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- W582807726 abstract "The behavior of the conjugate gradient algorithm in exact and finite precision arithmetic is described. Sharp upper bounds are given for the error at each step of an exact arithmetic conjugate gradient solution of a symmetric, positive definite system of linear equations. It is shown that a finite precision computation maintains close agreement with its exact arithmetic counterpart until (and unless) it reaches a step at which some error component(s) have been reduced by a much greater factor than the overall reduction in the norm of the error. Subsequent steps of the finite precision computation may closely resemble the previous steps of an exact arithmetic conjugate gradient calculation from a different initial guess. This resemblance is maintained until other error components are greatly reduced, at which point subsequent steps may resemble still earlier steps of another exact arithmetic calculation. As long as this agreement with various exact arithmetic calculations is maintained, the appropriate exact arithmetic error bounds can be applied to a finite precision computation. The computation may gradually lose agreement with any exact arithmetic calculation. If this occurs, convergence usually becomes slow and it can be guaranteed only that the rate of convergence is at least as fast as that of the steepest descent algorithm." @default.
- W582807726 created "2016-06-24" @default.
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- W582807726 date "1981-01-01" @default.
- W582807726 modified "2023-09-28" @default.
- W582807726 title "Convergence properties of the conjugate gradient algorithm in exact and finite precision arithmetic" @default.
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