Parallel interval multisplittings. (English) Zbl 0682.65016
The multisplitting concept of matrices which was defined by D. P. O’Leary and R. E. White [SIAM J. Algebraic Discrete Methods 6, 630-640 (1985; Zbl 0582.65018)] is adapted for its use in interval analysis. A method for solving systems of linear interval equations based on this multisplitting is worked out. The convergence behaviour of this method is investigated as well. A numerical example with a 24\(\times 24\)- interval matrix is given.
Reviewer: H.Ratschek
Keywords:
multisplitting; interval analysis; linear interval equations; convergence; numerical exampleCitations:
Zbl 0582.65018References:
| [1] | Alefeld, G.: On the convergence of some interval-arithmetic modifications of Newton’s method. SIAM J. Numer. Anal.21, 363-372 (1984) · Zbl 0536.65026 · doi:10.1137/0721027 |
| [2] | Alefeld, G., Herzberger, J.: Introduction to interval computations. New York: Academic Press 1983 · Zbl 0552.65041 |
| [3] | Barth, W., Nuding, E.: Optimale Lösung von Intervallgleichungssystemen. Computing12, 117-125 (1974) · Zbl 0275.65008 · doi:10.1007/BF02260368 |
| [4] | Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. New York: Academic Press 1979 · Zbl 0484.15016 |
| [5] | Kulisch, U., Miranker, W.: A new approach to scientific computation. New York: Academic Press 1983 · Zbl 0538.00023 |
| [6] | Mayer, G.: Enclosing the solution set of linear systems with inaccurate data by iterative methods based on incomplete LU-decompositions. Computing35, 189-206 (1985) · Zbl 0588.65027 · doi:10.1007/BF02260505 |
| [7] | Mayer, G.: Reguläre Zerlegungen und der Satz von Stein und Rosenberg für Intervallmatrizen. Habilitationsschrift. Karlsruhe 1986 |
| [8] | Mayer, G.: Comparison theorems for iterative methods based on strong splittings. SIAM J. Numer. Anal.24, 215-227 (1987) · Zbl 0614.65030 · doi:10.1137/0724018 |
| [9] | Mayer, G.: On the speed of convergence of some iterative processes. In: Greenspan, D., Rósza, P. (eds.), Colloq. Math. Soc. János Bolyai 50. Numerical Methods. Miskolc, 1986, pp. 207-228. Amsterdam: North-Holland 1988 |
| [10] | Neumaier, A.: New techniques for the analysis of linear interval equations. Linear Algebra Appl.58, 273-325 (1984) · Zbl 0558.65019 · doi:10.1016/0024-3795(84)90217-9 |
| [11] | Neumaier, A.: Further results on linear interval equations. Linear Algebra Appl.87, 155-179 (1987) · Zbl 0617.65023 · doi:10.1016/0024-3795(87)90164-9 |
| [12] | Neumann, M., Plemmons, R.J.: Convergence of parallel multisplitting iterative methods forM-matrices. Linear Algebra Appl.88/89, 559-573 (1987) · Zbl 0626.65025 · doi:10.1016/0024-3795(87)90125-X |
| [13] | O’Leary, D.P., White, R.E.: Multi-splittings of matrices and parallel solution of linear systems. SIAM J. Alg. Disc. Methods6, 630-640 (1985) · Zbl 0582.65018 · doi:10.1137/0606062 |
| [14] | Schneider, H.: Theorems onM-splittings of a singularM-matrix which depend on graph structure. Linear Algebra Appl.58, 407-424 (1984) · Zbl 0561.65020 · doi:10.1016/0024-3795(84)90222-2 |
| [15] | Schwandt, H.: Schnelle fast global konvergente Verfahren für die Fünf-Punkte-Diskretisierung der Poissongleichung mit Dirichletschen Randbedingungen auf Rechteckgebieten. Ph.D. Thesis, Techn. Univ. Berlin 1981 · Zbl 0474.65037 |
| [16] | Varga, R.S.: Matrix iterative analysis. Englewood Cliffs NJ: Prentice-Hall 1962 · Zbl 0133.08602 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
