In this paper, we proposed a family of r-points 1-block implicit methods with optimized region of stability. This family of methods is derived with Mathematical 10.4 software and the stability is investigated using boundary locus techniques. The block methods are consistence, zero stable, and Astable and satisfy other stability requirements which finds them suitable for stiff problems in ODEs. Numerical experiments are presented and results are compared with other block methods and exact solutions of some stiff ordinary differential equations. The methods have been found to show competitiveness with other numerical methods.

Various algorithm such as Doolittle, Crouts and Cholesky s have been proposed to factor a square matrix into a product of L and U matrices, that is, to find L and U such that A = LU; where L and U are lower and upper triangular matrices respectively. These methods are derived by writing the general forms of L and U and the unknown elements of L and U are then formed by equati ng the corresponding entries in A and LU in a systematic way. This approach for computing L and U for larger values of n will involve many sum of products and will result in n 2 equations for a matrix of order n. In this paper, we propose a straightforward method based on multipliers derived from modification of Gaussion elimination algorithm.

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