数值方法公式

k1 f (ti , yi ) h 1 k 2 f (ti , yi k1h) 1 1 2 2 yi 1 yi k1 k 2 h 2 2 y y k h i 2 i 1 Ralston’s Method Third-Order Runge-Kutta Method Nystrom Method k f (t , y )
NEWTONPOLYNOMIALS
Lagrange interpolation polynomials
给了 N+1 个点
幂函数拟合
最小二乘曲线及其证明
曲线拟合中的几个误差计算方法
组合梯形公式和辛普森的误差
数值积分
龙贝格积分的理查森改进
组合梯形公式
Example: Heun’s Method h = 0.5
Heun’s Method k1 f (ti , yi )
Midpoint Method
k1 f (ti , yi ) k 2 f (ti yi 1 3 3 h, yi k1h) 4 4 1 2 yi ( k1 k 2 )h 3பைடு நூலகம்3
Classical Third-order Runge-Kutta
(Newton-Raphson Theorem)
(Newton’s Iteration for Finding Square Roots)
NEWTON-RAPHSON
例:泰勒多项式为什么要计算到十五项误差符合要求?
矩 阵 转 换 为
L U
型
Acceleration of Newton-Raphson Iteraon
y1 (0) 2 y2 (0) 4 y (0) 1 3
In Vector Notation y f (t , y )
k1 f (ti , yi ) 1 1 k 2 f (ti h, yi k1h) 3 3 2 2 k3 f (ti h, yi k 2 h) 3 3 1 yi 1 yi ( k1 3k3 ) h 4
y f (t , y, y, y) t 4ty 3 y 5 y y (0) 2, y(0) 4, y(0) 1
2
let y1 y, y 2 y dy/dt, y3 y d 2 y / dt 2
f1 (t , y1 , y2 , y3 ) y2 y1 f 2 (t , y1 , y2 , y3 ) y3 y2 2 y3 f 3 (t , y1 , y2 , y3 ) t 4ty1 3 y2 5 y3
System of First-Order ODE-IVPs
k1 f (ti , yi ) Method 1 1 k 2 f (ti h, yi k1h) 2 2 k3 f (ti h, yi k1h 2k 2 h) 1 yi 1 yi (k1 4k 2 k3 )h 6
k 2 f (ti h, yi k1h)
2 2 k 2 f (ti h, yi k1h) 3 3 2 2 k3 f (ti h, yi k 2 h) 3 3 1 yi 1 yi ( 2k1 3k 2 3k3 ) h 8
1 i i
3 -Order Heun Method
rd
Nearly Optimum Method
k1 f (ti , yi ) 1 1 k 2 f (ti h, yi k1h) 2 2 3 3 k3 f (ti h, yi k 2 h) 4 4 1 yi 1 yi ( 2k1 3k 2 4k3 ) h 9
感谢：许天野、李泽龙、熊振东