matlab解方程组解析解

问题描述

对于一个如下形式的方程组
{ y 1 = f 1 ( x 1 , x 2 , . . . , x n ) y 2 = f 2 ( x 1 , x 2 , . . . , x n ) . . . y m = f m ( x 1 , x 2 , . . . , x n )

⎩⎪⎪⎪⎨⎪⎪⎪⎧​y1​=f1​(x1​,x2​,...,xn​)y2​=f2​(x1​,x2​,...,xn​)...ym​=fm​(x1​,x2​,...,xn​)​
需要求解其反函数，即
{ x 1 = g 1 ( y 1 , y 2 , . . . , y m ) x 2 = g 2 ( y 1 , y 2 , . . . , y m ) . . . x n = g m ( y 1 , y 2 , . . . , y m )

{x1=g1(y1,y2,...,ym)x2=g2(y1,y2,...,ym)...xn=gm(y1,y2,...,ym)" role="presentation" style="position: relative;">⎧⎩⎨⎪⎪⎪⎪⎪⎪x1=g1(y1,y2,...,ym)x2=g2(y1,y2,...,ym)...xn=gm(y1,y2,...,ym)$$\{\begin{array}{l}{x}_1=g_1(y_1,y_2,...,y_m)\\ x_2=g_2(y_1,y_2,...,y_m)\\ ...\\ x_n=g_m(y_1,y_2,...,y_m)\end{array}$$

⎩⎪⎪⎪⎨⎪⎪⎪⎧​x1​=g1​(y1​,y2​,...,ym​)x2​=g2​(y1​,y2​,...,ym​)...xn​=gm​(y1​,y2​,...,ym​)​

matlab求解

可以通过matlab的slove函数可以实现上述功能

syms x1 x2 ....  xn, y1 y2 ... ym
eq1 = y1 == f_1(x_1, x_2, ..., x_n)
eq2 = y2 == f_2(x_1, x_2, ..., x_n)
...
eqm = ym == f_m(x_1, x_2, ..., x_n)
[x1_, x2_, ..., xn_] = solve(eq1, eq2, eq3, ..., eqm, x_1, x_2, ..., x_n)
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具体实例

syms kappa d phi varepsilon_0 varepsilon_a varepsilon_b varepsilon_c
equ1 = varepsilon_a == kappa*d*cos(-phi) + varepsilon_0;
equ2 = varepsilon_b == kappa*d*cos(-phi+2/3*pi) + varepsilon_0;
equ3 = varepsilon_c == kappa*d*cos(-phi+4/3*pi) + varepsilon_0;

result = solve(equ1,equ2,equ3, kappa, phi, varepsilon_0)
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