In: Civil Engineering
Using Matlab, create a code that determines the highest real root of f(x)=x3-6x2+11x-6.1 using the Newton-Raphson method with x0=3.5 for three iterations. Verify that the process is quadratically convergent.
I found the code to get the highest real root (root for three iterations = 3.0473), however, I do not know how to verify that it is quadratically convergent.
Convergence of Newton-Raphson method:
Suppose is a root of
and
is an estimate of
s.t.
. Then by Taylor series
expansion we have,
for some between
and
.
By Newton-Raphson method, we know that
i.e.
Using(2*) in (1*) we get
Say
where denote the error in the solution
at n
and (n+1)
iterations.
Newton Raphson Method is said to
have quadratic convergence.