A new matrix product and its applications in partitioned matrix differentiation

Tracy, Derrick S.; Singh, Rana Pratap

doi:10.1111/j.1467-9574.1972.tb00199.x

Cited by 75 publications

(35 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Definition 2.8 [25] If A is an m×n matrix and B is a p×q matrix, then the Kronecker product on A and B, denoted as A B…”

Section: B Matrix Operationmentioning

confidence: 99%

A New Hypernetwork Model Based on Matrix Operation

Liu

2015

2015 10th International Conference on Intelligent Systems and Knowledge Engineering (ISKE)

View full text Add to dashboard Cite

In this paper, we propose a new hypernetwork model based on Tracy-Singh Product on the correlation matrix of hypergraph. Node degree, node hyperdegree, and hyperedge degree and their corresponding polynomials are introduced to describe this hypernetwork model. It is shown that this kind of hypernetworks is fractal as its correlation matrix is a fractal matrix. The fractal parameter is then given. What's more, this kind of hypernetworks is also small-world for the diameter won't exceed twice the diameter of primitive hypergraph. By a novel product of node degree polynomial, node hyperdegree polynomial and hyperedge degree polynomial, node degree, node hyperdegree and hyperedge degree are obtained.

show abstract

“…Definition 2.8 [25] If A is an m×n matrix and B is a p×q matrix, then the Kronecker product on A and B, denoted as A B…”

Section: B Matrix Operationmentioning

confidence: 99%

A New Hypernetwork Model Based on Matrix Operation

Liu

2015

2015 10th International Conference on Intelligent Systems and Knowledge Engineering (ISKE)

View full text Add to dashboard Cite

show abstract

“…where ⊛ denotes the Tracy-Singh product when R is partitioned by columns and R > is partitioned by rows [19]. The transpose of D is calculated as and E a i;j;n a k;l;n Â Ã ¼…”

Section: Variance Relationmentioning

confidence: 99%

Analysis of a reduced-communication diffusion LMS algorithm

et al. 2015

View full text Add to dashboard Cite

a b s t r a c tIn diffusion-based algorithms for adaptive distributed estimation, each node of an adaptive network estimates a target parameter vector by creating an intermediate estimate and then combining the intermediate estimates available within its closed neighborhood. We analyze the performance of a reduced-communication diffusion least mean-square (RC-DLMS) algorithm, which allows each node to receive the intermediate estimates from only a subset of its neighbors at each iteration. This algorithm eases the usage of network communication resources and delivers a trade-off between estimation performance and communication cost. We show analytically that the RC-DLMS algorithm is stable and convergent in both mean and mean-square senses. We also calculate its theoretical steady-state mean-square deviation. Simulation results demonstrate a good match between theory and experiment.

show abstract

“…To reveal the symmetry properties ofĥ 12 , we rewrite it as 43 (45) in terms of the following "block operators" that merely interchange the blocks XY of the wave function:…”

Section: B Symmetries Ofĥ 12mentioning

confidence: 99%

“…where the symbol • represents the Tracy-Singh product, 45,46 which is a generalization of the standard Kronecker product (⊗) for partitioned matrices. The multiplications between the "component operators" (e.g., c σ 1 · p 12 ) and the "block operators" (e.g., C 1 ) in Eq.…”

Section: B Symmetries Ofĥ 12mentioning

confidence: 99%

Relativistic explicit correlation: Coalescence conditions and practical suggestions

Shao

Liu

2012

The Journal of Chemical Physics

View full text Add to dashboard Cite

To set up the general framework for relativistic explicitly correlated wave function methods, the electron-electron coalescence conditions are derived for the wave functions of the Dirac-Coulomb (DC), Dirac-Coulomb-Gaunt (DCG), Dirac-Coulomb-Breit (DCB), modified Dirac-Coulomb (MDC), and zeroth-order regularly approximated (ZORA) Hamiltonians. The manipulations make full use of the internal symmetries of the reduced two-electron Hamiltonians such that the asymptotic behaviors of the wave functions emerge naturally. The results show that, at the coalescence point of two electrons, the wave functions of the DCG Hamiltonian are regular, while those of the DC and DCB Hamiltonians have weak singularities of the type r(12)(ν) with ν being negative and of O(α(2)). The behaviors of the MDC wave functions are related to the original ones in a simple manner, while the spin-free counterparts are somewhat different due to the complicated electron-electron interaction. The behaviors of the ZORA wave functions depend on the chosen potential in the kinetic energy operator. In the case of the nuclear attraction, the behaviors of the ZORA wave functions are very similar to those of the nonrelativistic ones, just with an additional correction of O(α(2)) to the nonrelativistic cusp condition. However, if the Coulomb interaction is also included, the ZORA wave functions become close to the large-large components of the DC wave functions. Note that such asymptotic expansions of the relativistic wave functions are only valid within an extremely small convergence radius R(c) of O(α(2)). Beyond this radius, the behaviors of the relativistic wave functions are still dominated by the nonrelativistic limit, as can be seen in terms of direct perturbation theory (DPT) of relativity. However, as the two limits α → 0 and r(12) → 0 do not commute, DPT is doomed to fail due to incorrect descriptions of the small-small component Ψ(SS) of the DC wave function for r(12) < R(c). Another deduction from the possible divergence of Ψ(SS) at r(12) = R(c) is that the DC Hamiltonian has no bound electronic states, although the last word cannot be said. These findings enrich our understandings of relativistic wave functions. On the practical side, it is shown that, under the no-pair approximation, relativistic explicitly correlated wave function methods can be made completely parallel to the nonrelativistic counterparts, as demonstrated explicitly for MP2-F12. Yet, this can only be achieved by using an extended no-pair projector.

show abstract

A new matrix product and its applications in partitioned matrix differentiation

Cited by 75 publications

References 7 publications

A New Hypernetwork Model Based on Matrix Operation

A New Hypernetwork Model Based on Matrix Operation

Analysis of a reduced-communication diffusion LMS algorithm

Relativistic explicit correlation: Coalescence conditions and practical suggestions

Contact Info

Product

Resources

About