Why does Matlab round this error to 0, even though I know it isn't precisely 0

I am trying to plot Ramanujan's Pi formula and observe how the errors are reducing as the number of iterations increases. However MATLAB simply outputs 0 after the first iteration even when I put the value of digits() to around 300. I know the formula doesn't calculate pi to a 100 decimals after the first iteration, in fact it's an additonal 8 decimal places per iteration after the first, yet it continues to output 0.0 as the value of the function - value of pi calculated to 300 digits. How can I fix this?

This is my code for the formula and the error function (I am using disp() at the end just to test the code).

% Ramanujan's Formula Plot %
rconst = sqrt(8)/9801;
rsum = 0;
rlim = 100;
plt3= 1:(rlim+1);

for n= 0:rlim    
    ctr = 1;
    f4n = factorial(4*n);
    fn = factorial(n);
    
    num = (f4n * (1103+26390*n));
    den = ((fn^4) * 396^(4*n));
    rcalc = (num/den);
    rtemp = (rconst * rcalc);
    rsum = (rsum + rtemp);
    rpi = (1/rsum);
    plt3(n+1) = vpa(rpi);
    ctr = ctr +1;   
end

plot(plt3);

for ii = 1:41
    var = abs(plt3(ii) - vpa(pi,334));
    phi = sym(var);
    disp(vpa(phi,334));
end

Answer 1

[Steven Lord]

All your calculations to compute rpi are being performed in double precision. By the time you call vpa on it it's too late, much like presenting a coupon at the store after the cashier has already taken your payment and given you your change back.

If you want to perform the calculations in arbitrary precision, start off by creating the data on which you want to operate as symbolic variables so you perform symbolic calculations.

rconst = sqrt(sym(8))/9801;
rsum = sym(0);

When you perform operations like / where one input is a double and the other a sym the result is performed symbolically. That's why I didn't need to convert 9801 into a sym; it will be automatically treated as a sym when the division is performed.

Another example later in the code:

ns = sym(n);    
f4n = factorial(4*ns);
fn = factorial(ns);

[John D'Errico]

The other reason why you did not need to convert 9801 to a sym is because MATLAB stores the exact value of an integer when it is no larger than 2^53-1.

[Anirudh Bhalekar]

Oh okay that makes sense! Its working now, thanks a lot!

Answer 2

[Matt J]

You need to pre-allocate as type 'sym',

plt3= sym(1:(rlim+1));

[Anirudh Bhalekar]

It's still giving 0 as the answer to the second iteration.