Roots of Equations - Numerical Methods in Engineering with MATLAB - page 159

Inspecting the plot of the function, wesuspect that the smallest positivezero is a
double root near x 2. Bisection and Brent's methodwouldnot workhere, since they

depend on the function changing its sign at the root. The same argument applies to
the function newtonRaphson .But there no reasonwhy the unrefinedversion of the
Newton-Raphsonmethod shouldnot succeed.We used the following program, which
prints the number of iterations in addition to the root:

=

function [root,numIter] = newton simple(func,dfunc,x,tol)

% Simple version of Newton-Raphson method used in Example 4.7.

_

if nargin < 5; tol = 1.0e6*eps; end
fori=1:30
dx = -feval(func,x)/feval(dfunc,x);
x=x+dx;
if abs(dx) < tol
root = x; numIter = i; return

end

end
root = NaN

The twofunctionscalledbythe program are

functiony=fex4 7(x)

% Function used in Example 4.7.
y=xˆ4-6.4*xˆ3+6.45*xˆ2+20.538*x-31.752;

_

functiony=dfex4 7(x)

% Function used in Example 4.7.
y = 4.0*xˆ3 - 19.2*xˆ2 + 12.9*x + 20.538;

_

Here are the results:

_ _ _

>> [root,numIter] = newton simple(@fex4 7,@dfex4 7,2.0)

root =
2.1000
numIter =
27

It canbe shown that near amultiple root the convergence of the Newton-Raphson
methodislinear, rather than quadratic, which explains the large number of iterations.
Convergence to amultiple root can bespeededupbyreplacing the Newton-Raphson