It is well-known that the pseudoinverse Newton Raphson method converges locally if the rank of the Jacobian matrix is constant.
From Figures 7 and 8 we can see that the triangularity of the M cp matrix is constant, regardless the geographical and/or temporal selection of the data.
The mass matrix is constant over time; the external forces P e are the reaction forces; the displacement induced internal forces Kx are the nodal forces resulting from the bar forces.
By applying Theorem 4.7 and Theorem 3.4(b), we give sufficient and also necessary conditions for the solutions of (1.4) to be asymptotically constant, if in addition each matrix is constant (independent of ).
In this work, three-dimensional reconstruction with WBP is addressed from a matrix perspective by formulating the problem as a set of sparse matrix-vector products, with the matrix being constant and shared by all the products.
Then, the rate matrix and the population projection matrix are constant over time.
As in the bifurcation analyses for Eq. (3) mentioned before, we assume that the deterministic parts of the perturbation as well as the diffusion matrix are constant, i.e. begin{aligned} mu_{i}(t):=mu_{i}inmathbb{R},quad text{for } iinlbrace3,4,5rbracequad text{and}quad varSigma(t)=varSigmainmathbb {R}^{3times3}.
Most phylogeny reconstruction methods for nucleotides, including those used here, assume that the relative rates of different nucleotide substitution types (i.e., the instantaneous rate matrix) are constant across the tree, and can be led astray if this assumption is violated.
As in the univariate case, the population will eventually follow the optimum with a constant lag (assuming the G-matrix is constant): (Jones et al. 2004; Gomulkiewicz and Houle 2009; Chevin 2013; Jones et al. 2012).
where (t) is the iteration order, ∇g(U) is the projected gradient of the function g with respect to U, while all the other matrices are constant, and α (t) is the step size to update the matrix.
Therefore, by iteratively updating each matrix using updating rule in Eq. (26), while the other matrices are constant, we can iteratively solve for W j, W 1d, W 2d, H 1, and H 2, in the sub-problems of Eqs.
