Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (...Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (0≤θ≤π/2,α,β>-1), as n→+∞. An accurate approximation with error bounds is also constructed for the zero θ n,s of P (α,β) n(cosθ)(α≥0,β>-1).展开更多
As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix A involve...As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix A involved is diagonalizable. The theoretical background is established, in which lower and upper error bounds for eigenvectors are new for both Arnoldi's method and a general perturbation problem, and furthermore these bounds are shown to be optimal and they generalize a classical perturbation bound due to W. Kahan in 1967 for A symmetric. The algorithms can adaptively determine the multiplicity of an eigenvalue and a basis of the associated eigenspace. Numerical experiments show reliability of the algorithms.展开更多
基金Supported by the Natural Science Foundation of Beijing
文摘Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (0≤θ≤π/2,α,β>-1), as n→+∞. An accurate approximation with error bounds is also constructed for the zero θ n,s of P (α,β) n(cosθ)(α≥0,β>-1).
文摘As is well known, solving matrix multiple eigenvalue problems is a very difficult topic. In this paper, Arnoldi type algorithms are proposed for large unsymmetric multiple eigenvalue problems when the matrix A involved is diagonalizable. The theoretical background is established, in which lower and upper error bounds for eigenvectors are new for both Arnoldi's method and a general perturbation problem, and furthermore these bounds are shown to be optimal and they generalize a classical perturbation bound due to W. Kahan in 1967 for A symmetric. The algorithms can adaptively determine the multiplicity of an eigenvalue and a basis of the associated eigenspace. Numerical experiments show reliability of the algorithms.
