Matrix Science Mathematic (MSMK)

The approximate equivalence of functions is defined with respect to Taylor series expansions first. Based on the approximate equivalence, approximation of a given function with other functional series is discussed. The theory is first applied to root finding iteration algorithms. Then the solution of ordinary differential equations is treated as the next application. Several numerical sample problems are considered. It is shown that the formalism can be effectively used in determining the roots of algebraic equations or finding the solutions of differential equations. To increase the precision of the solutions, higher order approximations should be taken.