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Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
In linear algebra and numerical analysis, a preconditioner of a matrix is a matrix such that has a smaller condition number than . It is also common to call T = P − 1 {\\displaystyle T=P^{-1}} the preconditioner, rather than P {\\displaystyle P} , since P {\\displaystyle P} itself is rarely explicitly available.
Richardson, L.F. (1910). "The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam".
Assume furthermore that a solution on any grid may be obtained with a given effort = from a solution on a coarser grid +. Here, ρ = N i + 1 / N i < 1 {\displaystyle \rho =N_{i+1}/N_{i}<1} is the ratio of grid points on "neighboring" grids and is assumed to be constant throughout the grid hierarchy, and K {\displaystyle K} is some constant ...
Using the pseudoinverse and a matrix norm, one can define a condition number for any matrix: = ‖ ‖ ‖ + ‖. A large condition number implies that the problem of finding least-squares solutions to the corresponding system of linear equations is ill-conditioned in the sense that small errors in the entries of A {\displaystyle A} can ...
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1: ρ ( D − 1 ( L + U ) ) < 1. {\displaystyle \rho (D^{-1}(L+U))<1.} A sufficient (but not necessary) condition for the method to converge is that the matrix A is strictly or irreducibly diagonally dominant .
The number of gradient descent iterations is commonly proportional to the spectral condition number of the system matrix (the ratio of the maximum to minimum eigenvalues of ), while the convergence of conjugate gradient method is typically determined by a square root of the condition number, i.e., is much faster.