Solutions of Algebraic Equation Systems

The matrix inversion J is performed by the Gauss elimination method applied to linear equations. The derivatives d//dy, are calculated from analytical expressions or numerical approximations. The approximation of the derivatives d//dy, with forward differences is 

The selection of the difference Ayj is critical the smaller the Ayj, the better the derivative approximation. However, if Ayj is smaller than the accuracy of the computer, the denominator in Equation A1.4 becomes 0. A criterion for the choice of Ay, is 

The choice of the initial estimate of y(o) is critical, as the equation system is solved with the Newton-Raphson method. In many cases, a solution of a problem is expected as a function of the parameter x. The parameter x can be chosen as a continuity parameter the solution for the equation system, obtained for a certain parameter value x,y(g ), can be used as an initial estimate for the next parameter value, x + Ax  

The convergence of algorithm (Equation Al.2) is quadratic in most cases. A termination criterion for the iteration of Equation (A1.2) is usually 

For a system with a single equation, the algorithm (Equation A 1.2) is reduced to 

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