﻿功能:一阶常微分方程组初值问题求解(3级6阶隐式龙格库塔法)

格式:

[y1,y2,...,yn,error]=ODESolveRK36(f,fy,t,y0,t0,Error0,n)
[y1,y2,...,yn,error]=ODESolveRK36(f,fy,t,y0,t0,Error0)
[y1,y2,...,yn,error]=ODESolveRK36(f,fy,t,y0,t0)
[y1,y2,...,yn,error]=ODESolveRK36(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]=ODESolveRK36(f,y,t,y0,t0)//回车后得到如下的结果
y1 =
[ 0.24796120374788    0.63318362555231    1.24695685782497    2.23957859439573    3.85865436130531    6.51224155626004    10.8729554167354    18.0496069400114    29.8701870187642    49.3484264436426 ]
y2 =
[ 1.15279862284044    1.45191437910827    1.98777507887159    2.90989864062703    4.46518502110440    7.06105319248924    11.3695407205754    18.4989359041449    30.2767566785260    49.7163058848203 ]
er =
[ 1.1473849040E-13    7.8309525313E-14    1.5179614166E-13    1.4507102656E-13    9.4091025885E-14    7.6373836234E-13    3.2734212194E-13    1.2395007664E-13    6.8305235359E-13    2.1839578556E-13 ]