I have post once here about this question but i had some problems and i stopped with solving it. Now i started again and i think i`ve made it better. Here is where i need help.
This is the system of nonlinear equations.
f(x) = 0 where:
$$ \left\{ \begin{array}{c} x_1^5+x_2^3+x_3^4+1 \\ x_1^2*x_2* x_3 \\ x_3^4-1 \end{array} \right. $$
The right side of the equation is 0.
A) Find manually all the zeroes of the system.
B) Calculate the Jacobian J(X). (Notice that J(x) is singular for x3 = 0)
C) Check this two starting solutions:
1) X0 = {-0.01, -0.01, -0.01}
2) X0 = {-0.1, -0.1, -0.1}
D) Calculate the determinants |J(X0)| and |J^-1(X0)| for the two starting solutions. Notice that Jacobians are almost singular, altough the starting solutions are not so far from the real solutions.
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This is the task that i need to solve. I started from here.
From this system we can see that $$ X_3 = \pm 1 $$ So going from this we can make 4 solutions.
$$I:x_3=1, x_2=0, x_1=-1$$ $$II:x_3=1, x_2=-1, x_1=0$$ $$III:x_3=-1, x_2=0, x_1=-1$$ $$IV:x_3=-1, x_2=-1, x_1=0$$
After this i started with the Jacobian. I have made this for the Jacobian matrix.
$$ \left[ \begin{array}{ccc} 5x_1^4 & 3x_2^2x_3^4 &4x_3^3x_2^3\\ 2x_1 x_2 x_3 & 1*x_1^2 x_3 & 1*x_1^2 x_2\\ 0 & 0 & 4x_3^3 \end{array} \right] $$
After the jacobian i calculate the determinants for all 4 solutions and I have:
I: the determinant is: 20.
II: the determinant is: 0.
III: the determinant is: 20.
IV: the determinant is: 0.
I stucked on this: When i need to see the two starting solutions C). And when i need to calculate their determinant. The first number that i calculate is 0.00000005 and i said okay its enough i am something wrong.
HINT: eliminating $$x_2,x_3$$ from your System we get for $x_1$ $$2x_1^2+x_1^7=0$$ you can start with: one variable of your System must be Zero. If we have $x_2=0$ then we get$$x_1^5+2=0$$ If we have $$x_1=0$$ then $$x_2^3+2=0$$ from my equation you will get $$x_1=0$$ or $$x_1^5+2=0$$ the whole number of solutions for $(x_1, x_2, x_3)$ are $$ (0, -\sqrt[3]2, -1), \quad (0, -\sqrt[3]2, 1), \quad (-\sqrt[5]2, 0, -1), \quad (-\sqrt[5]2, 0, 1)$$