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In this paper, we propose a modification of the accelerated projective steepest descent method for solving nonlinear inverse problems with an 1 constraint on the variable, which was recently proposed by Teschke and Borries (2010 Inverse Problems 26 025007). In their method, there are some parameters need to be estimated, which is a difficult task for many applications. We overcome this difficulty by introducing a self-adaptive strategy in choosing the parameters. Theoretically, the convergence of their algorithm was guaranteed under the assumption that the underlying mapping F is twice Fréchet differentiable together with some other conditions, while we can prove weak and strong convergence of the proposed algorithm under the condition that F is Fréchet differentiable, which is a relatively weak condition. We also report some preliminary computational results and compare our algorithm with that of Teschke and Borries, which indicate that our method is efficient.

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In matrix theory and numerical analysis there are two very famous and important results. One is Geršgorin circle theorem, the other is strictly diagonally dominant theorem. They have important application and research value, and have been widely used and studied. In this paper, we investigate generalized diagonally dominant matrices and matrix eigenvalue inclusion regions. A class of G-function pairs is proposed, which extends the concept of G-functions. Thirteen kind of G-function pairs are established. Their properties and characteristics are studied. By using these special G-function pairs, we construct a large number of sufficient and necessary conditions for strictly diagonally dominant matrices and matrix eigenvalue inclusion regions. These conditions and regions are composed of different combinations of G-function pairs, deleted absolute row sums and column sums of matrices. The results extend, include and are better than some classical results.

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