This paper introduces NESTA, a fast and accurate first-order method for solving sparse recovery problems in signal processing. Inspired by Nesterov's smoothing technique, NESTA improves the convergence properties of standard gradient-descent algorithms through a subtle averaging of sequences of iterates. The algorithm is computationally efficient, accurate, flexible, and robust, making it suitable for large-scale compressed sensing reconstruction problems. It can handle various types of recovery problems, including total-variation minimization and convex programs seeking to minimize the ℓ₁ norm of Wx under constraints. Comprehensive numerical experiments show that NESTA performs well compared to state-of-the-art methods, and it is particularly effective for signals with large dynamic ranges. The algorithm is also extended to solve total-variation problems and is shown to be efficient in this context. NESTA's accuracy is influenced by a smoothing parameter μ, which balances the accuracy of the smoothed approximation and the speed of convergence. The paper also discusses the use of continuation techniques to accelerate NESTA, which significantly improves its performance for high-accuracy recovery tasks. Numerical experiments demonstrate that NESTA achieves high precision, even for signals with extreme dynamic ranges, and that it outperforms other methods in terms of both speed and accuracy.This paper introduces NESTA, a fast and accurate first-order method for solving sparse recovery problems in signal processing. Inspired by Nesterov's smoothing technique, NESTA improves the convergence properties of standard gradient-descent algorithms through a subtle averaging of sequences of iterates. The algorithm is computationally efficient, accurate, flexible, and robust, making it suitable for large-scale compressed sensing reconstruction problems. It can handle various types of recovery problems, including total-variation minimization and convex programs seeking to minimize the ℓ₁ norm of Wx under constraints. Comprehensive numerical experiments show that NESTA performs well compared to state-of-the-art methods, and it is particularly effective for signals with large dynamic ranges. The algorithm is also extended to solve total-variation problems and is shown to be efficient in this context. NESTA's accuracy is influenced by a smoothing parameter μ, which balances the accuracy of the smoothed approximation and the speed of convergence. The paper also discusses the use of continuation techniques to accelerate NESTA, which significantly improves its performance for high-accuracy recovery tasks. Numerical experiments demonstrate that NESTA achieves high precision, even for signals with extreme dynamic ranges, and that it outperforms other methods in terms of both speed and accuracy.