TABLE OF CONTENTS As we will see in the subsequent chapters, it is usually most efficient to approximate the Newton step in some way. One way to do this is to approximate F ?( x n) in a way that not only avoids computation of the derivative, but also saves linear algebra work and matrix storage.
The price for such an approximation is that the nonlinear iteration converges more slowly; i.e., more nonlinear iterations are needed to solve the problem. However, the overall cost of the solve is usually significantly less, because the computation of the Newton step is less expensive.
One way to approximate the Jacobian is to compute F ? ( x 0) and use that as an approximation to F ? ( x n) throughout the iteration. This is the chord method or modified Newton method. The convergence of the chord iteration is not as fast as Newton's method. Assuming that the initial iteration is near enough to x*, the convergence is q-linear. This means that there is ? ? (0, 1) such that
for n sufficiently large. We can apply Theorem 1.2 to the chord method with ? = 0 and ?( x n) = O( e 0) and conclude that ? is proportional to the initial error. The constant ? is called the q-factor. The formal definition of q-linear convergence allows for faster convergence.
