Continuation method for nonlinear system of scalar and functional equations in dynamic spin-fluctuation theory

In many applied problems the model is described by a nonlinear system consisting of scalar and functional equations (integral equations, boundary-value problems, etc.). We present a continuation method that ensures accurate and fast self-consistent calculation. For a fixed parameter value, the system of equations is solved using a modified Gauss-Seidel method, which allows one to use a suitable numerical method for each part of the system. We apply the continuation method to calculating temperature dependence of magnetic characteristics in metals. The method removes hysteresis behaviour, typical to the variational Gaussian approximation, without the necessity of taking the higher-order terms into account.