Conjugate gradient method were choosed to solve the large sparse matrix equations induced by the FEM and computer language FORTRAN was used to programme the numerical simulation system software.

Conjugate gradient method, which can be easily computed and requires no matrix storage, is one of the most popular and useful method for solving large scale optimization problems.

It is studied factorizing a matrix over quaternion field to the product of two self - conjugate matrices . and some useful results are obtained.