We develop novel theory and algorithms for computing approximate solution to $Ax=b$, or to $A^TAx=A^Tb$, where $A$ is an $m \times n$ real matrix of arbitrary rank. First, we describe the {\it Triangle Algorithm} (TA), where given an ellipsoid $E_{A,\rho}=\{Ax: \Vert x \Vert \leq \rho\}$, in each iteration it either computes successively improving approximation $b_k=Ax_k \in E_{A,\rho}$, or proves $b \not \in E_{A, \rho}$. We then extend TA for computing an approximate solution or minimum-norm solution. Next, we develop a dynamic version of TA, the {\it Centering Triangle Algorithm} (CTA), generating residuals $r_k=b - Ax_k$ via iterations of the simple formula, $F_1(r)=r-(r^THr/r^TH^2r)Hr$, where $H=A$ when $A$ is symmetric PSD, otherwise $H=AA^T$ but need not be computed explicitly. More generally, CTA extends to a family of iteration function, $F_t( r)$, $t=1, \dots, m$ satisfying: On the one hand, given $t \leq m$ and $r_0=b-Ax_0$, where $x_0=A^Tw_0$ with $w_0 \in \mathbb{R}^m$ arbitrary, for all $k \geq 1$, $r_k=F_t(r_{k-1})=b-Ax_k$ and $A^Tr_k$ converges to zero. Algorithmically, if $H$ is invertible with condition number $\kappa$, in $k=O( (\kappa/t) \ln \varepsilon^{-1})$ iterations $\Vert r_k \Vert \leq \varepsilon$. If $H$ is singular with $\kappa^+$ the ratio of its largest to smallest positive eigenvalues, in $k =O(\kappa^+/t\varepsilon)$ iterations either $\Vert r_k \Vert \leq \varepsilon$ or $\Vert A^T r_k\Vert= O(\sqrt{\varepsilon})$. If $N$ is the number of nonzero entries of $A$, each iteration take $O(Nt+t^3)$ operations. On the other hand, given $r_0=b-Ax_0$, suppose its minimal polynomial with respect to $H$ has degree $s$. Then $Ax=b$ is solvable if and only if $F_{s}(r_0)=0$. Moreover, exclusively $A^TAx=A^Tb$ is solvable, if and only if $F_{s}(r_0) \not= 0$ but $A^T F_s(r_0)=0$. Additionally, $\{F_t(r_0)\}_{t=1}^s$ is computable in $O(Ns+s^3)$ operations.

翻译：我们开发了新颖的理论和算法，用于计算$Ax=b$或$A^TAx=A^Tb$的近似解，其中$A$是任意秩的$m \times n$实矩阵。首先，我们描述了**三角形算法**（TA），给定椭球$E_{A,\rho}=\{Ax: \Vert x \Vert \leq \rho\}$，在每次迭代中，它要么计算逐步改进的逼近$b_k=Ax_k \in E_{A,\rho}$，要么证明$b \not \in E_{A, \rho}$。然后我们将TA扩展到计算近似解或最小范数解。接下来，我们开发了TA的动态版本——**中心三角形算法**（CTA），通过简单公式$F_1(r)=r-(r^THr/r^TH^2r)Hr$的迭代生成残差$r_k=b - Ax_k$，其中当$A$为对称半正定矩阵时$H=A$，否则$H=AA^T$但无需显式计算。更一般地，CTA扩展为迭代函数族$F_t( r)$（$t=1, \dots, m$），满足：一方面，给定$t \leq m$和$r_0=b-Ax_0$（其中$x_0=A^Tw_0$且$w_0 \in \mathbb{R}^m$任意），对所有$k \geq 1$，有$r_k=F_t(r_{k-1})=b-Ax_k$且$A^Tr_k$收敛于零。算法上，若$H$可逆且条件数为$\kappa$，则经过$k=O( (\kappa/t) \ln \varepsilon^{-1})$次迭代，有$\Vert r_k \Vert \leq \varepsilon$。若$H$奇异且$\kappa^+$为其最大与最小正特征值之比，则经过$k =O(\kappa^+/t\varepsilon)$次迭代，要么$\Vert r_k \Vert \leq \varepsilon$，要么$\Vert A^T r_k\Vert= O(\sqrt{\varepsilon})$。若$N$为$A$的非零元素个数，则每次迭代需$O(Nt+t^3)$次运算。另一方面，给定$r_0=b-Ax_0$，假设其关于$H$的最小多项式次数为$s$。则$Ax=b$可解当且仅当$F_{s}(r_0)=0$。此外，$A^TAx=A^Tb$唯一可解当且仅当$F_{s}(r_0) \not= 0$但$A^T F_s(r_0)=0$。另外，$\{F_t(r_0)\}_{t=1}^s$可在$O(Ns+s^3)$次运算内计算。