This paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned.Such matrices arise in random matrix theory and require the use of extremely high pr...This paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned.Such matrices arise in random matrix theory and require the use of extremely high precision arithmetic.Surprisingly,we find that a group of commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach for this class of matrices.We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations.Our approach combines message passing and shared memory,achieving near-perfect scalability and high tolerance for network latency.We are thus able to find solutions for much larger matrices than previously possible,with the potential for extending this work to systems with greater levels of parallelism.The contributions of this work are in three areas:determination that a direct algorithm based on the secant method is more effective when extreme fixed-point precision is required than are the algorithms more typically used in parallel floating-point computations;the particular mix of optimizations required for extreme precision large matrix operations on a modern multi-core cluster,and the numerical results themselves.展开更多
We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices.To obtain these sufficient conditions,we construct new ...We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices.To obtain these sufficient conditions,we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices.As applications,we provide a unified way to produce inequalities for many combinatorial polynomials,such as the Eulerian polynomials,Schröder polynomials,and Narayana polynomials.展开更多
Sparse sums of Lorentzians can give good approximations to functions consisting of linear combination of piecewise continuous functions. To each Lorentzian, two parameters are as- signed： translation and scale. These...Sparse sums of Lorentzians can give good approximations to functions consisting of linear combination of piecewise continuous functions. To each Lorentzian, two parameters are as- signed： translation and scale. These parameters can be found by using a method for complex fre- quency detection in the frequency domain. This method is based on an alternating projection scheme between Hankel matrices and finite rank operators, and have the advantage that it can be done in weighted spaces. The weighted spaces can be used to partially revoke the effect of finite band-width filters. Apart from frequency extrapolation the method provides a way of estimating discontinuity locations.展开更多
基金This work is supported in part by the National Science Foundation under Award No.CCF-1217590 and NFS grant#CNS-0619337 and by FDCT 077/2012/A3.Any opinions,findings conclusions or recommendations expressed here are the authors and do not necessarily reflect those of the sponsors.
文摘This paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned.Such matrices arise in random matrix theory and require the use of extremely high precision arithmetic.Surprisingly,we find that a group of commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach for this class of matrices.We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations.Our approach combines message passing and shared memory,achieving near-perfect scalability and high tolerance for network latency.We are thus able to find solutions for much larger matrices than previously possible,with the potential for extending this work to systems with greater levels of parallelism.The contributions of this work are in three areas:determination that a direct algorithm based on the secant method is more effective when extreme fixed-point precision is required than are the algorithms more typically used in parallel floating-point computations;the particular mix of optimizations required for extreme precision large matrix operations on a modern multi-core cluster,and the numerical results themselves.
基金This work was supported in part by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China(Grant Nos.11522110,11971249).
文摘We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices.To obtain these sufficient conditions,we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices.As applications,we provide a unified way to produce inequalities for many combinatorial polynomials,such as the Eulerian polynomials,Schröder polynomials,and Narayana polynomials.
文摘Sparse sums of Lorentzians can give good approximations to functions consisting of linear combination of piecewise continuous functions. To each Lorentzian, two parameters are as- signed： translation and scale. These parameters can be found by using a method for complex fre- quency detection in the frequency domain. This method is based on an alternating projection scheme between Hankel matrices and finite rank operators, and have the advantage that it can be done in weighted spaces. The weighted spaces can be used to partially revoke the effect of finite band-width filters. Apart from frequency extrapolation the method provides a way of estimating discontinuity locations.
