Abstract
Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A
frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two
parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian
iterative method is three, and we design the base method in a way that we can maximize the convergence
order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four
systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence
order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m - 2) for
m = 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only
single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper
because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The
claimed convergence order is verified by computing the computational order of convergence for a system of
nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving
many nonlinear initial and boundary value problems.
Peer Reviewed