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I have a set of 5 non-linear equations:

$w_0 + w_1 + w_2 = 1$,

$w_0v_0 + w_1v_1 + w_2v_2 = a$,

$w_0v_0^2 + w_1v_1^2 + w_2v_2^2 = a^2 + b^2$,

$w_0v_0^3 + w_1v_1^3 + w_2v_2^3 = a^3 + 3ab^2$,

$w_0v_0^4 + w_1v_1^4 + w_2v_2^4 = a^4 + 6a^2b^2 + 3b^4$,

Where $v_0 = a$. I want to express the values of $w_0,w_1,w_2,v_1,v_2 $in terms of $a,b,$ and $c = w_1/w_2,$ where c is a root of a polynomial of c parametrized by a and b. I know I can do this for the case of $v_0 = 0$, but I can't manage to do it it for $v_0 = a$. Could anyone help me figure this out, or more generally to reduce the system to a root finding problem of a polynomial (that is not necessarily a polynomial of $c = w_1/w_2)$. Any tips would also be much appreciated.

saulspatz
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AnthonyPadua
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  • Hello and welcome to math.stackexchange. Note that each equation is linear in the $w_i$. Thus you can solve the first three equations in terms of $a = v_0, b, v_1, v_2$ using e.g. Cramer's Rule and substitute the result into the last two equations. This will give you two polynomial equations in the unknowns $v)1, v_2$. – Hans Engler Dec 12 '20 at 18:25

2 Answers2

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For $a \ne 0$, Mathematica gives the solutions $$ (w_1,w_2,w_2,v_0,v_1,v_2) = (0, \frac{1}{2},\frac{1}{2}, a, a \pm b, a \mp b) \, . $$

Hans Engler
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MATLAB has isolate command which does this kind of operations. You can arrange your equations according to any variable by using this command.

voo
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