用变步长梯形方法、变步Simpson方法和Romberg方法计算积分的matlab代码

在MATLAB中，你可以使用`integral`函数结合循环结构来实现变步长梯形法、变步长Simpson法和Romberg积分法。下面是一些基本示例：

1. 变步长梯形法（Trapzoidal Rule）:

function trapezoidal(f, a, b, n)
    h = (b - a) / n;
    trapz = (f(a) + f(b)) * h / 2; % 梯形面积公式

    for i = 1:n-1
        xi = a + i * h;
        trapz += f(xi);
    end
    disp(['Trapezoidal approximation: ', num2str(trapz)]);
end

呼叫方式: `trapezoidal(@(x)x.^2, 0, 1, 10);`

2. 变步长辛普森法则(Simpson's Rule):

function simpson(f, a, b, n)
    h = (b - a) / n;
    if mod(n, 2) == 0
        warning('Odd number of intervals required for Simpson''s rule.');
    end
    simpson_sum = f(a) + f(b);
    
    for i = 1:2:(n-1)
        xi = a + i * h;
        simpson_sum += 4 * f(xi);
    end
    
    for i = 2:2:(n-2)
        xi = a + i * h;
        simpson_sum += 2 * f(xi);
    end
    disp(['Simpson''s approximation: ', num2str(simpson_sum * h / 3)]);
end

呼叫方式: `simpson(@(x)x.^3, 0, 1, 10);`

3. Romberg积分法(Romberg Integration):

function romberg(f, a, b, tol)
    R = zeros(2, 1);
    R(1, 1) = (b - a) * (f(a) + f(b))/2;

    for n = 2:2:100  % 可调整精度等级
        m = floor((n - 1) / 2);
        for j = m:-1:1
            R(n, 1) = R(n, 1) + (R(2*j+1, 1) - R(2*j-1, 1)) / (4^j);
        end
        error = abs(R(n, 1) - R(n-1, 1));
        if error < tol
            break;
        end
    end
    disp(['Romberg approximation: ', num2str(R(n, 1))]);
end

呼叫方式: `romberg(@(x)x.^4, 0, 1, 1e-6);`