Data commit

Task/Pi/MATLAB/pi.m

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+function pi_str = piSpigot(N)
+% Return N digits of pi using Gibbons's first spigot algorithm.
+% If N is omitted, the digits are printed ad infinitum.
+% Uses the expansion
+%   pi = sum_{i=0} (i!)^2 2^{i+1} /(2i+1)!
+%      = 2 + 1/3 * ( 2 + 2/5 * (2 + 3/7 * ( 2 + 4/9 * ( ..... )))))
+%      = (2 + 1/3 *)(2 + 2/5 *)(2 + 3/7 *)...
+% where the terms in the last expression represent Linear Fractional
+% Transforms (LFTs).
+%
+% Requires the Variable Precision Integer (vpi) Toolbox
+%
+% Reference:
+% "Unbounded Spigot Algorithms for the Digits of Pi" by J. Gibbons, 2004
+% American Mathematical Monthly, vol. 113.
+if nargin < 1
+    N = Inf;
+    lineLength = 50;
+else
+    pi_str = repmat(' ',1,N);
+end
+
+q = vpi(1);
+r = vpi(0);
+t = vpi(1);
+k = 1; % If printing more than 3E15 digits, use k = vpi(1);
+
+i = 1;
+first_digit = true;
+while i <= N
+    threeQplusR = 3*q + r;
+    n = double(threeQplusR / t);
+    if q+threeQplusR < (n+1)*t
+        d = num2str(n);
+        if isinf(N)
+            fprintf(1,'%s', d);
+            if first_digit
+                fprintf(1,'.');
+                first_digit = false;
+                i = i+1;
+            end
+            if i == lineLength
+                fprintf(1,'\n');
+                i = 0;
+            end
+        else
+            pi_str(i) = d;
+        end
+        q = 10*q;
+        r = 10*(r-n*t);
+        i = i + 1;
+    else
+        t = (2*k+1)*t;
+        r = (4*k+2)*q + (2*k+1)*r;
+        q = k*q;
+        k = k + 1;
+    end
+end
+end