Basic properties of an inverse iteration algorithm for solving linear systems with positive definite matrices

The problem of solving positive definite systems of linear equations with slowly varying coefficients is considered. The problem is reduced to solving the equations of motion for mechanical systems with respect to accelerations when performing their numerical integration. A new iteration method of solution is proposed. It is shown that this method is a modification of the variable metric Powell–Broyden's method. Some conditions of convergence are obtained and the basic properties of the method are considered. It is proved that, in the case of exact arithmetic, the method converges after a finite number of iterations, which does not exceed the rank of the perturbation matrix for the linear system. A comparative efficiency of the method is shown by an example of solving the equation of motion for some particular mechanical systems.