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Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, (,) = [1]The parameter is usually a real scalar and the solution is an n-vector.
The primary computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated solutions (points) of one are continued to the other. This is a specialization of the more general method of numerical continuation.
In many-body physics, the problem of analytic continuation is that of numerically extracting the spectral density of a Green function given its values on the imaginary axis. It is a necessary post-processing step for calculating dynamical properties of physical systems from Quantum Monte Carlo simulations, which often compute Green function ...
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function.Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent.
The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for >, and its analytic continuation elsewhere.
PHCpack implements the homotopy continuation method. This solver computes the isolated complex solutions of polynomial systems having as many equations as variables. The third solver is Bertini, [17] [18] written by D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler. Bertini uses numerical homotopy continuation with adaptive precision.
Numerical continuation; Numerical differentiation; Numerical error; Numerical integration; Numerical method; Numerical methods in fluid mechanics; Numerical model of the Solar System; Deep backward stochastic differential equation method; Numerical smoothing and differentiation; Numerical stability; Nyström method
Periodic travelling waves correspond to limit cycles of these equations, and this provides the basis for numerical computations. The standard computational approach is numerical continuation of the travelling wave equations. One first performs a continuation of a steady state to locate a Hopf bifurcation point. This is the starting point for a ...