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In what follows, we assume that the functions f and R are nonlinear in y; on [a, b], they satisfy conditions that ensure the existence of a solution to problem (1.1),(1.2).

Let us examine the numerical solution of problem (1.1), (1.2) by the shooting method. The problem consists in reducing the boundary value problem (1.1), (1.2) to the initial value problem for system (1.1) subject to the conditions

(2.3)

On the Parametrization of Numerical Solutions ... 541

It is clear that, in this case, the solution to the initial value problem depends on x; i.e.,

Moreover, function (2.4) must satisfy the condition

F(x) R(y(a,x),y(b,x)) =0. (2.5)

The vectors can be determined from Eq. (2.5), which can be solved, for instance, by the iterative Newton's method. Then,

(2.6)

Here, x(k) is the value of x at the kth iteration step and * is the Jacobian matrix of system (2.5).