Householder's Method

Author

Eric W. Weisstein
March 24, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumericalMath/HouseholdersMethod.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HouseholdersMethod.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Iteration

householderit[n_][z_] := z-pow[n][z]/pow[n]'[z](1-pow[n][z]^2 pow[n]''[z]/
  2/pow[n]'[z]^2);

house=Compile[{{n, _Integer}, {z, _Complex}}, 
      Arg[FixedPoint[(n-1)/(2n^2)#(2n+#^n ) &, z, 35]]];

Simplify[z-z^n/D[z^n,z](1-(z^n)^2D[z^n,{z,2}]/2/D[z^n,z]^2)]

house[3,1.2+I]

house[3,1.0+I]

Roots[z^3==1,z]

N[%7]

FixedPointList[(n-1)/(2n^2)#(2n+#^n )& /. n->3 ,-.5+.8I, 20]

Graphics

r=2.5;
rstep=.02;
Show[GraphicsArray[Partition[
Table[ListDensityPlot[Table[house[n,b+a I],{a,-r,r,rstep},{b,r,-r,-rstep}],Mesh->False,
DisplayFunction->Identity,
PlotRange->All,
ColorFunction->Hue,
Frame->False],
{n,3,8}],3],
GraphicsSpacing->0.05]]//Timing