CSC_349_HANDOUT#16

CSC_349_HANDOUT#16 - COMPUTER SCIENCE 349A Handout Number...

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COMPUTER SCIENCE 349A Handout Number 16 GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING (Section 9.4) The Naïve Gaussian elimination algorithm will fail if any of the pivots K , , , ) 2 ( 33 ) 1 ( 22 11 a a a is equal to 0. Mathematically speaking, the algorithm only requires that the pivots be nonzero. However, as the algorithm fails when a pivot is exactly equal to 0, it often gives very poor numerical results (using floating-point arithmetic) when a pivot is close to 0. Example Consider the n = 2 linear system with augmented matrix 78 . 46 13 . 6 291 . 5 17 . 59 14 . 59 003 . 0 . The exact solution is = 1 10 x . However, using 4 decimal-digit floating-point rounding arithmetic and Naïve Gaussian elimination, a very inaccurate solution is computed: 6 2 2 ) 1 ( 2 6 2 2 ) 1 ( 22 10 1044 . 0 or 10 1044 )) 10 1044 ( 78 . 46 ( )) 17 . 59 1764 ( 78 . 46 ( 10 1043 . 0 or 10 1043 )) 10 1043 ( 13 . 6 ( )) 14 . 59 1764 ( 13 . 6 ( . 1764 ) 666 . 1763 ( ) 003 . 0 / 291 . 5 ( mult × × = × = × × × = × = × = = l l l l l l K l l f f f b f f f a f f That is, the reduced linear system (in which the coefficient matrix has been reduced to upper triangular form) is × × 2 2 10 1044 10 1043 0 . 0 17
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CSC_349_HANDOUT#16 - COMPUTER SCIENCE 349A Handout Number...

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