3- The numerical method

The method of spline collocation at Gaussian points is implemented in COLSYS to solve the mixed system of d nonlinear differential equations of orders

where the sought solution u(x)=(u_1(x), ... ,u_d(x)) is an isolated solution vector. The system is subjet to m*=m_1+...+m_d nonlinear multipoint separated boundary conditions

where \zeta_j is the location ofthe j-th boundary (or side) condition. The problem is solved on a sequence of meshes until a user-specified set of tolerances is satisfied.

Nonlinear problem are solving using the damped Newton's method of quasi-linearization. Thus, at each iteration a linearized problem is solved by collocation as described above. The damping or relaxation factor is controlled by a scheme that is a slight modification of that suggested by Deuflhard (THU-Math-7627, Munchen, Germany) . If the problem (1)-(2) is not prespecified by the user as being "sensitive", and if nonlinear convergence on a mesh has just been obtained, then on the next mesh a modified Newton method with a fixed Jacobian is performed as long as the residual monotonically decreases at a sufficiently rapid rate.