Increasing number of decimal places with FixedPoint

Question

I've tried:

In[169]:= newton3[x_] := N[1/2 (x + 3/x)];
FixedPointList[newton3, 1.0]

Out[170]= {1., 2., 1.75, 1.73214, 1.73205, 1.73205, 1.73205}

Of course:

In[164]:= N[Sqrt[3], 20]

Out[164]= 1.7320508075688772935

How do you use the FixedPointList to increase the number of decimal places?

Answer 1

The problem is not FixedPointList - the problem is that Mathematica by default doesn't display that many digits. Consider for example

FixedPoint[newton3, 1.0] // InputForm

which displays several more decimals. Another option is to increase PrintPrecision:

SetOptions[InputNotebook[], PrintPrecision -> 10]

This increases the number of decimals shown for all values in the current notebook.

Style[FixedPoint[newton3, 1.0], PrintPrecision -> 10]

does the same for the particular expression.

Finally we may do the following to make sure the problem isn't with FixedPoint:

FixedPoint[newton3, 1.0] == Sqrt[3]
(* True *)

Answer 2

You can also change the number of digits in the solution by setting smaller the criterion for convergence through SameTest, as in:

FixedPointList[(# + 3/# )/2 &, 1`20, SameTest -> (Abs[#1 - #2] < 1*^-10 &)]


{1.0000000000000000000, 2.0000000000000000000, 1.7500000000000000000, \
1.7321428571428571429, 1.7320508100147275405, 1.7320508075688772953, \
1.7320508075688772935}