Lecture+20--Partial+pivoting - Operation Count Important...

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Operation CountImportant issue for a large matrix Cramer’s rule:Using expansion by minors: (n+1)! M/D;Using Gauss elimination for Determinants: O(n4/3) M/DGauss elimination routine uses on the order of O(n3/3) operationsBack-substitution uses O(n2/2)n(n+1)!n3/3103628800333202.4329E+182667408.1592E+4721333Cramer
Why there are 2n3/3 Operations?
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Operation CountØNumber of flops (floating-point operations) for Naive Gauss eliminationØComputation time increase rapidly with nØMost of the effort is incurred in the elimination stepØImprove efficiency by reducing the elimination effort%....%.%.859910676106861000000106761000539866666768155010000671550100588766780510070510nEliminatiotoDuePercentage32nFlopsTotalonSbustitutiBacknElimination8683×××
Partial PivotingProblems with Gauss elimination division by zeroround off errorsill conditioned systemsUse “Pivoting” to avoid thisFind the row with largest absolute coefficient below the pivot elementSwitch rows (“partial pivoting”)complete pivoting switch columns also (rarely used)
Round-off ErrorsA lot of rounding with more than n3/3operationsMore important - error is propagatedFor large systems (more than 100 equations), round-off error can be very important (machine dependent)Ill conditioned systems - small changes in coefficients lead to large changes in solutionRound-off errors are especially important for ill-conditioned systems

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