Biorthogonal Polynomials and the Bordering Method for Linear Systems
نویسنده
چکیده
The problem of solving a system of linear equations is equivalent to the computation of biorthogonal polynomials and to the bordering method which is a procedure for solving recursively a sequence of linear systems with increasing dimensions. In some cases, the biorthogonal polynomials can be computed recursively thus leading to procedures for solving linear systems. The particular cases of Hankel and Toeplitz matrices is treated in details.
منابع مشابه
On Squaring Krylov Subspace Iterative Methods for Nonsymmetric Linear Systems
The Biorthogonal Lanczos and the Biconjugate Gradients methods have been proposed as iterative methods to approximate the solution of nonsymmetric and indefinite linear systems. Sonneveld [19] obtained the Conjugate Gradient Squared by squaring the matrix polynomials of the Biconjugate Gra dients method. Here we square the Biorthogonal Lanczos, the Biconjugate Residual and the Biconjugate Orth...
متن کاملDuality, Biorthogonal Polynomials and Multi–Matrix Models
The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V1, V2 in two different variables, these kernels may be expressed in terms of finite dimensional “windows” s...
متن کاملOn the squared unsymmetric Lanczos method
The biorthogonal Lanczos and the biconjugate gradient methods have been proposed as iterative methods to approximate the solution of nonsymmetric and indefinite linear systems. Sonneveld (1989) obtained the conjugate gradient squared by squaring the matrix polynomials of the biconjugate gradient method. Here we square the unsymmetric (or biorthogonal) Lanczos method for computing the eigenvalue...
متن کاملApproximation of Gaussian by Scaling Functions and Biorthogonal Scaling Polynomials
The derivatives of the Gaussian function, G(x) = 1 √ 2π e−x 2/2, produce the Hermite polynomials by the relation, (−1)mG(m)(x) = Hm(x)G(x), m = 0, 1, . . . , where Hm(x) are Hermite polynomials of degree m. The orthonormal property of the Hermite polynomials, 1 m! ∫∞ −∞Hm(x)Hn(x)G(x)dx = δmn, can be considered as a biorthogonal relation between the derivatives of the Gaussian, {(−1)nG(n) : n = ...
متن کاملQ-Hermite Polynomials and Classical Orthogonal Polynomials
We use generating functions to express orthogonality relations in the form of q-beta integrals. The integrand of such a q-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegő and leads naturally to the Al-Salam-Chihara p...
متن کامل