The file L17.m contains all MATLAB instructions:
diary L17.txt % We apply Newton's method on the system % exp(x) - y = 0, % x*y - exp(x) = 0, % starting at (0.9,2.5). x0 = 0.9; y0 = 2.5; % start point a = [exp(x0) -1; y0 - exp(x0) x0] b = -[exp(x0) - y0; x0*y0 - exp(x0)] dxy = a\b x1 = x0 + dxy(1) y1 = y0 + dxy(2) % we cannot resist taking another step a = [exp(x1) -1; y1 - exp(x1) x1] b = -[exp(x1) - y1; x1*y1 - exp(x1)] dxy = a\b x2 = x1 + dxy(1) y2 = y1 + dxy(2) % and we continue : a = [exp(x2) -1; y2 - exp(x2) x2] b = -[exp(x2) - y2; x2*y2 - exp(x2)] dxy = a\b x3 = x2 + dxy(1) y3 = y2 + dxy(2) % the exact solution is (1,exp(1)): error0 = [x0 - 1; y0 - exp(1)]; norm(error0) error1 = [x1 - 1; y1 - exp(1)]; norm(error1) error2 = [x2 - 1; y2 - exp(1)]; norm(error2) error3 = [x3 - 1; y3 - exp(1)]; norm(error3) diary off
The file produced by running L17.m in MATLAB is L17.txt, displayed below:
a =
2.4596 -1.0000
0.0404 0.9000
b =
0.0404
0.2096
dxy =
0.1091
0.2280
x1 =
1.0091
y1 =
2.7280
a =
2.7432 -1.0000
-0.0152 1.0091
b =
-0.0152
-0.0097
dxy =
-0.0091
-0.0097
x2 =
1.0000
y2 =
2.7183
a =
2.7184 -1.0000
-0.0001 1.0000
b =
1.0e-03 *
-0.1129
0.0244
dxy =
1.0e-04 *
-0.3255
0.2443
x3 =
1.0000
y3 =
2.7183
ans =
0.2401
ans =
0.0133
ans =
4.0702e-05
ans =
2.2546e-09
Observe the quadratic convergence: the errors go from 10^(-1), 10^(-2), 10^(-5), to 10^(-9). This means that the number of correct decimal places doubles in each step.