| Systems of linear equations arise in the solutions of many continuous mathematical models. Finding exact solutions to such models is very tedious, sometimes impossible. Instead of finding exact solutions to continuous problems such as ordinary differential equations, partial differential equations, and optimization problems, computer simulations are used to provide approximate solutions.; In the numerical solution of ordinary differential equations with separated boundary conditions, almost block diagonal (ABD) linear systems are encountered, using a technique of spline collocation. ABD linear systems are one type of the many systems of linear equations. This thesis deals exclusively with two types of almost block diagonal linear systems arising from B-splines, and monomial splines. An explanation and a description of spline functions, collocation methods, and the structure of ABD systems is given. The motivation of this thesis however, is to provide and discuss two MATLAB packages, COLPACK and ABDPACK, which are programmed efficiently to give solutions for the two ABD systems, and condition number estimation for the B-spline basis system. In consequence, two MATLAB test packages, (COLTESTPACK and ABDTESTPACK) are developed to test, COLPACK and ABDPACK. The numerical results obtained from the test packages, demonstrate that the two packages give reasonable approximations to the exact solutions, and a good condition number estimation. |
