**Summary:** This paper presents an iterative re-weighted least squares (IRLS) algorithm for sparse recovery, which is used to find sparse solutions to underdetermined linear systems. The algorithm is based on the idea of iteratively minimizing a weighted least squares objective, where the weights are updated based on the current solution. The key idea is to use a sequence of weights that adaptively adjust to the current estimate of the sparse solution, leading to faster convergence. The paper begins by introducing the problem of sparse recovery, where a sparse vector $ x \in \mathbb{R}^N $ is to be recovered from a measurement $ y = \Phi x $, where $ \Phi $ is an $ m \times N $ matrix with $ m < N $. Under certain conditions, specifically the Restricted Isometry Property (RIP), the sparse vector can be uniquely recovered using $ \ell_1 $-minimization, which is a convex relaxation of the non-convex $ \ell_0 $-minimization problem. The IRLS algorithm is introduced as an alternative to $ \ell_1 $-minimization. It works by iteratively solving a weighted least squares problem, where the weights are updated based on the current solution. The weights are chosen such that they emphasize the non-zero entries of the current estimate, leading to a more accurate solution. The algorithm is shown to converge to the sparse solution under the RIP conditions, and it exhibits linear convergence when the solution is sufficiently close to the true sparse vector. The paper also discusses the convergence properties of the IRLS algorithm, showing that it converges to the unique $ \ell_1 $-minimizer under the RIP conditions. Furthermore, it demonstrates that the algorithm can be modified to handle non-convex $ \ell_\tau $-norm minimization for $ 0 < \tau < 1 $, which can lead to superlinear convergence rates. The paper concludes with a detailed analysis of the convergence and rate of convergence of the IRLS algorithm, showing that it can recover sparse solutions efficiently under the RIP conditions. The algorithm is shown to converge to the true sparse solution when it exists, and it exhibits exponential convergence when the solution is sufficiently close to the true sparse vector. The results are supported by theoretical analysis and numerical examples.**Summary:** This paper presents an iterative re-weighted least squares (IRLS) algorithm for sparse recovery, which is used to find sparse solutions to underdetermined linear systems. The algorithm is based on the idea of iteratively minimizing a weighted least squares objective, where the weights are updated based on the current solution. The key idea is to use a sequence of weights that adaptively adjust to the current estimate of the sparse solution, leading to faster convergence. The paper begins by introducing the problem of sparse recovery, where a sparse vector $ x \in \mathbb{R}^N $ is to be recovered from a measurement $ y = \Phi x $, where $ \Phi $ is an $ m \times N $ matrix with $ m < N $. Under certain conditions, specifically the Restricted Isometry Property (RIP), the sparse vector can be uniquely recovered using $ \ell_1 $-minimization, which is a convex relaxation of the non-convex $ \ell_0 $-minimization problem. The IRLS algorithm is introduced as an alternative to $ \ell_1 $-minimization. It works by iteratively solving a weighted least squares problem, where the weights are updated based on the current solution. The weights are chosen such that they emphasize the non-zero entries of the current estimate, leading to a more accurate solution. The algorithm is shown to converge to the sparse solution under the RIP conditions, and it exhibits linear convergence when the solution is sufficiently close to the true sparse vector. The paper also discusses the convergence properties of the IRLS algorithm, showing that it converges to the unique $ \ell_1 $-minimizer under the RIP conditions. Furthermore, it demonstrates that the algorithm can be modified to handle non-convex $ \ell_\tau $-norm minimization for $ 0 < \tau < 1 $, which can lead to superlinear convergence rates. The paper concludes with a detailed analysis of the convergence and rate of convergence of the IRLS algorithm, showing that it can recover sparse solutions efficiently under the RIP conditions. The algorithm is shown to converge to the true sparse solution when it exists, and it exhibits exponential convergence when the solution is sufficiently close to the true sparse vector. The results are supported by theoretical analysis and numerical examples.