Positivity of Integrable Algorithms for Computing Matrix Eigenvalues and Singular Values

• 作 者: Yoshimasa Nakamura
• 机 构: GraduateSchoolofInformatics,KyotoUniversity,Japan
• 出 处: International Conference on Orthogonal Polynomials,Integrabl
• 发表日期: 2014年10期

It is known [1] that the 3-terms recurrence relation and the Christoffel transformation(CT)for orthogonal polynomials(OPs)induce a discrete deformation of the recurrence coefficients which is remarkably equivalent to a discrete-time integrable system such as the Toda lattice and the Lotka-Volterra chain.If and only if the parameter in the CT is less than or equal to the lower bound of the interval of orthogonality,the new moment functional defined by the CT is positive-definite [2] and then the recurrence coefficients are kept positive.On the other hand,the discrete Toda lattice is closely related to the quotient difference(qd)algorithm for tridiagonal matrix eigenvalues and continued fractions [3] and the discrete Lotka-Volterra chain is useful to design a new numerical algorithm,named dLV,for computing bidiagonal matrix singular values [4].One of the beautiful features of these ”integrable algorithms” is a highly relative accuracy of computed eigenvalues and singular values received from the positivity of solutions of the discrete-time integrable systems [5].In this talk I will give a brief review on an intimate relationship between the positivity of moment functional of OPs and the relative accuracy of the resulting integrable algorithms through the positivity of Hankel determinant of moments.