Frobenius norm

Note that the expression on the right hand side of the inequality (A.27) is called the Frobenius norm of a matrix, HA1 ... 

The MNM method further modifies the Hessian matrix based on the PD scheme, and it guarantees the matrix to be positive semidefinite. This process is fundamentally equivalent to modifying the molecular interactions defined in the original force field. In MNM, all PD s are changed to their nearest (in terms of the Frobenius norm) SPSD matrices ffi. 

Pearson s linear correlation coefficient, 239 perturbation analysis, pairwise decomposition scheme Frobenius norm, 235 RTB Hessian matrix, 234 symmetric positive semidefinite (SPSD) matrix, 237... 

Chebyshev s and Holder s norms define the matrix norms Halloo, IIA lip. Also, the Frobenius norm (F norm) is a customary matrix norm, A f = IfliyP), ... 

In this paper, we examine the electron correlation of one-dimensional and quasi-one-dimensional Hubbard models with two sets of approximate iV-representability conditions. While recent RDM calculations have examined linear [20] as well as 4 x 4 and 6x6 Hubbard lattices [2, 57], there has not been an exploration of ROMs on quasi-one-dimensional Hubbard lattices with a comparison to the one-dimensional Hubbard lattices. How does the electron correlation change as we move from a one-dimensional to a quasi-one-dimensional Hubbard model How are these changes in correlation reflected in the required A -repre-sentability conditions on the 2-RDM One- and two-par-ticle correlation functions are used to compare the electronic structure of the half-filled states of the 1 x 10 and 2x10 lattices with periodic boundary conditions. The degree of correlation captured by approximate A -repre-sentability conditions is probed by examining the one-particle occupations around the Fermi surfaces of both lattices and measuring the entanglement with a size-extensive correlation metric, the Frobenius norm squared of the cumulant part of the 2-RDM [23]. 

We measure correlation explicitly by calculating the squared Frobenius norm of the cumulant portion of the 2-RDM... 

Fig. 6 Frobenius norm squared of the cumulant part of the 2-RDM is shown for the (a) 2 x 10 and (b) 1x10 lattices where the 2-RDMs are computed with variational 2-RDM calculations with DQG and DQGT conditions...

The inter-molecular distance between two molecules A and B, each represented by an n X n LDM, is defined as the Frobenius norm of the difference matrix, that is ... 

Remark 1. Note that in contrast to problem (14) a weighted Frobenius norm (weighted by the matrix is used in problem (23). 

Several norms of a matrix are defined in the existing literature. The Frobenius norm is defined as ... 

The previously undefined symbols in Equation 22 will now be defined as follows rg = [Pg.529]

The low-dimensional embedding of both the previously learnt data, and the newly added data, is then improved using an iterative method similar to Ritz acceleration [34], The basic premise is to refine the low-dimensional embedding using the updated structure information found in F ew. The frill algorithm is displayed in Algorithm 1 of [33]. The termination criteria for this refinement step is to either continue for a set number of iterations, or terminate when the error—measured as the Frobenius norm between the previous iteration s estimate and that of the current iteration—falls below a certain tolerance threshold. 

---