NON-LINEAR EQUATIONS
                         ====================

The following is a MATLAB session, in which several root finding methods
are applied to find a root of the function cubic. Its derivative, cubic_der,
is used in Newton's Method. Note that the programs do not use "natural"
stopping criteria (hence, for example, the division by zero with the
Secant Method.) Notice how much faster Newton's Method and the Secant
Method are relative to the Bisection Method.

==============================================================================

function y=cubic(x)
y=x^3-x-6;
end;

==============================================================================

function y=cubic_der(x)
y=3*x^2-1;
end;

==============================================================================

>> format long
>> bisection('cubic',0,3,50)

ans =

  Columns 1 through 4 

   1.50000000000000   2.25000000000000   1.87500000000000   2.06250000000000

  Columns 5 through 8 

   1.96875000000000   2.01562500000000   1.99218750000000   2.00390625000000

  Columns 9 through 12 

   1.99804687500000   2.00097656250000   1.99951171875000   2.00024414062500

  Columns 13 through 16 

   1.99987792968750   2.00006103515625   1.99996948242188   2.00001525878906

  Columns 17 through 20 

   1.99999237060547   2.00000381469727   1.99999809265137   2.00000095367432

  Columns 21 through 24 

   1.99999952316284   2.00000023841858   1.99999988079071   2.00000005960464

  Columns 25 through 28 

   1.99999997019768   2.00000001490116   1.99999999254942   2.00000000372529

  Columns 29 through 32 

   1.99999999813735   2.00000000093132   1.99999999953434   2.00000000023283

  Columns 33 through 36 

   1.99999999988358   2.00000000005821   1.99999999997090   2.00000000001455

  Columns 37 through 40 

   1.99999999999272   2.00000000000364   1.99999999999818   2.00000000000091

  Columns 41 through 44 

   1.99999999999955   2.00000000000023   1.99999999999989   2.00000000000006

  Columns 45 through 48 

   1.99999999999997   2.00000000000001   1.99999999999999   2.00000000000000

  Columns 49 through 50 

   2.00000000000000   2.00000000000000

>> newton('cubic','cubic_der',3,10)

ans =

  Columns 1 through 4 

   3.00000000000000   2.30769230769231   2.04181989484242   2.00092462158247

  Columns 5 through 8 

   2.00000046599634   2.00000000000012   2.00000000000000   2.00000000000000

  Columns 9 through 11 

   2.00000000000000   2.00000000000000   2.00000000000000

>> secant('cubic',3,4,10)

Warning: Divide by zero

ans =

  Columns 1 through 4 

   3.00000000000000   4.00000000000000   2.50000000000000   2.27200000000000

  Columns 5 through 8 

   2.05723296766887   2.00753962342663   2.00022971811026   2.00000094194724

  Columns 9 through 12 

   2.00000000011802   2.00000000000000   2.00000000000000                NaN

>> quit

 670 flops.