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In this paper, we further investigate the single proper nonnegative splittings and double proper nonnegative splittings of rectangular matrices. Two convergence theorems for the single proper nonnegative splitting of a semimonotone matrix are derived, and more comparison results for the spectral radii of matrices arising from the single proper nonnegative splittings and double proper nonnegative splittings of different rectangular matrices are presented. The obtained results generalize the previous ones, and it can be regarded as the useful supplement of the results in [Comput. Math. Appl., 67: 136–144, 2014] and [Results. Math., 71: 93–109, 2017].

Based on the relaxed alternating positive semi-definite splitting (RAPSS) preconditioner, in this paper, a new preconditioner, called variant of relaxed alternating positive semi-definite splitting (VRAPSS) preconditioner, is presented and discussed. Spectral properties of the VRAPSS preconditioned matrix are analyzed in detail. Numerical experiments are provided to verify the efficiency of the VRAPSS preconditioner.

The comparison results for K-double splittings of one K-monotone matrix are established in the literatures. As comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners, we propose some comparison results for K-nonnegative double splittings of different K-monotone matrices in this note. The obtained results generalize the previous ones.

Picard-type methods are efficient methods for solving the absolute value equation $ Ax-|x| = b $. To further improve the performance of Picard iteration method, a new inexact Picard iteration method, named Picard-SHSS iteration method, is proposed to solve the absolute value equation. The sufficient condition for the convergence of the proposed method for solving the absolute value equation is given. A numerical example is given to demonstrate the effectiveness of the new method.

In this paper, new preconditioned GAOR methods are proposed for solving a class of $ 2 \times 2 $ block structure linear systems arising from the weighted linear least squares problems. Comparison theorems are derived. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods in the previous literatures whenever these methods are convergent. A numerical example is given to confirm our theoretical results.

In this paper, a new preconditioner for a class of $$2\times 2$$ block linear systems is proposed. The proposed new preconditioner is a better approximation to the original coefficient matrix than the previous ones. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed. Finally, two numerical examples are provided to show the effectiveness of the proposed preconditioner.

In this paper, we investigate structured backward errors for two kinds of generalized saddle point systems. The explicit formulas of structured backward errors are derived, which are useful for tes...

Based on the single-step iteration (SSI) method for the non-Hermitian positive definite linear systems, we propose a Uzawa-SSI method for solving the saddle point problems with non-Hermitian positive definite (1, 1) block in this paper. The convergence and semi-convergence properties of the Uzawa-SSI method, respectively, for nonsingular and singular cases are analyzed. Numerical examples with experiments are given to show the robustness and the efficiency of Uzawa-SSI method.

For nonsymmetric saddle point problems, the fast shift-splitting method and its generalized form are studied recently. In this paper, a general fast shift-splitting iteration method is proposed for solving nonsymmetric saddle point problems. The convergence and the semi-convergence of the proposed iteration method, respectively, for nonsingular and singular nonsymmetric saddle point problems are studied, respectively. Meanwhile, the spectral properties of the corresponding preconditioned matrix are analyzed in details. Numerical examples with experiments are given to show the robustness and the efficiency of the proposed method.

In this paper, based on the simplified Hermitian and skew-Hermitian (SHSS) preconditioner, a new preconditioner, called the generalized SHSS (GSHSS) preconditioner, is considered to solve the generalized saddle point problem. We prove that the GSHSS iteration method is convergent if the iteration parameters satisfy appropriate conditions. In addition, it is proved that all eigenvalues of the GSHSS preconditioned matrix are real and non-unit eigenvalues are located in a positive interval. Numerical experiment is provided to show the effectiveness of the GSHSS preconditioner.

We study a preconditioned generalized shift-splitting iteration method for solving saddle point problems. The unconditional convergence theory of the preconditioned generalized shift-splitting iteration method is established. When the splitting matrix is used as a preconditioner, we analyze eigenvalue distribution of the preconditioned saddle point matrix. It is proved that complex eigenvalues having nonzero imaginary parts of the preconditioned matrix are located in an intersection of two circles and the real parts of all eigenvalues of the preconditioned matrix are located in a positive interval. Numerical experiments are used to verify our theoretical results and illustrate effectiveness of the proposed iteration method and the corresponding splitting preconditioner.

Recently, a class of generalized shift-splitting iteration methods were proposed for solving nonsingular nonsymmetric saddle point problems (Cao et al., 2015). In this paper, the generalized shift-splitting iteration method is extended to solve singular nonsymmetric saddle point problems. Semi-convergence of the generalized shift-splitting iteration method is carefully analyzed, which shows that the iterative sequence generated by the generalized shift-splitting iteration method converges to a solution of the singular saddle point problem unconditionally. Numerical results verify the robustness and efficiency of the generalized shift-splitting iteration method.

In this paper, a new relaxed splitting preconditioner is proposed for the generalized saddle point problems. The proposed preconditioner is based on a new matrix splitting of the coefficient matrix and is easier to be implemented than the dimensional splitting preconditioner. Eigenvalue distribution of the preconditioned matrix is analyzed. An upper bound of the degree of the minimal polynomial of the preconditioned matrix is discussed. Finally, some numerical experiments are illustrated to show the efficiency of the new preconditioner.

Comparison theorems between the spectral radii of different matrices are useful tools for judging the efficiency of preconditioners. In this paper, some comparison theorems for the spectral radii of matrices arising from proper splittings of different semimonotone matrices are presented.

In 2009, Zheng and Miao (B. Zheng and S.-X. Miao, Two new modified Gauss-Seidel methods for linear system with M-matrices, J. Comput. Appl. Math. 233 (2009), 922-930) considered the modified Gauss-Seidel method for solving M-matrix linear system with the pre- conditioner Pmax. In this paper, we consider the modified Gauss-Seidel method for solving the linear system with the generalized preconditioner Pmax(�), and study its convergent properties when the coefficient ma- trix is an H-matrix. Numerical experiments are performed with different examples, and the numerical results verify our theoretical analysis.

In this paper, the generalized stationary iterative (GSI) method is studied for solving the saddle point problems. The convergence of this method is studied under suitable restrictions on iteration parameters. Numerical example for solving Stokes equation is presented to show the superiority of GSI method.

In 2002, H. Kotakemori et al. proposed the modified Gauss–Seidel (MGS) method for solving the linear system with the preconditioner P=I+Smax [H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+Smax) J. Comput. Appl. Math. 145 (2002) 373–378]. Since this preconditioner is constructed by only the largest element on each row of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the nth row. In the present paper, to deal with this drawback, we propose two new preconditioners. The convergence and comparison theorems of the modified Gauss–Seidel methods with these two preconditioners for solving the linear system are established. The convergence rates of the new proposed preconditioned methods are compared. In addition, numerical experiments are used to show the effectiveness of the new MGS methods.

Comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners. For single splittings of different monotone matrices, Elsner et al. (Linear Algebra Appl. 363:65–80, 2003) gave out comparison theorems for spectral radii. For double splittings, some convergence and comparison theorems of a monotone matrix are presented by Shen et al. (Comput. Math. Appl. 51:1751–1760, 2006). In this note we give the comparison theorem for the spectral radii of matrices arising from double splittings of different monotone matrices.

In this paper, the block SOR iterative methods are studied for n×n fuzzy linear systems and the corresponding convergence theorems are also given out. We know that the coefficient matrix S of the augmented system SX=Y is consistently ordered when S1 is nonsingular, and in this case the optimal parameter ω of the block SOR method is obtained. Numerical examples are presented to illustrate the theory.

In this paper, we present an efficient numerical algorithm, namely block homotopy perturbation method, for solving fuzzy linear systems based on homotopy perturbation method. Some numerical examples are given to show the efficiency of the algorithm.

In this note, we discuss the convergence behavior of a modified inexact Uzawa algorithm for solving generalized saddle point problems, which is an extension of the result obtained in a recent paper [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math., 46 (2003) 157-171]., {"references":["Z.-Z. Bai, Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008) 2900-2932.","M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005) 1-137.","J.H. Bramble, J.E. Pasciak, A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997) 1072-1092.","Y. Cao, Y. Lin, Y. Wei, Nolinear Uzawa methods for solving nonsym-metric saddle point problems, J. Appl. Math. Comput., 21 (2006) 1-21.","Z.-H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math., 46 (2003) 157-171.","X.-L. Cheng, The inexact Uzawa algorithm for saddle point problem, Appl. Math. Letters, 13 (2000) 1-3.","H.C. Elman, G.H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31(1994) 1645-1661.","B. Fischer, R. Ramage, D.J. Silvester, A.J. Wathen, Minimum residual methods for augmented systems, BIT Numer. Math., 38 (1998) 527-543.","G.H. Golub, [] X. Wu, J.-Y. Yuan, SOR-like methods for augmented systems, BIT Numer. Math., 41 (2001) 71-85.\n[10] Y. Lin, Y. Wei, A convergence analysis of the nonlinear Uzawa algorithm for saddle point problems, Appl. Math. Letters, 20 (2007) 1094-1098. \n[11] Y. Lin, Y. Wei, Fast corrected Uzawa methods for solving symmetric saddle point problems, Calcolo, 43 (2006) 65-82.\n[12] S.-X. Miao, K. Wang, On generalized stationary iterative method for solving the saddle point problems, J. Appl. Math. Comput., DOI: 10.1007/s 12190-009-0369-8.\n[13] X. Wu, B.P.B. Silva, J.-Y. Yuan, Conjugate gradient method for rank deficient saddle point problems, Numer. Algorithms, 35 (2004) 139-154. \n[14] J.H. Yun, S.W. Kim, Generalized stationary iterative method for solving linear systems, Korean J. Comput. Appl. Math., 5 (1998) 341-349.\n[15] B. Zheng, Z.-Z. Bai, X. Yang, On Semi-Convergence of Parameterized Uzawa Methods for Singular Saddle Point Problems, Linear Algebra Appl., 431 (2009) 808-817."]}

In this paper, the Gaussian type quadrature rules for fuzzy functions are discussed. The errors representation and convergence theorems are given. Moreover, four kinds of Gaussian type quadrature rules with error terms for approximate of fuzzy integrals are presented. The present paper complements the theoretical results of the paper by T. Allahviranloo and M. Otadi [T. Allahviranloo, M. Otadi, Gaussian quadratures for approximate of fuzzy integrals, Applied Mathematics and Computation 170 (2005) 874-885]. The obtained results are illustrated by solving some numerical examples., {"references":["J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, second ed.,\nSpringer, New York, 1991.","L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-352.","L.A. Zadeh, The concept of a linguistic variable and its applications to\napproximate reasoning, Inform. Sci. 8 (1975) 199-249, 301-357 9 (1975)\n43-80.","C-X. Wu, M. Ma, Embedding problem of fuzzy number space: Part I,\nFuzzy Sets and Systems 44 (1991) 33-38.","S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems\n24 (1987) 319-330.","M.L. Puri, D. Ralescu, Fuzzy random variables, J. Math. Anal. Appl.\n114 (1986) 409-442.","R. Goetschel, W. Voxman, Elementary calculus, Fuzzy Sets and Systems\n18 (1986) 31-43.","T. Allahviranloo, Newton Cot-s methods for integation of fuzzy functions,\nApplied Mathematics and Computation 166 (2005) 339-348.","T. Allahviranloo, M. Otadi, Gaussian quadratures for approximate of\nfuzzy integrals, Applied Mathematics and Computation 170 (2005) 874-\n885.\n[10] M. Ma, M. Friedman, A. Kandel, Numerical solutions of fuzzy differential\nequations, Fuzzy Sets and Systems 105 (1999) 133-138.\n[11] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24\n(1987) 301-317.\n[12] D. Dubois, H. Prade, Towards fuzzy differential calculus, Fuzzy Sets\nand Systems 8 (1982) 1-7, 105-116, 225-233.\n[13] M. Friedman, M. Ma, A. Kandel, Numerical solutions of fuzzy differential\nand integral equations, Fuzzy Sets and Systems 106 (1999) 35-48.\n[14] M. Friedman, M. Ma, A. Kandel, Numerical methods for calculating the\nfuzzy integral, Fuzzy Sets and Systems 83 (1996) 57-62.\n[15] S. Abbasbandy, T. Allahviranloo, Numerical solutionof fuzzy differential\nequation by Talor method, Journal of Computational Method in Applied\nMathematics 8 (2002) 43-53.\n[16] C.-X. Wu, M. Ma, On embedding problem of fuzzy number spaces,\nFuzzy Sets and Systems 44 (1991) 33-38; 45 (1992) 189-202; 46 (1992)\n281-286.\n[17] J.Y. Park, Y.C. Kwan, J.V. Jeong, Existence of solutions of fuzzy integral\nequations in Banaeh spaces, Fuzzy Sets and Systems 72 (1995) 373-378.\n[18] Byung Soo Moon, A tuning algorithm for the fuzzy logic controllers,\nEUFIT -95 620-624.\n[19] R. Palm, D. Driankov, Fuzzy inputs, Fuzzy Sets and Systems 70 (1995)\n315-335."]}