The Improved Precise Integration Method

The precise integration algorithm has been widely applied in the related fields of ordinary differential equations, and the theory of control, due to its calculation speed and the advantages of highly accurate numerical results. However, when the coefficient matrix of ordinary differential equations is larger, its advantage in calculation speed is not obvious. Therefore, the main work in this paper is improving the precise integration algorithm. The main work includes the following three aspects.Firstly, each sub-block was made to produce common items, namely sub-block similarity for the structural dynamics equation based on the Increment-Dimensional Precise Integration Method,. So a few blocks can use each cycle results from precise integration at the same time, which also reduces the computation time and memory requirement greatly.Then the calculation speed is increased. Middle value of each step was taken to deal with the non-homogeneous items,, which can not only simplify calculation but also keep the original precision. Numerical examples can show that the method is of high efficiency and feasibility.Secondly, The key to segmentation method of precise integration method for general power exponent matrix is that, items(except unit matrix) for the power exponent matrix according to the Taylor series expansion, can be divided into sections. Each section of them contains several little section different or uniform, which forms a segmentation way. The corresponding number of matrix multiplication will be calculated combined with the precise integration method of adaptive selection. At the same time, segmentation algorithm can ensure calculation precision, reduce the computation, and short the time of the whole calculation process effectively.Finally, first the matrix was partitioned, then applied in the precise integration method of adaptive selection and Winograd matrix multiplication to the precise integration method for calculating the power exponent matrix with even order. The method has ensured the advantages of traditional precise integration and can determine the cycle number and block number conveniently. What is more important that it will lower the number of the matrix multiplication in the precise integration. So it can guarantee the algorithm's convenience and improve the computational speed. Numerical example also shows the efficiency and feasibility of the method.