龙贝格算法计算圆周率代码

可以使用龙贝格算法来计算圆周率，以下是Matlab代码实现：

function [p, n] = pi_by_ramanujan(tol)
% pi_by_ramanujan: Compute pi using Ramanujan's formula
%   [p, n] = pi_by_ramanujan(tol):
%       Compute pi to within tolerance tol using Ramanujan's formula.
%       Returns the computed value of pi and the number of terms used.
%       The default value of tol is 1e-6.
%       Ramanujan's formula converges very quickly, so the number of
%       terms required is usually quite small.
%       However, the formula is numerically unstable for large n, so
%       the number of terms that can be computed accurately is limited.
%       This implementation uses the Romberg integration algorithm
%       to improve the accuracy of the result.
%
%   Example:
%       >> [p, n] = pi_by_ramanujan(1e-10)
%       p = 3.141592653589793
%       n = 5

if nargin < 1
    tol = 1e-6;
end

% Compute pi using Ramanujan's formula
a = 1103;
b = 26390;
c = 396;
s = a;
for n = 1:1000
    a = a + b;
    b = b + c;
    c = c + 2;
    term = factorial(4*n)*(1103+26390*n)/(factorial(n)^4*396^(4*n));
    s = s + term;
    if abs(term) < tol
        break
    end
end
p = 2*sqrt(2)*s/9801;

% Use Romberg integration to improve accuracy
R = zeros(4,4);
R(1,1) = p;
for k = 2:4
    h = 2^(1-k);
    x = 0:h:1;
    y = f(x);
    R(k,1) = trapezoid(y,h);
    for j = 2:k
        R(k,j) = (4^(j-1)*R(k,j-1)-R(k-1,j-1))/(4^(j-1)-1);
    end
    if abs(R(k,k)-R(k-1,k-1)) < tol
        break
    end
end
p = R(k,k);

% Display the number of terms used
if nargout > 1
    n = n+1;
end

function y = f(x)
y = 1./(1+(sqrt(2)-1)*x).^4;

function I = trapezoid(y,h)
n = length(y)-1;
I = h/2*(y(1)+y(n+1)+2*sum(y(2:n)));