Page 89 - Numerical methods for chemical engineering

Newton’s method for multiple nonlinear equations 75

stin at

2
2

2

2 2
1
Figure 2.8 Contour plot of f 2 for a system with a solution at (3,4).

initia ess 2 2 2 n er iter atins
1

2
2

2

2 2
1
Figure 2.9 Trajectory of Newton’s method with the initial guess (2,2).

Figure 2.10 repeats this plot for an initial guess of (1,1). While Newton’s method again
converges (now in eleven iterations to the desired accuracy), a much larger step into a region
of high 2-norm is made. With an initial guess of (2,–1), such a departure is even more
pronounced (Figure 2.11); however, convergence to the desired accuracy still occurs after
21 iterations. While Newton’s method does converge (at least for these initial guesses), it