Nicholas J. Higham and Desmond J. Higham. Large growth factors in Gaussian elimination with pivoting. SIAM J. Matrix Anal. Appl., 10(2): 155--164, April 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....matrix elements are moved to the diagonal for use in elimination. Pivoting is a necessity in practice. Although partial pivoting is unstable in theory, fifty years worth of experience has suggested that it is usually stable in practice [7] though very recently this has come into question again [10, 23, 30]) However, even partial pivoting interrupts the algorithm s data flow and introduces data movement that can slow high performance computers. This research partially supported by NSF grant IRI 8917907. Pivoting introduces complexity into the otherwise straightforward nested loop Gaussian ....

....the bound ij j 2 k Gamma1 max i;j j a ij j holds, and is actually attained by certain matrices. This bound is a very poor guarantee of accuracy for moderate or large n. Recently several naturally arising problems have been identified for which partial pivoting does give poor numerical results [10, 30]. 2.3 Block Gaussian Elimination Many matrix operations can be expressed in block form, in terms of submatrices [5, 7] For example, the following block Gaussian elimination step can be used recursively on square submatrices of a nonsingular square matrix, assuming that the first block A is ....

N.J. Higham, D.J. Higham, "Large Growth Factors in Gaussian Elimination with Pivoting", SIAM J. Matrix Anal. Appl. 10:3, 155-164, April 1989.

....with t digit, base fi floating point 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 ....

....factors, stressing that concern may be warranted for error growth in real problems when Gaussian Elimination with partial pivoting is used. For example, Wright [42] finds exponential error growth for a specific family of matrices arising in two point boundary value problems. Both [14, p. 122] and [23] present many other recent results on growth factors. Von Neumann and Goldstine noted that when A is symmetric positive definite, a ii a ii and [26, p.1066] b a ii b a ii for 1 k i n, and putting this together with the fact that max i a ii = max i;j ja ij j when A is symmetric ....

N.J. Higham, D.J. Higham, "Large Growth Factors in Gaussian Elimination with Pivoting", SIAM J. Matrix Anal. Appl. 10:3, 155--164, April 1989.

.... 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 ....

....j n X k=j 1 ja (j Gamma1) j;k j (21) and since the matrix is an upper Hessenberg matrix then one of the sum in the right hand side of (21) involves elements of the matrix A. 2 For what concerns the complete pivoting, most of the results about the growth factor are reported by Higham [8], which gives a lower bound for ae W;c n and evaluates it for Hadamard and orthogonal matrices. We report in Fig. 2 the computed growth factors ae W;c n (dashed line) and ae c n (dash dotted line) for the Hadamard matrices H 1 = 1 1 Gamma1 1 ; H j 1 = H j H j GammaH j H j ....

N. J. Higham, D. J. Higham, Large Growth Factors in Gaussian Elimination with Pivoting, SIAM J. Matrix Anal. Appl. 10 (1989), 155-164.

....matrix elements are moved to the diagonal for use in elimination. Pivoting is a necessity in practice. Although partial pivoting is unstable in theory, fifty years worth of experience has suggested that it is usually stable in practice [7] though very recently this has come into question again [10, 23, 30]) However, even partial pivoting interrupts the algorithm s data flow and introduces data movement that can slow high performance computers. This research partially supported by NSF grant IRI 8917907. Copyright c fl1994, 1995 D. Stott Parker, Dinh Le 2 Pivoting introduces complexity into the ....

....j b a (k) ij j 2 k Gamma1 max i;j j a ij j holds, and is actually attained by certain matrices. This bound is a very poor guarantee of accuracy for moderate or large n. Recently several naturally arising problems have been identified for which partial pivoting does give poor numerical results [10, 30]. 2.3 Block Gaussian Elimination Many matrix operations can be expressed in block form, in terms of submatrices [5, 7] For example, the following block Gaussian elimination step can be used recursively on square submatrices of a nonsingular square matrix, assuming that the first block A is ....

N.J. Higham, D.J. Higham, "Large Growth Factors in Gaussian Elimination with Pivoting", SIAM J. Matrix Anal. Appl. 10:3, 155-164, April 1989.

.... grows exponentially if partial pivoting is used, but in practice ae oe is usually comparable to the modest growth that results when complete pivoting is used (which is approximately 10 in practice) 17, page 69] Further discussion and new results regarding the growth factor ae oe can be found in [21] and [26] where kE Delta b Xk 1 Phi 8k (oe) k 3 Delta ae oe Delta O( 2 ) Psi Delta k b Xk 1 : A similar analysis can be done for solving (17) to obtain b Y. But b X yields the first column of b S (oe) z) with residual error E Delta b X and b Y yields the remaining ....

N. J. Higham and D. J. Higham, Large growth factors in Gaussian elimination with pivoting, SIAM Journal on Matrix Analysis and Applications, 10 (1989), pp. 155--164.

.... 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 ....

Higham N.J. and D.J. Higham (1989) Large growth factors in Gaussian Elimination with pivoting, SIAM Journal on Matrix Analysis and Applications 10, No 2, pp. 155-164.

....floating point operations are 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 ....

....factors, stressing that concern may be warranted for error growth in real problems when Gaussian Elimination with partial pivoting is used. For example, Wright [42] finds exponential error growth for a specific family of matrices arising in two point boundary value problems. Both [14, p. 122] and [23] present many other recent results on growth factors. Von Neumann and Goldstine noted that when A is symmetric positive definite, a (k 1) ii a (k) ii and [26, p.1066] b a (k 1) ii b a (k) ii for 1 k i n, and putting this together with the fact that max i a ii = max i;j ja ij j when ....

N.J. Higham, D.J. Higham, "Large Growth Factors in Gaussian Elimination with Pivoting", SIAM J. Matrix Anal. Appl. 10:3, 155--164, April 1989.

....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 large growth factors are identified by Foster [64] and Wright [131] while Higham and Higham [86] identify matrices for which any pivoting strategy gives growth factors of order n. An experimental investigation by Trefethen and Schreiber [126] suggests that the behaviour of the growth factor can be explained by statistical means, but no theorems are proved. Yeung and Chan [133] prove ....

Nicholas J. Higham and Desmond J. Higham. Large growth factors in Gaussian elimination with pivoting. SIAM J. Matrix Anal. Appl., 10(2): 155--164, April 1989.

....is usually discontinuous when there is a tie in the choice of pivot element, because then an arbitrarily small change in A can alter the pivot sequence. We applied the maximizer MDS to f , this time starting with the orthogonal matrix A 2 IR 4 Theta4 with a ij = 2= p 2n 1) sin(2ij= 2n 1) [34], for which ae 4 (A) 2:32. After 29 iterations and 1169 function evaluations the maximizer converged to a matrix B with ae 4 (B) 5:86. We used this matrix to start the maximizer AD (described in section 3) it took five iterations and 403 function evaluations to converge to the matrix C = 2 ....

....A = 2 6 4 1 1 Gamma1 1 1 Gamma1 Gamma1 1 1 Gamma1 Gamma1 Gamma1 1 3 7 5 identified by Wilkinson [59] as yielding the maximum possible growth ae n = 2 n Gamma1 for partial pivoting. The whole set of matrices A 2 IR n Thetan for which ae n (A) 2 n Gamma1 is described in [34], and C is one of these matrices, modulo the convergence tolerance. These examples, and others presented below, illustrate the following attractions of using direct search methods to aid the understanding of algorithms in matrix computations. 1) The simplest possible formulation of optimization ....

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Nicholas J. Higham and Desmond J. Higham, Large growth factors in Gaussian elimination with pivoting, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 155--164.

....It is easy to see that ae n (A) can be arbitrarily large, and so GE without pivoting is unstable in general. For partial pivoting, ae p n is almost invariably small in practice (ae p n 10, say) but a parametrized family of matrices is known for which it achieves its maximum of 2 n Gamma1 [26]. The situation is similar for ae c n , except that a much smaller upper bound is known, and it has been conjectured that ae c n (A) n for real A. Recent work has shed more light on the behaviour of ae p n and ae c n . Higham and Higham [26] present several families of real matrices from ....

....for which it achieves its maximum of 2 n Gamma1 [26] The situation is similar for ae c n , except that a much smaller upper bound is known, and it has been conjectured that ae c n (A) n for real A. Recent work has shed more light on the behaviour of ae p n and ae c n . Higham and Higham [26] present several families of real matrices from practical applications for which ae p n (A) and ae c n (A) are about n=2. These examples show that moderately large growth factors can be achieved on non contrived matrices. Trefethen and Schreiber [43] develop a statistical model of the average ....

N.J. Higham and D.J. Higham, Large growth factors in Gaussian elimination with pivoting, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 155--164.

....elimination with partial pivoting; these include Wilkinson s classic example [45, p. 212] gfpp(7) ans = 1 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 as well as all members of the 2 n Gamma1 class described by Higham and Higham [27]. The following extract uses the toolbox routine gecp to evaluate the growth factor for complete pivoting on the Wilkinson matrix. n = 20; A = gfpp(n) L, U] lu(A) Partial pivoting. max(max(abs(U) max(max(abs(A) 2(n 1) Approximation to growth factor. ans = 524288 524288 ....

....[L, U, P, Q, rho] gecp(A) rho rho = 2 As the output shows, complete pivoting is perfectly stable for these matrices. However, several of the matrices produced by orthog yield relatively large growth for complete pivoting: growth of order n=2 for real data, or n for a particular complex matrix [27]. n = 50; for k = 2 1 1 2 3] A = orthog(n, k) L, U, P, Q, rho] gecp(A) fprintf( g n , rho) end 25.3116 24.7028 25.6214 25.3296 50 A = hadamard(64) L, U, P, Q, rho] gecp(A) rho rho = 64 It is easy to show that complete pivoting suffers growth of at least n for an n Theta n Hadamard ....

Nicholas J. Higham and Desmond J. Higham. Large growth factors in Gaussian elimination with pivoting. SIAM J. Matrix Anal. Appl., 10(2):155--164, April 1989.

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N.J.Higham and D.J.Higham, Large growth factors in Gaussian elimination with pivoting, SIAM J. Matrix Anal. Appl., 10 (1989), 155--164.

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N.J.Higham and D.J.Higham, Large growth factors in Gaussian elimination with pivoting, SIAM J. Matrix Anal. Appl., 10 (1989), 155--164. bitemHH

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