recursive sqrt(2) ratio function in matlab / octave

Question

I'm trying to create a recursive sqrt(2) ratio function in matlab / octave. I can create the recursive pattern in a spreadsheet see numbers and the corresponding formulas that create them below.

I'm having trouble getting the function to work correctly in matlab / octave. The a and b values should match the spreadsheet images above.

function f = rtsqrt2(a,b,n)          
  for rr=1:n
    rr
    if (rr==1)
      ab_num(rr,:)=[a,b];

    elseif (rr==2)
      a=a+b+b;
      b=1+b;
      ab_num(rr,:)=[a,b];

    elseif 
      a=a+b+b;
      b=a+b;
      ab_num(rr,:)=[a,b];

    end
  end
  ab_num
end

when I do a rtsqrt2(1,1,10)

The first two lines are correct but on the third line I get 7,9 instead of what I should get 7,5 and things get worse from there. How can I fix it so that my recursive function follows the pattern in the first image.

Please note I'm using octave 4 which is similar to matlab and that I plan on using more than just 10 recursions (10 recursions were used just to test with)

Answer 1

Here's an implementation that fits your description. Note the computation (as you described) is iterative not recursive.

I've included some extra bits in the function to make and display the table.

function x = rtsqrt2(a, b, n)
%% Compute the square root of 2 iteratively
%
% n = number of iterations
% a, b  are the initial values

%Initialise the vector A and B to hold the computation for each step
A = zeros(1, n);
A(1) = a;
B = zeros(1, n);
B(1) = b;
C = zeros(1, n);

% Compute the values of A and B iteratively
for k = 2 : n
    A(k) = A(k-1) + 2 * B(k-1);
    B(k) = A(k-1) + B(k-1);
end

% Compute the entire C vector (a/b) in a vectorised manner
C = A ./ B;
% Create a table to display the table like you'd see in excel
TBL = table(A', B', C');
TBL.Properties.VariableNames = {'A' 'B' 'C'};
disp(TBL)

% sqrt(2)
x = C(end);

Here's the function running:

>>> rtsqrt2(1, 1, 10)
 A       B        C   
____    ____    ______

   1       1         1
   3       2       1.5
   7       5       1.4
  17      12    1.4167
  41      29    1.4138
  99      70    1.4143
 239     169    1.4142
 577     408    1.4142
1393     985    1.4142
3363    2378    1.4142


ans =

1.4142