Interval method is used for the inclusion of a zero of a function. The Ehrman(EHR) method considers the Newton’s iteration in finding the root of a function. This method is modified by using the mid point in the procedure and improved method has a faster convergence rate and less processing time. In this paper, the convergence analysis and the numerical results are shown.

Interval method is used for the inclusion of a zero of a function. The Ehrman(EHR) method considers the Newton's iteration in finding the root of a function. This method is modified by using the mid point in the procedure and improved method has a faster convergence rate and less processing time. In this paper, the convergence analysis and the numerical results are shown.

In this paper we present a deterministic and continuous model for predator - prey population model based on Lotka-Volterra model. The model is then developed by considering time delay and the two populations are subjected to constant effort of harvesting. We study analytically the necessary conditions of harvesting to ensure the existence of the equilibrium points and their stabilities. The methods used to analyze the stability are linearization and by investigation the eigenvalues of the Jacobian matrix. The results show that there exists a globally asymptotically stable equilibrium point in the positive quadrant for the model with and without harvesting. The time delay can induce instability and a Hopf bifurcation can occur. The stable equilibrium point for the model with harvesting is then related to profit function problem. We found that there exists a critical value of the effort that maximizes the profit and the equilibrium point also remains stable. This means that the predator and prey populations can live in coexistence and give maximum profit although the two populations are harvested with constant effort of harvesting.

In this paper, the result are established in the following four ways: First, we present a general representation for the weighted Drazin inverse Ad,W of an arbitrary rectangular matrix A ∈ Mm,n involving Moore-Penrose inverse, which reduces to the well-known result if the matrix A is a square and W = In . Second, we find represenations for

We study the numerical performance of a matrices storage free quasi- Newton method for large-scale optimization, which we call the F-BFGS method. We compare its performance with that of the limited memory BFGS, L-BFGS methods developed by Nocedal (1980) and the conjugate gradient methods. The F-BFGS method is very competitive due to its low storage requirement and computational labor and also able to solve large-scale problems with 106 variables successfully while other methods fail.

In this paper, a limited modified BFGS method for solving large- scale unconstrained optimization problems is proposed. The proposed algorithm generates quasi-Newton directions using a modified BFGS method suggested by Biggs (1973). The modified BFGS method is then extended to the limited memory version. In order to use only minimum storage for the modification, the modification is only applied to the last BFGS corrections. Numerical results indicate that an improvement is achieved.

Abstrak Pengusikan ke atas hampiran matriks songsanganHessian berpangkat- dua yang simetri dan tentu positif dipertimbangkan. Kita usik setiap komponen dalam matriks tersebut dengan suatu skalar nyata. Kita mula dengan anali- sis secara teliti untuk mendapatkan batas pengusikan norma demi norma yang membawa kepada matriks nombor syarat terturun dan tentu positif. Contoh berangka dikemukakan untuk mengkaji kesan pengusikan tersebut. Katakunci Hampiran songsangan Hessian pangkat-dua yang simetri dan tentu positif, batas-batas pengusikan norma demi norma. Abstract Perturbation to a symmetric positive definite of the rank-two ap- proximate inverse Hessian matrix is considered. We perturb every component in the considered matrix by a real scalar. We begin the work by giving a thorough analysis to abtain the normwise perturbation bounds that lead to a reduced con- dition number and positive definite matrix. Numerical examples are presented to study the eect of our perturbation.

In this paper we present a deterministic and continuous model for predator - prey population model based on Lotka-Volterra model. The model is then developed by considering time delay and the two populations are subjected to constant effort of harvesting. We study analytically the necessary conditions of harvesting to ensure the existence of the equilibrium points and their stabilities. The methods used to analyze the stability are linearization and by investigation the eigenvalues of the Jacobian matrix. The results show that there exists a globally asymptotically stable equilibrium point in the positive quadrant for the model with and without harvesting. The time delay can induce instability and a Hopf bifurcation can occur. The stable equilibrium point for the model with harvesting is then related to profit function problem. We found that there exists a critical value of the effort that maximizes the profit and the equilibrium point also remains stable. This means that the predator and prey populations can live in coexistence and give maximum profit although the two populations are harvested with constant effort of harvesting.