We give estimates of the infinity norm of the inverses of matrices of monotone type and totally positive matrices.Keywords: matrix of monotone type, M -matrix, totally positive matrix, diagonal dominance, inverse matrix, norm of a matrix In many problems of numerical analysis there is a need to estimate some norm of the matrix A −1 , the inverse to a given nonsingular matrix A = (a ij ) ∈ R n×n . It is usually not difficult to compute (or estimate) the norm of A. The estimation of the norm of A −1 = (a ij ) is a far more difficult task for any norm when A −1 is not known explicitly.A sufficiently simple method for estimation of the ∞-norm of the inverse is known for the matrices with diagonal dominance. This method was proposed in [1] in connection with the necessity to derive some error bounds of interpolation by cubic splines. Many papers have appeared, since then the estimate from [1] was improved and refined in various particular cases. Especially many articles in this area were devoted to the so-called matrices of monotone type, M -matrices.In this paper we consider a broad class of matrices and extend some results on the norm estimation for the inverses of M -matrices to the class. Furthermore, similar estimates are derived for totally positive matrices.Denote bythe diagonal dominance of each row, and putAs usual, we call a matrix A diagonally dominant provided that R * (A) 0.In spite of the availability of many papers refining (2), the estimate (2) is, however, attainable. Namely, if the main diagonal entries of A are positive, all off-diagonal entries are nonpositive, and the diagonal dominance of all rows is the same; then the norm A −1 ∞ is found exactly.
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