Newton&'s Method

Definition (Order of Convergence)Assume that converges top,and set.If two positive constantsexist, and



then the sequence is said to converge topwith order of convergence R.The numberAis called the asymptotic error constant.The casesare given specialconsideration.

(i)If, the convergence ofis called linear.

(ii)If, the convergence ofis called quadratic.

Theorem (Convergence Rate for Newton-Raphson Iteration)Assume that Newton-Raphson iteration produces a sequence that converges to the rootpof the function.

Ifpis a simple root, then convergence is quadratic andforksufficiently large.

Ifpis a multiple root of orderm,then convergence is linear andforksufficiently large.

Reduce the volume of printout.

After you have debugged you program and it is working properly, delete the unnecessary print statements

and

Concise Program for the Newton-Raphson Method

Now test this subroutine using the function in Example 1.

Error Checking in the Newton-Raphson Method

Error checking can be added to the Newton-Raphson method.Here we have added a third parameterto the subroutine which estimate the accuracy of the numerical solution.

The following subroutine call uses a maximum of 20 iterations, just to make sure enough iterations are performed.However, it will terminate when the difference between consecutive iterations is less than.By interrogatingkafterward we can see how many iterations were actually performed.

Various Scenarios