外文翻譯---高斯消去法是穩(wěn)定的反對(duì)角占優(yōu)矩陣-其他專業(yè)-全文預(yù)覽
【正文】 E with no pivoting as applied to A is the same as GE with partial pivoting. In fact, relation (6) means that the entry with the largest modulus in the entire matrix A belongs to the main diagonal. The same is true for all Schur plements A/A??? . Hence, to bound )n A（? , we have to examine only the behavior of the diagonal entries in the course of elimination. Assume that )(,max tretsr aM ? is attained when r = s = i。 ?? ? , The last two equalities are particular cases of a general formula that connects minors in B and A (see formula (33) in [2, Chapter1]). We say that B∈ Mn(C) is a (row) diagonally dominant matrix (. matrix, for short) if (3) ,n,1i,bb nij 1jijiii ??? ???? Where 0≤ ? ＜ 1, i = 1,? ,n. The quantity (4) ? = ni1max ?? i? will be called the dominance factor of B. Lemma 2. Let B be a . matrix, and let 1B = A = (aij ): Then, for i = 1,…,n, (5) ，iiib)1(de t BB I??? where Bi is the cofactor of bii, and (6) .aa iijji ｉ， ?。??? ? Both assertions of the lemma can be found in [3, Sections 4, 6, and 7]. Inequality (6) says that, in each column of the inverse matrix A。 [6]這是某些積分方程的解 , 一種 時(shí)間序列方法的數(shù)值微分 [7], 以及 某些物理問題涉及 的 耦合振蕩器 [8]. 參考書籍 1. . Higham, Factorizing plex symmetric matrices with positive de_nite real and imaginary parts, Math. Comp. 67 (1998), 1591{1599. MR 99a:65049 2. . Gantmacher, The Theory of Matrices, Chelsea, New York, 1959. MR 21:6372c 3. . Ostrowski, Note on bounds for determinants with dominant principal diagonal, Proc. Amer. Math. Soc. 3 (1952), 26{30. MR 14:611c 4. . Horn and . Johnson, Matrix Analysis, Cambridge University Press, 1985. MR 87e:15001 5. M. Neumann and . Plemmons, Backward error analysis for linear systems associated with inverses of Hmatrices, BIT 24 (1984), 102{112. MR 85f:65027 6. . Willoughby, The inverse Mmatrix problem, Linear Algebra Appl. 18 (1977), 75{94. MR 57:12561 7. R. Andersen and R. Bloom_eld, A time series approach to numerical di_erentiation, Technometrics 16 (1974), 69{75. MR 53:1889 GAUSSIAN ELIMINATION 657 8. . Le_, Correlation inequalities for coupled oscillators, J. Math. Phys. 12 (1971), 569{578. MR 43:8322 9. . Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996. MR 97a:65047 School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada Faculty of Computational Mathematics and Cyberics, Moscow State University, 119992 Moscow, Russia Email address: License GAUSSIAN ELIMINATION IS STABLE FOR THE INVERSE OF A DIAGONALLY DOMINANT MATRIX ALAN GEORGE AND KHAKIM D. IKRAMOV Abstract. Let B∈ Mn (C) be a row diagonally dominant matrix, ., ,n,1i,bb nij 1jijiii ??? ???? where 0≤ ? ＜ 1， i = 1,? ,n, with ? = ni1max ?? i? , We show that nopivoting is necessary when Gaussian elimination is applied to 1??BA , Moreover,the growth factor f
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