The approach of a numerical solution toward a final, accurate answer through iteration.Convergence in numerical analysis describes the process by which an iterative algorithm approaches a final, accurate solution through repeated refinement. Each iteration brings the approximation closer to the true answer, and convergence is achieved when further iterations produce negligible change. This concept underpins virtually all computational physics, from quantum chemistry calculations to machine learning training.
The formal study of convergence dates to the 19th century with mathematicians like Augustin-Louis Cauchy, who defined rigorous criteria for sequences and series. Numerical convergence became central to computing in the mid-20th century as digital computers enabled iterative methods for complex problems.
An iterative method starts with an initial guess and applies a mathematical operation repeatedly, generating a sequence of values. Convergence occurs when this sequence approaches a limit—the solution—within a desired tolerance. The rate of convergence (linear, quadratic, etc.) determines how quickly the algorithm reaches accuracy. Techniques like error estimation and adaptive step sizes help ensure and accelerate convergence.
Convergence is essential in computational chemistry for optimizing molecular structures, in neural network training where loss functions must converge, and in solving differential equations for weather prediction and engineering simulations.
In some chaotic systems, tiny changes in initial conditions can prevent convergence entirely, a phenomenon known as sensitive dependence on initial conditions.
| \kappa'(\theta) | variable | transformed convergence — convergence |
| \kappa(\theta) | variable | original convergence — convergence |
| \lambda | variable | transformation parameter — dimensionless parameter |
| kappa | variable | weak lensing convergence — convergence |
| theta | variable | angular position on the sky — angular position radian |
| chi | variable | comoving distance to the source — comoving distance metre |
| chi' | variable | comoving distance along the line of sight — comoving distance metre |
| q | variable | lensing efficiency function — lensing efficiency |
| a | variable | scale factor — scale factor |
| f_K | variable | comoving angular diameter distance — comoving angular diameter distance metre |
| delta | variable | matter density contrast — density contrast |
| H_0 | constant | current expansion rate of the universe — hubble constant = 2.2e-18 per second |
| Omega_m | constant | ratio of matter density to critical density today — matter density parameter = 0.315 |
| c | constant | speed of light in vacuum — speed of light = 299792458 metre per second |
| ∫ | operator | summation over infinitesimal elements — integral |
| \tilde{\kappa} | variable | Fourier transform of convergence — convergence |
| \tilde{\kappa}_B | variable | Fourier transform of B-mode convergence — b mode convergence |
| \tilde{\gamma}_1 | variable | Fourier transform of first shear component — shear component |
| \tilde{\gamma}_2 | variable | Fourier transform of second shear component — shear component |
| k_1 | variable | First component of wave vector — wave vector component radian per metre |
| k_2 | variable | Second component of wave vector — wave vector component radian per metre |
| k | variable | Magnitude of wave vector — wave number radian per metre |
