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For example, the 2-norm condition number of the matrix above is about 4.8 ... The condition number of the n × n Hilbert matrix grows as ((+) /) ...
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.
Matrices with large condition numbers can cause numerically unstable results: small perturbation can result in large errors. Hilbert matrices are the most famous ill-conditioned matrices. For example, the fourth-order Hilbert matrix has a condition of 15514, while for order 8 it is 2.7 × 10 8. Rank
The smallest singular value of a matrix A is σ n (A). It has the following properties for a non-singular matrix A: The 2-norm of the inverse matrix (A −1) equals the inverse σ n −1 (A). [2]: Thm.3.3 The absolute values of all elements in the inverse matrix (A −1) are at most the inverse σ n −1 (A). [2]: Thm.3.3
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 ...
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.
The Riemann–Hilbert problem deformation theory is applied to the problem of stability of the infinite periodic Toda lattice under a "short range" perturbation (for example a perturbation of a finite number of particles). Most Riemann–Hilbert factorization problems studied in the literature are 2-dimensional, i.e., the unknown matrices are ...
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra. Thus R 0 {\displaystyle R_{0}} is an Artinian ring , which is a k -vector space of dimension P (1) , and Jordan–Hölder theorem may be used for proving that P (1) is the degree of the ...