Abstract of Thesis

Acceleration via Newton method

Automated systems software specialty

6 CAUCHY PROBLEM SOLVING FOR ODE VIA USING NEWTON METHOD

In order to reduce time of the Cauchy problem (4.1) iterative methods with highest rate of convergence should be used.

The essence of a Newton method consists in searching of the next approximated solution by the formula (6.1). Factor defines the correction data for required numerical solution calculation on the next iteration [5].

(6.1)

Linear equation set that tie together correction data values _n,i (0 < i < k+1) and numerical solutions u_n,i (0 < i < k+1) on current iteration s is defined by the formula (6.2). Values _n,i (0 < i < k+1) are mentioned as unknown variables.

(6.2)

Using of Newton method in the multipoint method (for the correction data calculation on current iteration) determines finding solution of the linear equation set listed below. Every method used for solution finding of linear equation set (direct or iterative) assumes appropriate source data distribution among the processor units of the computer network.

Thus, execution time of parallel algorithm defined by formulas (6.1) – (6.2) and used for the Cauchy problem solving that depends on the chosen algorithm of linear equations set solving. In addition communicational overhead of the potential data transfers should be taken into consideration.