Ioan R. Ciric

By Ioan R. Ciric

The solution of certain differential equations is expressed using a special type of matrix series and is directly related to the solution of general systems of algebraic equations. Efficient formulae for matrix exponentials are derived in terms of rapidly convergent series of the same type. They are essential for two new solution methods, especially beneficial for large linear systems, namely an iterative method and a method based on an exact matrix product formula. The computational complexity of these two methods is analysed, and for both of them, the number of matrix exponential-vector multiplications required for an imposed accuracy can be predetermined in terms of the system condition. The total number of arithmetic operations involved is roughly proportional to n2, where n is the matrix dimension. The common feature of all the series in the results presented is that starting with a first term that is already well-conditioned, each subsequent term is computed by multiplication with an even better conditioned matrix, tending quickly to the identity matrix. This contributes substantially to the stability of the numerical computation. A very efficient method based on the numerical integration of a special kind of differential equations, applicable to even ill-conditioned systems, is also presented.

Part of the book: Functional Calculus

By Ioan R. Ciric

Various ordinary differential equations of the first order have recently been used by the author for the solution of general, large linear systems of algebraic equations. Exact solutions were derived in terms of a new kind of infinite series of matrices which are truncated and applied repeatedly to approximate the solution. In these matrix series, each new term is obtained from the preceding one by multiplication with a matrix which becomes better and better conditioned tending to the identity matrix. Obviously, this helps the numerical computations. For a more efficient computation of approximate solutions of the algebraic systems, we consider new differential equations which are solved by simple techniques of numerical integration. The solution procedure allows to easily control and monitor the magnitude of the residual vector at each step of integration. A related iterative method is also proposed. The solution methods are flexible, permitting various intervening parameters to be changed whenever necessary in order to increase their efficiency. Efficient computation of a rough approximation of the solution, applicable even to poorly conditioned systems, is also performed based on the alternate application of two different types of minimization of associated functionals. A smaller amount of computation is needed to obtain an approximate solution of large linear systems as compared to existing methods.

Part of the book: Matrix Theory