In this paper, the system of Fredholm integral equations of the second kind is investigated by using a modified degenerate kernel  method (MDKM). To construct a MDKM the source function is approximated by the same way of producing degenerate kernel. The interpolation is used to make the needed approximations. Lagrange polynomials are adopted for the interpolation. The equivalency of  proposed m...

The multiplication in Fpn can be performed using a polynomial version of Montgomery multiplication (Montgomery, 1985). In (Bajard et al., 2003) Bajard et al. improved this method by using a Lagrange representation: the elements of Fpn are represented by their values at a fixed set of points. The costly operations in this new algorithm are the two changes of Lagrange representation which require...

and there are no inequality constraints (i.e. there are no fi(x) i = 1, . . . , m). We simply write the p equality constraints in the matrix form as Cx− d = 0. The basic idea in Lagrangian duality is to take the constraints in (1) into account by augmenting the objective function with a weighted sum of the constraint functions. We define the Lagrangian L : R ×R ×R → R associated with the proble...

One of the well-known problems in evolutionary search for solving optimization problem is the premature convergence. The general constrained optimization techniques such as hybrid evolutionary programming, two{phase evolutionary programming, and Evolian algorithms are not safe from the same problem in the rst phase. To overcome this problem, we apply the sharing function to the Evolian algorith...

The Lagrange multiplier method for combining observations and models (i.e., the adjoint method or “4DVAR”) has been avoided or approximated when the numerical model is highly nonlinear or chaotic. This approach has been adopted primarily due to difficulties in the initialization of low-dimensional chaotic models, where the search for optimal initial conditions by gradient-descent algorithms is ...