The class of generalized α-matrices is presented by Cvetkovi?, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove th...The class of generalized α-matrices is presented by Cvetkovi?, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove that a generalized irreducible α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices. As applications of the obtained results, three regions including all the eigenvalues of a matrix are given.展开更多
Let fs,t（m, n） be the number of （0, 1） - matrices of size m × n such that each row has exactly s ones and each column has exactly t ones （sm = nt）. How to determine fs,t（m,n）？ As R. P. Stanley has obser...Let fs,t（m, n） be the number of （0, 1） - matrices of size m × n such that each row has exactly s ones and each column has exactly t ones （sm = nt）. How to determine fs,t（m,n）？ As R. P. Stanley has observed （Enumerative Combinatorics I （1997）, Example 1.1.3）, the determination of fs,t（m, n） is an unsolved problem, except for very small s, t. In this paper the closed formulas for f2,2（n, n）, f3,2（m, n）, f4,2（m, n） are given. And recursion formulas and generating functions are discussed.展开更多
In 1992, Brualdi and Jung first introduced the maximum jump number M(n, k), that is, the maximum number of the jumps of all (0, 1)-matrices of order n with k 1's in each row and column, and then gave a table about...In 1992, Brualdi and Jung first introduced the maximum jump number M(n, k), that is, the maximum number of the jumps of all (0, 1)-matrices of order n with k 1's in each row and column, and then gave a table about the values of M(n, k) when 1 ≤ k ≤ n ≤ 10. They also put forward several conjectures, including the conjecture M(2k - 2, k) = 3k - 4 + [k-2/2]. In this paper, we prove that b(A) ≥ 4 for every A ∈ A(2k - 2, k) if k ≥ 11, and find another counter-example to this conjecture .展开更多
We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product(MVP)for N-body problems in d dimensions,namely efficient H^(2)_(∗)(fully nested algorithm,i.e.,H^(2)matrix a...We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product(MVP)for N-body problems in d dimensions,namely efficient H^(2)_(∗)(fully nested algorithm,i.e.,H^(2)matrix algorithm)and(H^(2)+H)_(∗)(semi-nested algorithm,i.e.,cross of H^(2)and H matrix algorithms).The efficient H^(2)_(∗)and(H^(2)+H)_(∗)hierarchical representations are based on our recently introduced weak admissibility condition in higher dimensions(Khan et al.,J.Comput.Phys.2024),where the admissible clusters are the far-field and the vertex-sharing clusters.Due to the use of nested form of the bases,the proposed hierarchical matrix algorithms are more efficient than the non-nested algorithms(H matrix algorithms).We rely on purely algebraic low-rank approximation techniques(e.g.,ACA(Bebendorf et al.,Computing 2003)and NCA(Bebendorf et al.,Numer.Math.2012;Gujjula and Ambikasaran,arXiv:2203.148322022;Zhao et al.,IEEE Trans.Microw.Theory Tech.2019))and develop both algorithms in a black-box(kernel-independent)fashion.The initialization time of the proposed algorithms scales quasi-linearly,i.e.,complexity O(Nlog^(α)(N)),α≥0 and small.Using the proposed hierarchical representations,one can perform theMVP that scales at most quasi-linearly.Another noteworthy contribution of this article is that we perform a comparative study of the proposed algorithms with different algebraic(NCA or ACA-based compression)fast MVP algorithms(e.g.,H^(2),H,etc.)in 2D and 3D(d=2,3).The fast algorithms are tested on various kernel matrices and applied to get fast iterative solutions of a dense linear system arising from the discretized integral equations and radial basis function interpolation.The article also discusses the scalability of the algorithms and provides various benchmarks.Notably,all the algorithms are developed in a similar fashion in C++and tested within the same environment,allowing for meaningful comparisons.The numerical results demonstrate that the proposed algorithms are competitive to the NCA-based standard H^(2)matrix algorithm(where the admissible clusters are the far-field clusters)with respect to the memory and time.The C++implementation of the proposed algorithms is available at https://github.com/riteshkhan/H^(2)weak/.展开更多
The maxinmum jump number M(n, k) over a class of n×n matrices of zerosand ones with constant row and column sum k has been investigated by Brualdi andJung in [1] where they proposed the conjectureM(2k, k + 1) = 3...The maxinmum jump number M(n, k) over a class of n×n matrices of zerosand ones with constant row and column sum k has been investigated by Brualdi andJung in [1] where they proposed the conjectureM(2k, k + 1) = 3l - 1 + [k-1/2]In this note, we give two counter-examples to this conjecture.展开更多
文摘The class of generalized α-matrices is presented by Cvetkovi?, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove that a generalized irreducible α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices. As applications of the obtained results, three regions including all the eigenvalues of a matrix are given.
文摘Let fs,t（m, n） be the number of （0, 1） - matrices of size m × n such that each row has exactly s ones and each column has exactly t ones （sm = nt）. How to determine fs,t（m,n）？ As R. P. Stanley has observed （Enumerative Combinatorics I （1997）, Example 1.1.3）, the determination of fs,t（m, n） is an unsolved problem, except for very small s, t. In this paper the closed formulas for f2,2（n, n）, f3,2（m, n）, f4,2（m, n） are given. And recursion formulas and generating functions are discussed.
基金Hainan Natural Science Foundation of Hainan (10002)
文摘In 1992, Brualdi and Jung first introduced the maximum jump number M(n, k), that is, the maximum number of the jumps of all (0, 1)-matrices of order n with k 1's in each row and column, and then gave a table about the values of M(n, k) when 1 ≤ k ≤ n ≤ 10. They also put forward several conjectures, including the conjecture M(2k - 2, k) = 3k - 4 + [k-2/2]. In this paper, we prove that b(A) ≥ 4 for every A ∈ A(2k - 2, k) if k ≥ 11, and find another counter-example to this conjecture .
文摘We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product(MVP)for N-body problems in d dimensions,namely efficient H^(2)_(∗)(fully nested algorithm,i.e.,H^(2)matrix algorithm)and(H^(2)+H)_(∗)(semi-nested algorithm,i.e.,cross of H^(2)and H matrix algorithms).The efficient H^(2)_(∗)and(H^(2)+H)_(∗)hierarchical representations are based on our recently introduced weak admissibility condition in higher dimensions(Khan et al.,J.Comput.Phys.2024),where the admissible clusters are the far-field and the vertex-sharing clusters.Due to the use of nested form of the bases,the proposed hierarchical matrix algorithms are more efficient than the non-nested algorithms(H matrix algorithms).We rely on purely algebraic low-rank approximation techniques(e.g.,ACA(Bebendorf et al.,Computing 2003)and NCA(Bebendorf et al.,Numer.Math.2012;Gujjula and Ambikasaran,arXiv:2203.148322022;Zhao et al.,IEEE Trans.Microw.Theory Tech.2019))and develop both algorithms in a black-box(kernel-independent)fashion.The initialization time of the proposed algorithms scales quasi-linearly,i.e.,complexity O(Nlog^(α)(N)),α≥0 and small.Using the proposed hierarchical representations,one can perform theMVP that scales at most quasi-linearly.Another noteworthy contribution of this article is that we perform a comparative study of the proposed algorithms with different algebraic(NCA or ACA-based compression)fast MVP algorithms(e.g.,H^(2),H,etc.)in 2D and 3D(d=2,3).The fast algorithms are tested on various kernel matrices and applied to get fast iterative solutions of a dense linear system arising from the discretized integral equations and radial basis function interpolation.The article also discusses the scalability of the algorithms and provides various benchmarks.Notably,all the algorithms are developed in a similar fashion in C++and tested within the same environment,allowing for meaningful comparisons.The numerical results demonstrate that the proposed algorithms are competitive to the NCA-based standard H^(2)matrix algorithm(where the admissible clusters are the far-field clusters)with respect to the memory and time.The C++implementation of the proposed algorithms is available at https://github.com/riteshkhan/H^(2)weak/.
基金Supported by the Science Foundation of Hainan(10002)
文摘The maxinmum jump number M(n, k) over a class of n×n matrices of zerosand ones with constant row and column sum k has been investigated by Brualdi andJung in [1] where they proposed the conjectureM(2k, k + 1) = 3l - 1 + [k-1/2]In this note, we give two counter-examples to this conjecture.
