Lecture+20--Partial+pivoting

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

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Unformatted text preview: Operation Count ❚ Important 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/D ❚ Gauss elimination routine uses on the order of O ( n3/3 ) operations ❚ Back-substitution uses O ( n2/2 ) n (n+1)! n3/3 10 3628800 333 20 2.4329E+18 2667 40 8.1592E+47 21333 Cramer Why there are 2 n 3 /3 Operations? Total operation counts for elimination stage = 2 n 3 /3 + O ( n2 ) Total operation counts for back substitution stage = n 2 + O ( n ) i(i-1) = (n3 –n )/3 2nd row 3rd row nth row 2nd to nth col . + Outer Loop Inner Loop / * & /- 1 n i = ∑ k i flops flops 1 2 ( 1) ( 1)( 1) n n n n n →-- + 2 3 ( 2)( 1) ( 2)( ) n n n n n →--- 1 ( )( 1) ( )( 2) k k n n k n k n k n k + →-- +-- + 1 (1)(2) (1)(3) n n n- → Operation Count Ø Number of flops (floating-point operations) for Naive Gauss elimination Ø Computation time increase rapidly with n Ø Most of the effort is incurred in the elimination step Ø Improve efficiency by reducing the elimination effort % . . . . % . % . 85 99 10 67 6 10 68 6 1000000 10 67 6 1000 53 98 666667 681550 10000 671550 100 58 87 667 805 100 705 10 n Eliminatio to Due Percentage 3 2n Flops Total on Sbustituti Back n Eliminatio n 8 6 8 3 × × × Partial Pivoting Problems with Gauss elimination ❚ division by zero ❚ round off errors ❚ ill conditioned systems Use “Pivoting” to avoid this ❚ Find the row with largest absolute coefficient below the pivot element ❚ Switch rows (“partial pivoting”) ❚ complete pivoting switch columns also (rarely used) Round-off Errors ❚ A lot of rounding with more than n3/3 operations ❚ More important - error is propagated ❚ For 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 solution ❚ Round-off errors are especially important for ill- conditioned systems Ill-conditioned System 2x1 – x2 = 3 2.1x1 – x2 = 3 ❚...
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Lecture+20--Partial+pivoting - Operation Count ❚...

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