System of quadratic algebraic equations

I have this problem

$H_i(x_1,x_2,\dots, x_N) = a_{ijk} x_j x_k + b_{ij} x_j + c_i = 0 \quad 1\leq i \leq N$

And I need to show that applying Newton-Raphson can fail to find even one real solution of this system for any N.

Well if I apply Newton-Raphon, first I calculate the gradient:

$\frac{\partial H_i}{\partial x_l} = a_{ilk} x_k + a_{ijl} x_j + b_{il} = 0$

and solve for

$\boldsymbol{\Delta x} = -( \frac{\partial \boldsymbol{H}}{\partial \boldsymbol{x}})^{-1} \boldsymbol{H}$

Evaluating the gradient and the function at the initial estimation. The gradient is a second-order tensor. I do not know how this could fail. Maybe if the initial estimation did not converge, but that's a matter of choosing a good one. I assume the system has a solution, since the question focuses on finding the solutions, not on whether these exist or not. Thanks in advance.

• Is this question a homework problem? – Geoff Oxberry Jan 10 '14 at 1:52

This is a generalization of the scalar case $$f(x) = a x^2 + b x + c = 0$$. The derivative $$f'(x) = 2 a x + b$$ may be zero at a solution, which happens if and only if $$b^2 = 4 a c$$ (in which case the graph of $$f$$ only touches the $$x$$-axis but does not cross it).