自适应辛普森法

Posted 2021-03-14 lhm-

自适应辛普森法通过(Simpson)公式，用二次函数来拟合，实现时用二分递归来自动控制区间分割的大小，既保证精度，又保证速度

(Simpson)公式推导

[int_a^bf(x)dx]
[approxint_a^bAx^2+Bx+C]
[=frac{A}{3}(b^3-a^3)+frac{B}{2}(b^2-a^2)+C(b-a)]
[=frac{(b-a)}{6}(2Ab^2+2Aab+2Aa^2+3Bb+3Ba+6C)]
[=frac{(b-a)}{6}(Aa^2+Ba+C+Ab^2+Bb+C+4A(frac{a+b}{2})^2+4B(frac{a+b}{2})+4C)]
[=frac{(b-a)}{6}(f(a)+f(b)+4f(frac{a+b}{2}))]

求(displaystyle{int_l^rfrac{cx+d}{ax+b}mathrm{d}x})

(code:)

double f(double x)
{
    return (c*x+d)/(a*x+b);
}
double simpson(double l,double r)
{
    double mid=(l+r)/2;
    return (r-l)*(f(l)+f(r)+4*f(mid))/6;
}
double sol(double l,double r,double eps)
{
    double mid=(l+r)/2;
    double a=simpson(l,mid),b=simpson(mid,r),m=simpson(l,r);
    if(fabs(a+b-m)<=eps) return a+b+(a+b-m);
    return sol(l,mid,eps/2)+sol(mid,r,eps/2);
}