Newton's method is as follows: rn Write a function newton.m that is the fixed point iteration for Newton's method where f.m and df.m contai the functions f(x) and f'(x) respectively. (Just program the derivative you do not have to use Matlab to find the derivative.) Solve the following using Newton's method r3256 4-28 100 in(r) Note that Newton's method is generally seen as "superior to other methods because Newton's method converges quadratically, whereas the other methods converge linearly The experimental rate of convergence for an iterative method may be foun by finding NOTE: In Matlab, log(r) is the natural logarithm, and exp(r) is the natural exponential. Find the experimental rate of convergence for each example above, usig 1-4 iterations only For each example, produce a table that looks like the following table: 0.735759 4.261170- 2.67 2 0.694042 8.9511M4 2.22 3 0.693148 4.004998- 2.10 0.693147 8.015810e-14 2.05 Table 1: Iterations and convergence rates for Newton's approximation to ex-2, 1 Now, consider the problem, ez-z = l this is a root of multiplicity 2. This is because 0 solves the equation, and if we differentiate the equation ex-1 0, z = 0 solves that equation as well. In the case of roots of multiplicity greater than one, Newton's method only converges linearly, not quadratically Finally, solve the problem sin(z)/z = 1 here z 0 is not really a zero (why?) but Newton's method will give0 as an answer. Note that the "zero" also has a multiplicity of 2 We can fix this problem by modifying Newton's method. For a zero of multiplicity m, the modified Newton's method is given by fan) f'(Tn) The multiplicity of the root,m must be known ahead of time. NOTE: Your hd in should include five tables like Table 1 one for each example Newton's method is as follows: rn Write a function newton.m that is the fixed point iteration for Newton's method where f.m and df.m contai the functions f(x) and f'(x) respectively. (Just program the derivative you do not have to use Matlab to find the derivative.) Solve the following using Newton's method r3256 4-28 100 in(r) Note that Newton's method is generally seen as "superior to other methods because Newton's method converges quadratically, whereas the other methods converge linearly The experimental rate of convergence for an iterative method may be foun by finding NOTE: In Matlab, log(r) is the natural logarithm, and exp(r) is the natural exponential. Find the experimental rate of convergence for each example above, usig 1-4 iterations only For each example, produce a table that looks like the following table: 0.735759 4.261170- 2.67 2 0.694042 8.9511M4 2.22 3 0.693148 4.004998- 2.10 0.693147 8.015810e-14 2.05 Table 1: Iterations and convergence rates for Newton's approximation to ex-2, 1 Now, consider the problem, ez-z = l this is a root of multiplicity 2. This is because 0 solves the equation, and if we differentiate the equation ex-1 0, z = 0 solves that equation as well. In the case of roots of multiplicity greater than one, Newton's method only converges linearly, not quadratically Finally, solve the problem sin(z)/z = 1 here z 0 is not really a zero (why?) but Newton's method will give0 as an answer. Note that the "zero" also has a multiplicity of 2 We can fix this problem by modifying Newton's method. For a zero of multiplicity m, the modified Newton's method is given by fan) f'(Tn) The multiplicity of the root,m must be known ahead of time. NOTE: Your hd in should include five tables like Table 1 one for each example
