﻿功能:一阶常微分方程组初值问题求解(2级3阶隐式龙格库塔法)

格式:
[y1,y2,...,yn,error]=ODESolveRK23(f,fy,t,y0,t0,Error0,n)
[y1,y2,...,yn,error]=ODESolveRK23(f,fy,t,y0,t0,Error0)
[y1,y2,...,yn,error]=ODESolveRK23(f,fy,t,y0,t0)
[y1,y2,...,yn,error]=ODESolveRK23(f,fy,t,y0)
f:用符号变量存储的每个微分表达式,每个表达式以逗号分隔。注意默认的自变量名称为"t"
fy:符号变量存储的对应f当中每个表达式的变量名称
t:为矩阵变量或者数值,表示要求的此时刻的值
y0:为矩阵变量或者数值,表示边界条件值,这个y0对应fy当中的个数
t0:数值,对应t0时的y0,此值默认为0
Error0:变步长控制的相对误差,默认1E-12
n:步长等分最大深度,默认为20

y1,y2,...,yn:对应fy里的变量,表示返回值
error:返回的残差平方和

参考:现代应用数学手册.计算与数值分析卷[M].清华大学出版社,2007:583 

例子:

/*
已知:
y1'=3*y1+2*y2
y2'=4*y1+y2
y1(0)=0
y2(0)=1
求:t=0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1时刻的y1,y2的值
*/

f="3*y1+2*y2,4*y1+y2";
y="y1,y2";
y0=[0 1];
t0=0;
t=0.1:0.1:1;
[y1,y2,er]=ODESolveRK23(f,y,t,y0,t0)//回车后得到如下的结果
y1 =
[ 0.24796117467927    0.63318360161715    1.24695682455792    2.23957856190591    3.85865433515430    6.51224169220907    10.8729553679802    18.0496069100131    29.8701871472795    49.3484264038256 ]
y2 =
[ 1.15279859415378    1.45191435534588    1.98777504573645    2.90989860820792    4.46518499498459    7.06105332834911    11.3695406718377    18.4989358741527    30.2767568070271    49.7163058450062 ]
er =
[ 2.1250557930E-13    1.4509275973E-13    2.8127710478E-13    2.6884175426E-13    1.7437979066E-13    8.8508480271E-14    6.0668805424E-13    2.2973452187E-13    7.9139859899E-14    4.0478995232E-13 ]