N. I. M. Gould. On growth in Gaussian elimination with complete pivoting. SIAM J. Matrix Anal. Appl., 12(2):354--361, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Recall that the growth factor is defined by ae n = max i;j;k ja ij j max i;j ja ij j where a ij (k = 1: n) are the intermediate elements occurring during the elimination on A 2 IR . The long standing conjecture that ae n n for complete pivoting is now known to be false. Gould [70] found a counterexample in floating point arithmetic and Edelman modified it to create a counterexample in exact arithmetic [57] 58] By how much ae n can exceed n for complete pivoting is not known. Examples that can occur in practical applications where partial pivoting yields exponentially ....

N. I. M. Gould. On growth in Gaussian elimination with complete pivoting. SIAM J. Matrix Anal. Appl., 12(2):354--361, 1991.

.... about the results of Gaussian Elimination [36] Many of these revolve around its average and worst case roundoff error behavior, which has resisted analysis for fifty years [37] The worst case error growth for Gaussian Elimination and complete pivoting has only recently been determined for n 16 [15], and until [7] in 1988 was determined only for n 4. Because the roundoff errors are bounded by the sizes of intermediate results in b A [10] we can begin to solve these mysteries if we can say explicitly what Gaussian Elimination produces. 2 Results of Gaussian Elimination as Ratios of ....

....arithmetic is 2 fi when floating point operations are rounded, and fi when they are truncated. The bound implicitly assumes fi 1=n. This growth factor is usually estimated informally, because how it can be bounded is still a famous open question; see [7, 23, 37] and the recent work [15]. Growth factors often appear along with condition numbers of matrices in error bounds for linear systems [14] Recently there has been a good deal of research on growth factors, stressing that concern may be warranted for error growth in real problems when Gaussian Elimination with partial ....

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N. Gould, "On Growth in Gaussian Elimination with Complete Pivoting", SIAM J. Matrix Anal. Appl. 12:2, 354--361, April 1991.

....that examples where g(n) grows exponentially with n may arise in applications, e.g. for linear systems arising from boundary value problems. Even for complete pivoting, it has not been proved that g(n) is bounded by a polynomial in n. Wilkinson [79] showed that g(n) n (log n) 4 O(1) and Gould [39] showed that g(n) n is possible for n 12; there is still a large gap between these results. Thus, to be sure that Gaussian elimination satisfies Definition 2.1, we must restrict A to the class of matrices for which g is On (1) In practice this is not a problem, because g can easily be checked ....

N. Gould, On growth in Gaussian elimination with complete pivoting, SIAM J. Matrix Anal. Appl. 12 (1991), 354--361.

....and alternating directions performed much better than the textbooks might lead one to expect. Parallel direct search methods, such as those in [15] seem particularly attractive for tackling difficult problems such as maximizing the growth factor for Gaussian elimination with complete pivoting [23]. Acknowledgements. I thank Des Higham and Nick Trefethen for their many helpful comments on this work. ....

N. I. M. Gould, On growth in Gaussian elimination with complete pivoting, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 354--361.

.... for the backward and forward errors (see [11] for a complete description) Also the growth factor has been extensively analyzed, since it is considered the only term that may increase the upper bound (1) Upper and lower bounds for ae W n have been established for certain class of matrices in [3, 4, 7, 8, 11, 15], and the interest about the behavior of the LU analyzed in finite precision is still alive. Following the same approach used in [1] for the cyclic reduction algorithm, in this paper we present a new backward error analysis for the LU factorization. The obtained upper bound depends on a quadratic ....

N. Gould, On Growth in Gaussian Elimination with Complete Pivoting, SIAM J. Matrix Anal. Appl. 12 (1991), 354-361.

.... a jp = a jp =a ip and a jq = a jq Gamma a iq a jp =a ip , and check if the following stability criteria is satisfied ja ip j max 1kn ja kp j and j a jq j max 1kn ja kq j; 14) with the same as in (11) The conditions (11) and (14) are supposed to prevent large growth factor [21, 25] when solving equations with A 0 or its transpose. The larger (the closer to one) the more stable the equations with A 0 become. Let us observe that conditions (11) and (14) can easily be verified during the reordering phase and that they add very little computational effort. They introduce a ....

Gould N.I.M. (1991) On growth in Gaussian Elimination with complete pivoting, SIAM Journal on Matrix Analysis and Applications 12, No 2, pp. 354-361.

.... about the results of Gaussian Elimination [36] Many of these revolve around its average and worst case roundoff error behavior, which has resisted analysis for fifty years [37] The worst case error growth for Gaussian Elimination and complete pivoting has only recently been determined for n 16 [15], and until [7] in 1988 was determined only for n 4. Because the roundoff errors are bounded by the sizes of intermediate results in b A (k) 10] we can begin to solve these mysteries if we can say explicitly what Gaussian Elimination produces. Copyright c fl1994, 1995 D. Stott Parker 3 2 ....

....rounded, and fi 1 Gammat when they are truncated. The bound implicitly assumes fi 1 Gammat 1=n. Copyright c fl1994, 1995 D. Stott Parker 11 This growth factor is usually estimated informally, because how it can be bounded is still a famous open question; see [7, 23, 37] and the recent work [15]. Growth factors often appear along with condition numbers of matrices in error bounds for linear systems [14] Recently there has been a good deal of research on growth factors, stressing that concern may be warranted for error growth in real problems when Gaussian Elimination with partial ....

[Article contains additional citation context not shown here]

N. Gould, "On Growth in Gaussian Elimination with Complete Pivoting", SIAM J. Matrix Anal. Appl. 12:2, 354--361, April 1991.

....examples where g(n) grows exponentially with n may arise in applications, e.g. for linear systems arising from boundary value problems. Even for complete pivoting, it has not been proved that g(n) is bounded by a polynomial in n. Wilkinson [82] showed that g(n) n (log n) 4 O(1) and Gould [38] recently showed that g(n) n is possible for n 12; there is still a large gap between these results. Thus, to be sure that Gaussian elimination satisfies Definition 1, we must restrict A to the class of matrices for which g is O n (1) In practice this is not a problem, because g can easily ....

N. Gould, "On growth in Gaussian elimination with complete pivoting", SIAM J. Matrix Anal. Appl. 12 (1991), 354--361.

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N.Gould, On growth in Gaussian elimination with complete pivoting, SIAM J. Matrix Anal. Appl., 12 (1991), 354--361.

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N. Gould, "On growth in Gaussian elimination with complete pivoting", SIMAX 12 (1991), 354--361.

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