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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.
that is, as a Gramian matrix for powers of x. It arises in the least squares approximation of arbitrary functions by polynomials. The Hilbert matrices are canonical examples of ill-conditioned matrices, being notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8 × 10 5.
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".
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 .
For example, even if t is as large as –10000000000, the root α 1 only changes from 1 to about 0.99999991779380 (which is very close to the first order approximation 1 + t/19! ≈ 0.99999991779365). Similarly, the other small roots of Wilkinson's polynomial are insensitive to changes in t. Example.
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.