写一个高斯求积公式计算0到1上sinx/x

要使用高斯求积公式（也称为辛普森法则）来近似积分 `∫(0,1) sin(x)/x dx`，我们需要先定义函数、确定积分区间和选择适当的阶数。辛普森法则适用于二次多项式插值，因此我们通常用于三个节点（这里是等间距的节点）。以下是实现这一计算的Matlab代码：

function simpsonIntegral = gaussQuadratureSinXOverX(a, b, n)
% a and b are the limits of integration (0 and 1 in this case)
% n is the number of intervals for Simpson's rule

if n ~= round(n) || n < 2
    error('Number of intervals must be an integer greater than or equal to 2.');
end

% Define the function to integrate
f = @(x) sin(x)./x;

% Set up the interval and nodes for Simpson's rule
dx = (b - a) / (2 * n);
x = a + dx * (0:2:n - 1);

% Calculate the integral using Simpson's rule
integral = (dx/3) * sum(f(x(1:end-2)) + 4*f(x(2:end-1)) + f(x(end)));

simpsonIntegral = integral;
end

% Call the function with the desired parameters
a = 0;            % Lower limit of integration
b = 1;            % Upper limit of integration
n_intervals = 8;  % Number of intervals for Simpson's rule

integral_approximation = gaussQuadratureSinXOverX(a, b, n_intervals);
disp(['Approximate integral value: ' num2str(integral_approximation)])

运行这段代码后，它将输出 `∫(0,1) sin(x)/x dx` 的辛普森法近似值。