This paper is aimed at discussing and comparing the performance of standard method with its hybrid method of the same step number for the solution of first order initial value problems of ordinary differential equations. The continuous formulation for both methods was obtained via interpolation and collocation with the application of the shifted Legendre polynomials as approximate solution which was evaluated at some selected grid points to generate the discrete block methods. The order, consistency, zero stability, convergent and stability regions for both methods were investigated. The methods were then applied in block form as simultaneous numerical integrators over non-overlapping intervals. The results revealed that the hybrid method converges faster than the standard method and has minimum absolute error values.

Considered in this paper are two basic methods of approximating the solutions of nonlinear systems of algebraic equations. The Steepest Descent method was presented as a way of obtaining good and sufficient initial guess (starting value) which is in turn used for the Broyden's method. Broyden's method on the other hand replaces the Newton's method which requires the use of the inverse of the Jacobian matrix at every new step of iteration with a matrix whose inverse is directly determined at each step by up-dating the previous inverse. The result obtained by this method revealed that the setbacks encountered in computing the inverse of the Jacobian matrix at every step number is eliminated hence saving human effort and computer time. The obtained results also showed that the number of steps that is reduced when compared to Newton's method used on the same problem.

This paper presents the derivation techniques of block method for solving higher order initial value problems of ordinary differential equations directly. The method was developed via interpolation and collocation of the shifted Legendre polynomials as basis function. The method is capable of providing the numerical solution at several points simultaneously.