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The Accelerated and Modified Newton Methods

Example.

Use the accelerated Newton's method to find the double root,of the cubic polynomial.Use the starting value

Solution.

Graph the function.

[Graphics:../Images/NewtonAccelerateMod_gr_124.gif]

[Graphics:../Images/NewtonAccelerateMod_gr_127.gif]

Since thedouble roothas orderdouble root,the accelerated Newton-Raphson iteration formulag[x]is

Investigate quadratic convergence at the double root,using the starting value

First, do the iteration one step at a time.
Type each of the following commands in a separate cell and execute them one at a time.

Notice that convergence is much faster than the standard Newton-Raphson iteration.

At the double rootwe can explore the ratio .

[Graphics:../Images/NewtonAccelerateMod_gr_157.gif]

Therefore, the accelerated Newton--Raphson iteration is converging quadratically.

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