This paper aims to select the best value of the parameter $\rho$ from a general set of linear multistep formulae which have the potential for efficient implementation. The $\rho$ -Diagonally Implicit Block Backward Differentiation Formula ( $\rho$ -DIBBDF) was proposed to approximate the solution for stiff Ordinary Differential Equations (ODEs) to achieve the research objective. The selection of $\rho$ for optimal stability properties in terms of zero stability, absolute stability, error constant and convergence are discussed. In the diagonally implicit formula that uses a lower triangular matrix with identical diagonal entries, allowing a maximum of one lower-upper (LU) decomposition per integration stage to be performed will result in substantial computing benefits to the user. The numerical results and plots of accuracy indicate that the $\rho$ -DIBBDF method performs better than the existing fully and diagonally Block Backward Differentiation Formula (BBDF) methods.

A two-point block backward difference technique is established for solving nonlinear Riccati differential equations directly. The proposed method is coded using a variable order step size (VOS) algorithm. The advantage of the two-point block method is its programmability to implement parallel programming techniques. Combination of the block method and VOS algorithm allows for a significant reduction of computation cost in comparison to conventional methods. With an added advantage of the recursive relationship between integration coefficients of different orders, the proposed two-point block method provides efficient computation without loss of accuracy.