Solving general multivariable quadratic equations

Question

Consider the variables $\mathbf{x}\in\mathbb{R}^n$ and the known coefficients $\mathbf{A}_i \in \mathbb{R}^{n\times n}, \mathbf{b}_i \in \mathbb{R}^n,$ and $c_i \in \mathbb{R}$ for $i=1,2,\cdots, n$. They satisfy an $n$-equation system given by: $$ \mathbf{x}^T\mathbf{A}_i \mathbf{x} + \mathbf{b}_i^T \mathbf{x} + c_i = 0, \quad i=1, 2,\cdots,n. $$ Can anyone suggest a general solver that can be used for this type of system? I'm not well-versed in algebra and would appreciate it if you could provide me with some method names that I can use.

Answer 1

$ \def\e{\varepsilon} \def\p{\partial} \def\LR#1{\left(#1\right)} \def\BR#1{{\large\tt[}#1{\large\tt]}} \def\op#1{\operatorname{#1}} \def\trace#1{\op{Tr}\LR{#1}} \def\qiq{\quad\implies\quad} \def\grad#1#2{\frac{\p #1}{\p #2}} \def\c#1{\color{red}{#1}} \def\Sk{\sum_{k=1}^n} $Let $\{\e_k\}$ denote the standard basis vectors and define the following variables $$\eqalign{ A_k^s &= \tfrac12\LR{A_k+A_k^T} \qquad\{\,{\rm symmetric\;part\;of}\:A_k\} \\ f_k &= {x^TA_k^s x \,+\, b_k^Tx \,+\, c_k} \\ y_k &= {2A_k^s x} \\ Y &= \BR{y_1\;y_2\;\cdots\;y_n}^T \;\equiv\; \Sk\e_k\,y_k^T \\ B &= \BR{b_1\;b_2\;\cdots\;b_n}^T \\ }$$ so that $\{b_k,y_k\}$ are the $k^{th}\,$ rows of $\,\{B,Y\}$ respectively.

Calculate the differential of $f_k$ $$\eqalign{ df_k &= \LR{2A_k^s x + b_k}^T dx \;\equiv\; \LR{y_k+b_k}^T dx \\ }$$ Summing with the $\{\e_k\}$ yields a system of non-linear equations (NLEs) and its Jacobian $$\eqalign{ f &= \Sk \e_k\,f_k = 0 \quad &\{ {\rm NLEs} \} \\ df &= \Sk \e_k\,df_k \\&= \LR{Y+B}\cdot dx \\ \grad{f}{x} &= \LR{Y+B} \;=\; J \quad &\{ {\rm Jacobian} \} \\ }$$ Now you can solve the problem using Newton's Method $$\large\eqalign{ x_+ &= x - J^{-1}f \\ }$$