1 (x) = (x 2 /10)*(J 1 (x) + J 3 (x)), in which J are spherical Bessel function normally write as $j_1 (x)$ and $j_3(x)$ Methods 1 serial expansion: $j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} -... Read more

Inside the nucleus of every cell, DNA doesn’t float freely. It’s wound, folded, bundled into a dense tangle of proteins and genetic material called chromatin, which compacts roughly two metres of DNA into a space just a few millionths of a metre across. Within that tangle are tiny domains, each about 100 nanometres wide (smaller than the wavelength of visible light, for context), and these domain…

Has any one any idea to solve this equation J[1, x] = (x^2/10)*(J[1, x] + J[3, x]), in which J are spherical Bessel function normally write as j_1 (x) and j_3(x) Methods 1 serial expansion: j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} - \dfrac{(x)^3}{30} +... Read more

We introduce the Controlled Perturbation Algorithm (CPA) for escaping saddle points in generic non‑convex optimization problems. The key idea is to use two adaptive perturbations per coordinate, evaluate their directional derivatives, and deterministically select a descent direction – all without computing second or higher order derivatives. We also define the Non‑Descent Direction Approximation …

A couple days ago I wrote a post about turning a trick into a technique, finding another use for a clever way to construct simple, accurate approximations. I used as my example approximating the Bessel function J(x) with (1 + cos(x))/2. I learned via a helpful comment on Mathstodon that my approximation was the first-order […] The post Approximating even functions by powers of cosine first appear…

This paper presents an exhaustive historical and mathematical survey of the seventeen Permanent Axioms underlying the first machine-verified Coq formalisation of a global regularity result for the three-dimensional incompressible Navier-Stokes equations on the periodic torus T³. The formalisation establishes subcritical energy estimates for arbitrary smooth initial data in the Sobolev regularity …

Friends, another possible route to solution of above problem when explaining variables are correlated is to somehow orthogonalize the explaining variables and then use the orthogonalized variables in our analysis. I had thought about this yesterday but problem is that every hermite polynomial component has its own correlation matrix and there is no single correlation matrix that would have to be …

This paper proposes a structural reinterpretation of limits, derivatives, and partial differential equations based on the notion of convergence modes. In classical analysis, limits are defined through asymptotic conditions (e.g., ε–δ formulations), which characterize correctness but do not specify the operational structure by which convergence is realized. In practice, however—whether in numerica…