#先上代码后补笔记#

#可以直接复制粘贴使用的MATLAB函数！#

1. 定步长牛顿-柯茨积分公式

function [ integration ] = CompoInt( func, left, right, step, mode )

% 分段积分牛顿-柯茨公式；

% 输入5个参数：被积函数(FUNCTIONHANDLE)'func'，积分上下界'left'/'right'，步长'step'，

% 模式mode共三种('midpoint','trapezoid','simpson')；

% 返回积分值；

switch mode

% 仅仅是公式不同

case 'midpoint'

node = left; integration = 0;

while (node + step <= right) % 按照给定步长开始分段积分

pieceInt = step*(func(node + step/2)); % 使用中点积分公式

integration = integration + pieceInt; node = node + step;

end

if (node < right) % 补齐最后一段积分

pieceInt = (right - node)*(func((node + right)/2));

integration = integration + pieceInt;

end

case 'trapezoid'

node = left; integration = 0;

while (node + step <= right)

pieceInt = step*(func(node) + func(node + step))/2; % 使用梯形公式

integration = integration + pieceInt; node = node + step;

end

if (node < right)

pieceInt = (right - node)*(func(node) + func(right))/2;

integration = integration + pieceInt;

end

case 'simpson'

node = left; integration = 0;

while (node + step <= right)

pieceInt = step*(func(node) + 4*func(node + step/2) + func(node + step))/6; % 使用辛普森公式

integration = integration + pieceInt; node = node + step;

end

if (node < right)

pieceInt = (right - node)*(func(node) + 4*func((node + right)/2) + func(right))/6;

integration = integration + pieceInt;

end

end

2. 自适应分段积分方法

通过函数内调用自身的方法使得积分函数显得简洁明快。因为需要调用自身计算原积分的一段，必须传入参数length标识原积分上下限总长，通过left, right和length三个参数才能够得到某一段的要求精度。

function [ integration ] = AdaptInt( func, left, right, prec, length )

% 自适应分段积分函数AdaptInt；

% 输入五个参数：被积函数(句柄)func，积分上下限right，left，要求精度prec，积分总长length；

% 输出一个参数：积分值integration；

trapeInt = (right - left)*(func(left) + func(right))/2;

midInt = (right - left)*func((left + right)/2);

err = (trapeInt - midInt)/3; % 由中点公式和梯形公式差估算误差

if (abs(err) < prec && (right - left) < length/5) % 如果误差符合要求，则使用辛普森公式计算较精确结果

integration = (right - left)*(func(left) + 4*func((left + right)/2) + func(right))/6;

else % 否则，二分该段，分别进行自适应分段积分

integration = AdaptInt(func, left, (left + right)/2, prec/2, length) + AdaptInt(func, (left + right)/2, right, prec/2, length);

end

end