Transcription of 4 Iterative Methods for Solving Linear Systems
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4 Iterative Methods for Solving Linear Systems
4 IterativeMethodsforSolvingLinearSystemsI terative Methods formally yield the solution x of a Linear system after aninfinite number of steps. At each step they require the computationof theresidual of the system. In the case of a full matrix, their computationalcost istherefore of the order ofn2operations for each iteration, to be compared withan overall cost of the order of23n3operations needed by direct Methods . Itera-tive Methods can therefore become competitivewith direct Methods providedthe number of iterations that are required toconverge (within a prescribedtolerance) is either independent ofn or scales sublinearly with respect the case of large sparse matrices, as discussed in Section , directmethods may be unconvenient due to the dramatic fill-in, although extremelyefficient direct solverscan be devised on sparse matrices featuringspecialstructures like, for example, those encountered in the approximation ofpartialdifferential equations (see Chapters 12 and 13).
4 Iterative Methods for Solving Linear Systems Iterative methods formally yield the solution x of a linear system after an infinite number of steps.
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