A method of solving a system of linear algebraic equations where is a positive-definite (symmetric) matrix. This is at the same time a direct and an iterative method: for any initial approximation, it converges after a finite number of iterations to give the exact solution. In this method the matrix of the system does not change in the process of calculation and at every iteration it is only used to multiply a vector. Therefore, the order of systems that can be solved on computers is high, being determined by the amount of numerical information needed to specify the matrix.
The conjugate-gradient method is related to a class of methods in which for a solution a vector that minimizes some functional is taken. To calculate this vector an iterated sequence is constructed that converges to the minimum point.
This method and its analogues have many different names, such as the Lanczos method, the Hestenes method, the Stiefel method, etc. Of all the methods for minimizing a functional, the conjugate-gradient method is best in strategic layout: it gives the maximal minimization after steps