The computation and communication complexity of a parallel banded system solver
ACM Transactions on Mathematical Software (TOMS)
A bisection method for systems of nonlinear equations
ACM Transactions on Mathematical Software (TOMS)
ACM Transactions on Mathematical Software (TOMS)
Computational geometry: an introduction
Computational geometry: an introduction
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Solving large nonlinear systems of equations by an adaptive condensation process
Numerische Mathematik
Iterative Algorithms for Solution of Large Sparse Systems of Linear Equations on Hypercubes
IEEE Transactions on Computers
Terminal attractors in neural networks
Neural Networks
Introduction to parallel computing: design and analysis of algorithms
Introduction to parallel computing: design and analysis of algorithms
A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems
SIAM Journal on Scientific Computing
On nonlinear generalized conjugate gradient methods
Numerische Mathematik
Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms
ACM Transactions on Mathematical Software (TOMS)
Algorithm-Based Error-Detection Schemes for Iterative Solution of Partial Differential Equations
IEEE Transactions on Computers
Accelerated Inexact Newton Schemes for Large Systems of Nonlinear Equations
SIAM Journal on Scientific Computing
Iterative solution of nonlinear equations in several variables
Iterative solution of nonlinear equations in several variables
Numerical Solution of Nonlinear Equations
ACM Transactions on Mathematical Software (TOMS)
Testing Unconstrained Optimization Software
ACM Transactions on Mathematical Software (TOMS)
Scientific Computing and Differential Equations: An Introduction to Numerical Methods
Scientific Computing and Differential Equations: An Introduction to Numerical Methods
An Overlaying Technique for Solving Linear Equations in Real-Time Computing
IEEE Transactions on Computers
Parallel Minimal Norm Method for Tridiagonal Linear Systems
IEEE Transactions on Computers
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
ACM Transactions on Internet Technology (TOIT)
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We propose solving nonlinear systems of equations by function optimization and we give an optimal algorithm which relies on a special canonical form of gradient descent. The algorithm can be applied under certain assumptions on the function to be optimized, that is, an upper bound must exist for the norm of the Hessian, whereas the norm of the gradient must be lower bounded. Due to its intrinsic structure, the algorithm looks particularly appealing for a parallel implementation. As a particular case, more specific results are given for linear systems. We prove that reaching a solution with a degree of precision $\varepsilon$ takes $\Theta(n^2 k^2 \log {\frac{k}{\varepsilon}})$, $k$ being the condition number of ${{\schmi A}}$ and $n$ the problem dimension. Related results hold for systems of quadratic equations for which an estimation for the requested bounds can be devised. Finally, we report numerical results in order to establish the actual computational burden of the proposed method and to assess its performances with respect to classical algorithms for solving linear and quadratic equations.