# What is Numerical algorithms: Definition and 18 Discussions

The Numerical Algorithms Group (NAG) is a software and services company which provides methods for the solution of mathematical and statistical problems, and offers services to users of High performance computing (HPC) systems. Its products and services are employed by tens of thousands of users from Global 500 companies, universities, supercomputing sites and numerous independent software vendors. As a not-for-profit organization, NAG reinvests its surpluses into the research and development of its products and services, and the fostering of new numerical and scientific talent. NAG serves its customers from offices in Oxford, Manchester, Chicago, and Tokyo, through staff in France and Germany, and via a global network of distributors.

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1. ### I Make Intelligent Initial Guesses for Newton-Raphson Programmatically

I'm writing code to numerically solve a single variable equation, currently with Newton Raphson's method. Right now, I'm just using an initial guess of 1, and reporting a failure if it doesn't converge. While it usually works, it does of course fails for many functions with asymptotes or other...
2. ### A Solving this first-order differential equation for neutron abundance

The time rate of change of neutron abundance ##X_n## is given by $$\frac{dX_n}{dt} = \lambda - (\lambda + \hat\lambda)X_n$$ where ##\lambda## is neutron production rate per proton and ##\hat\lambda## is neutron destruction rate per neutron. Given the values of ##\lambda## and ##\hat\lambda## at...
3. ### Calculate tilt angle of a bar lifted via two ropes on fixed points

Summary: How to calculate the equilibrium angle of a bar that is lifted on its two ends with ropes attached to fixed lifting points? Hello and good day all, First of all I would like to apologize in advance for my english, I am not a native speaker so some grammar errors may be left. My...
4. B

### Hi I'm Jayvhie, Mechanical Engineer from PH

◎ As Structural Design Engineer, Highly-motivated individual with a demonstrated history in Finite Element Analysis to solve complex structural engineering problems, applying numerical methods approach and view solution results (Post- Processing) to optimize the design of a Ship to Shore (STS)...
5. ### How to judge the singularity of a matrix in numerical method?

Summary: different methods give different results. They are not consistent. Summary: different methods give different results. They are not consistent. I use two different methods to detect whether a matrix is singular. The result of calculating the determinant of a 9-order square matrix is...
6. ### Visual Python Pendulum: Solving the Forces

Hello and thanks in advance for your help. For about a week now, I've been trying to write what should be a simple python program. The idea is first to write a program for a simple harmonic pendulum, then adapt it to a spring pendulum. However, in order to do this, I have to write the simple...
7. ### A N-Body Simulation using symplectic integrators

Hi, I hope I am in the right section of the forum. I was trying to understand the following algorithm: https://benchmarksgame-team.pages.debian.net/benchmarksgame/program/nbody-python3-1.html and particulary this part: def advance(dt, n, bodies=SYSTEM, pairs=PAIRS): for i in range(n)...
8. ### A Implicit Euler method with adaptive time step and step doubling

For Initial Value problems I want to implement an ODE solver for implicit Euler method with adaptive time step and use step doubling to estimate error. I have found some reading stuff about adaptive time step and error estimation using step doubling but those are mostly related to RK methods. I...
9. ### A How to choose the number of particles per site in Fermionic DMRG?

I am doing DMRG (in traditional formalism, not MPS) for Hubbard model H = -t ∑i ∑σci,σ ci+1,σ + U∑ini,σni,σ- In every iteration we add two sites to the system, but how do we set that how many particles are allowed in the system?
10. ### "Shooting Method" for simulating a Particle in an Infinite Square Well

Hello! I am trying to write a program that solves the Schrodinger Equation for a particle in an infinite square well. I did a lot of research regarding the methods that could be used to accomplish this. I am writing this program in Matlab. The method I am using is called the Shooting Method. In...
11. ### A Computing solutions to the radial Schroedinger equation?

Hi all, I'm trying to compute the solutions to a general case for a Schroedinger equation with a radial potential but I'm stuck on a rather small detail that I'm not sure about. It's well known that I can perform the change of variables to spherical coordinates and express the radial part of the...
12. ### A How do I supply arpack drivers with all starting vectors?

I am using arpack (the dsdrv1 driver) to iteratively solve the eigenvalue problem Ax = λx I am interested in the first m eigenvectors, and I have very good initial approximations for these vectors, so I would like to use my m starting vectors as an initial guess. However...
13. ### A Efficiently Computing Eigenvalues of a Sparse Banded Matrix

I have a Hamiltonian represented by a penta-diagonal matrix The first bands are directly adjascent to the diagonals. The other two bands are N places above and below the diagonal. Can anyone recommend an efficient algorithm or routine for finding the eigenvalues and eigenvectors?
14. ### I Numerical Calculus of Variations

I attempt to solve the brachistochrone problem numerically. I am using a direct method which considers the curve ##y(x)## as a Lagrange polynomial evaluated at fixed nodes ##x_i##, and the time functional as a multivariate function of the ##y_i##. The classical statement of the problem requires...
15. ### A Numerical integration of motion

Hi,I'd like to build a simulation (realtime) of space ships near a black hole (neutral, still or rotating possibly). Key features would be: 1) the ships are test particles that do not affect the metric a) possibly test rigid-bodies with GR consistent rotational DOF. 2) the ships can fire...
16. ### A Imaginary time propagation to find eigenfunctions

Hi, I have been trying to use imaginary time propagation to get the ground state and excited states eigen function but the results I got is different from the analytical solution. I know that to get excited states, I should propagate 2 or more orthogonal functions depending on the number of...
17. ### A Monte Carlo Wavefunction Methods

From the Theory of Open Quantum Systems; the Euler scheme is given by: ##\psi_{k+1} = \psi_{k} + D_1(\psi_k)\Delta t + D_2(\psi_k) \Delta W_k## and is a scheme of order 1. What does the order of convergence mean? From my understanding higher order schemes require fewer interations to give a...
18. ### Numerical algorithms for finding an eigenvector

All matrices A\in\mathbb{C}^{n\times n} have at least one eigenvector z\in\mathbb{C}^n. I'm interested to know what kind of algorithms there are for the purpose of finding an eigenvector. I noticed that \frac{|z^{\dagger} A z|}{\|Az\|} = 1\quad\quad\quad\quad (1) holds only when z is an...