A Special Form of the GMRES Algorithm
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Resource Overview
A specialized variant of the GMRES algorithm, designed for solving large-scale sparse matrix equations, with implementation insights on Krylov subspace generation and least-squares optimization.
Detailed Documentation
The GMRES algorithm is a specialized iterative method for solving large-scale sparse matrix equations. It constructs a Krylov subspace—generated from an initial vector and a sequence of matrix-vector products—to approximate the solution. During each iteration, the algorithm expands the Krylov subspace and applies a least-squares minimization to compute the optimal approximate solution. This approach typically outperforms traditional direct methods for large problems due to its memory efficiency and avoidance of matrix factorization. Notably, the GMRES algorithm can also be adapted to solve linearized systems derived from nonlinear equations. Key implementation steps include Arnoldi iteration for orthogonal basis construction and QR decomposition for solving the least-squares problem efficiently.
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