Add to ORKGLet $A \in \mathbb{R}^{n \times n}$ be invertible, $x \in \mathbb{R}^n$ unknown and $b =Ax $ given. We are interested in approximate solutions: vectors $y \in \mathbb{R}^n$ such that $\|Ay - b\|$ is small. We prove that for all $0< \varepsilon <1 $ there is a composition of $k$ orthogonal projections onto the $n$ hyperplanes generated by the rows of $A$, where $$k \leq 2 \log\left(\frac{1}{\varepsilon} \right) \frac{ n}{ \varepsilon^{2}}$$ which maps the origin to a vector $y\in \mathbb{R}^n$ satisfying $\| A y - Ax\| \leq \varepsilon \cdot \|A\| \cdot \| x\|$. We note that this upper bound on $k$ is independent of the matrix $A$. This procedure is stable in the sense that $\|y\| \leq 2\|x\|$. The existence proof is based on a probabilistic...
Let A be a real m by n matrix, and b a real m-vector. Consider estimating x from an orthogonally inv...
We consider the problem of approximate solution ex of a linear system Ax = b over the reals, such th...
.The authors consider the numerical solution of Ax=f, where A is a bounded invertible linear operato...
Let A0x=b0 be a consistent (but possibly unknown) linear algebraic system of m equations in n unknow...
AbstractFor 1 ⩽ p ⩽ ∞, the lp-approximate solutions of Ax = b are the minimizers of ‖Ax − b‖p, where...
summary:For a large system of linear algebraic equations $A_x=b$, the approximate solution $x_k$ is ...
We consider inexact linear equations y ≈ Φα where y is a given vector in R n, Φ is a given n by m ma...
Given two n×n matrices A and A0 and a sequence of subspaces {0}=V0 ⊂ · · · ⊂ Vn = Rn with dim(Vk) = ...
AbstractLet A0x=b0 be a consistent (but possibly unknown) linear algebraic system of m equations in ...
AbstractLet the linear system Ax=b be rectangular but solvable. If A is a large sparse matrix, then ...
AbstractConsider the system, of linear equations Ax = b where A is an n × n real symmetric, positive...
The information-based study of the optimal solution of large linear systems is initiated by studying...
AbstractOur randomized preprocessing enables pivoting-free and orthogonalization-free solution of ho...
By combining our weakly randomized preconditioning with aggrega-tion and other known and novel techn...
The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b....
Let A be a real m by n matrix, and b a real m-vector. Consider estimating x from an orthogonally inv...
We consider the problem of approximate solution ex of a linear system Ax = b over the reals, such th...
.The authors consider the numerical solution of Ax=f, where A is a bounded invertible linear operato...
Let A0x=b0 be a consistent (but possibly unknown) linear algebraic system of m equations in n unknow...
AbstractFor 1 ⩽ p ⩽ ∞, the lp-approximate solutions of Ax = b are the minimizers of ‖Ax − b‖p, where...
summary:For a large system of linear algebraic equations $A_x=b$, the approximate solution $x_k$ is ...
We consider inexact linear equations y ≈ Φα where y is a given vector in R n, Φ is a given n by m ma...
Given two n×n matrices A and A0 and a sequence of subspaces {0}=V0 ⊂ · · · ⊂ Vn = Rn with dim(Vk) = ...
AbstractLet A0x=b0 be a consistent (but possibly unknown) linear algebraic system of m equations in ...
AbstractLet the linear system Ax=b be rectangular but solvable. If A is a large sparse matrix, then ...
AbstractConsider the system, of linear equations Ax = b where A is an n × n real symmetric, positive...
The information-based study of the optimal solution of large linear systems is initiated by studying...
AbstractOur randomized preprocessing enables pivoting-free and orthogonalization-free solution of ho...
By combining our weakly randomized preconditioning with aggrega-tion and other known and novel techn...
The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b....
Let A be a real m by n matrix, and b a real m-vector. Consider estimating x from an orthogonally inv...
We consider the problem of approximate solution ex of a linear system Ax = b over the reals, such th...
.The authors consider the numerical solution of Ax=f, where A is a bounded invertible linear operato...