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)(編輯修改稿)
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【文章內(nèi)容簡(jiǎn)介】 re three main classes of matrix for which it is known to be safe not to pivot when puting an LU factorization: matrices diagonally dominant by rows or columns, Hermitian positive definite matrices, and totally nonnegative matrices. Then the author proceeds to “identify another class of matrices with this highly desirable property:plex symmetric matrices whose real and imaginary parts are both positive definite. In this short article we extend the set of matrices having this property to include matrices whose inverses are matrices diagonally dominant by rows or columns, and we show that the growth factor for such matrices is bounded by two. The reader will find the proof in Section 3, with preliminary material required for the proof contained in Section 2. 2. Preliminary facts Let A∈ Mn(C), the set of n n plex matrices. For an index set ? ?n,1??? , we denote the principal submatrix of A that lies in the rows and columns indexed by? as A(? ) and its plementary principal submatrix as A( 39。a ).The following lemma is of crucial importance in Section 3. Received by the editor February 13, 2021 and, in revised form, August 1, 2021. 2021 Mathematics Subject Classification. Primary 65F10. Key words and phrases. Gaussian elimination, growth factor, diagonally dominant matrices, Schur plement. This work was supported by Natural Sciences and Engineering Research Council of Canada grant OGP0008111. Lemma 1. Let A∈ Mn(C) be a nonsingular matrix, and B = 1A . Let ? be a subset of ? ?n,1? .The inequality (1) )(d e t)(d e td e t /??? AAA ? with a positive scalar ? holds if and only if the similar inequality (2) )(d e t)(d e td e t /??? Ｂ ?。拢?? holds for the matrix B: Proof. Inequality (2) is just another form of (1). This can be seen from the relations AB det1det ?, AAB de t )(de t)(de t 39。?? ? , AAB de t )(de t)(de t 39。 ?? ? , 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。 the element with the largest modulus is on the main diagonal. Suppose that a nonsingular nbyn matrix A with nonvanishing leading principal minors undergoes Gaussian elimination with no pivoting. After k steps of the elimination have been pleted, we have an order nk matrix that has yet to be process
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