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Iterative methods for nearly singular linear systems
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DOI：10.1137/S106482759834634X.png)
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Abstract
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Iterative methods are developed and studied for near-singular linear systems Cx = b. Our approach, called the transformed minimal residual algorithm (TMRES), is derived from any convergent iterative scheme Sx(k+1) = Tx(k) + b associated with a splitting C = S-T. In each step of TMRES, the transformed residual S-1 (b-Cx) is minimized over a Krylov space generated by S-1T. The original iterative scheme typically converges slowly when C is nearly singular, while a Krylov space generated by S-1T often contains a much better approximation to a solution. TMRES is algebraically equivalent to the generalized minimal residual algorithm (GMRES) preconditioned by S-1, although there are numerical differences since a different matrix S-1C is used to generate the Krylov space in preconditioned GMRES. Special attention is given to sparsity and convergence issues related to linear systems of the form (AA(T) +sigma I) x = b, where sigma greater than or equal to 0.
Keywords:
singular linear system
ill-conditioned system
Krylov space
matrix splitting
preconditioning
generalized minimal residual
successive overrelaxation
Gauss Seidel
conjugate gradients
linear programming
sparse matrices
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