>> x[1] = 1;
>> for i=1:4, x(i+1) = 5 + x(i) - x(i)^2; end
I1 is not convergent
>> x

x =

           1           5         -15        -235      -55455

>> for i=1:4, x(i+1) = 5/x(i); end
I2 is not convergent
>> x

x =

     1     5     1     5     1

>> for i=1:20, x(i+1) = 1 + x(i) - x(i)^2/5; end
I3 is convergent
>> x

x =

  Columns 1 through 7 

    1.0000    1.8000    2.1520    2.2258    2.2350    2.2360    2.2361

  Columns 8 through 14 

    2.2361    2.2361    2.2361    2.2361    2.2361    2.2361    2.2361

  Columns 15 through 21 

    2.2361    2.2361    2.2361    2.2361    2.2361    2.2361    2.2361

I3 is converging to sqrt(5) 
>> err = sqrt(5) - x

err =

  Columns 1 through 6 

  1.2361e+000  4.3607e-001  8.4068e-002  1.0289e-002  1.1074e-003  1.1716e-004

  Columns 7 through 12 

  1.2371e-005  1.3061e-006  1.3789e-007  1.4557e-008  1.5368e-009  1.6225e-010

  Columns 13 through 18 

  1.7129e-011  1.8083e-012  1.9096e-013  2.0428e-014  2.2204e-015            0

  Columns 19 through 21 

            0            0            0

>> for i=1:19, ratio(i) = err(i+1)/err(i); end

the ratio's bounded with n.  So I3 is linearly convergent.  The ratio
is going to g'(sqrt(5)), as expected.

>> ratio

ratio =

  Columns 1 through 6 

  3.5279e-001  1.9279e-001  1.2239e-001  1.0763e-001  1.0579e-001  1.0560e-001

  Columns 7 through 12 

  1.0558e-001  1.0557e-001  1.0557e-001  1.0557e-001  1.0557e-001  1.0557e-001

  Columns 13 through 18 

  1.0557e-001  1.0560e-001  1.0698e-001  1.0870e-001            0          NaN

  Column 19 

          NaN

>> 1 - 2*sqrt(5)/5

ans =

  1.0557e-001


>> for i=1:10, ratio(i) = err(i+1)/err(i)^(1.2); end

the ratio's growing with n.  So I3 is not convergent with rate p=1.2.
>> ratio

ratio =

  Columns 1 through 6 

  3.3815e-001  2.2760e-001  2.0082e-001  2.6882e-001  4.1267e-001  6.4550e-001

  Columns 7 through 10 

  1.0118e+000  1.5862e+000  2.4868e+000  3.8989e+000

>> for i=1:10, ratio(i) = err(i+1)/err(i)^2; end

the ratio's growing with n.  So I3 is not quadratically convergent.
>> ratio

ratio =

  Columns 1 through 6 

  2.8541e-001  4.4210e-001  1.4558e+000  1.0461e+001  9.5535e+001  9.0134e+002

  Columns 7 through 10 

  8.5340e+003  8.0832e+004  7.6564e+005  7.2523e+006

now test I4.
>> for i=1:10, x(i+1) = 1/2*(x(i) + 5/x(i)); end
>> x

x =

  Columns 1 through 6 

  1.0000e+000  3.0000e+000  2.3333e+000  2.2381e+000  2.2361e+000  2.2361e+000

  Columns 7 through 11 

  2.2361e+000  2.2361e+000  2.2361e+000  2.2361e+000  2.2361e+000

>> err = sqrt(5) - x

>> err =

  Columns 1 through 6 

  1.2361e+000 -7.6393e-001 -9.7265e-002 -2.0273e-003 -9.1814e-007 -1.8829e-013

  Columns 7 through 11 

            0            0            0            0            0

>> for i=1:10, ratio(i) = err(i+1)/err(i); end
the ratio's bounded with n.  So I4 is linearly convergent.
>> ratio

ratio =

  Columns 1 through 6 

 -6.1803e-001  1.2732e-001  2.0843e-002  4.5290e-004  2.0508e-007            0

  Columns 7 through 10 

          NaN          NaN          NaN          NaN

>> for i=1:10, ratio(i) = err(i+1)/abs(err(i))^(1.2); end
the ratio's bounded with n.  So I4 is convergent with rate p=1.2
>> ratio

ratio =

  Columns 1 through 6 

   5.9238e-01   1.3437e-01   3.3217e-02   1.5654e-03   3.3063e-06            0

  Columns 7 through 10 

          NaN          NaN          NaN          NaN


>> for i=1:10, ratio(i) = err(i+1)/err(i)^2; end

the ratio's bounded with n.  So I4 is quadratically convergent.
>> ratio

ratio =

  Columns 1 through 6 

 -5.0000e-001 -1.6667e-001 -2.1429e-001 -2.2340e-001 -2.2336e-001            0

  Columns 7 through 10 

          NaN          NaN          NaN          NaN