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A remark on solving large systems of equations in function spaces. (English) Zbl 0759.65033

The authors consider a system \(x_ i=T_ i(x_ i)\), \(i=1,\dots,N\), where \(T_ i\), \(i=1,\dots,N\) is supposed to be a splitting of a nonlinear operator \(T\) mapping a complete metric space \((X,d)\) into itself. \(T_ i\) operates on a complete metric space \((X_ i,d_ i)\) with values in \((X,d)\). The above system is supposed to be equivalent to the fixed point equation \(y=T(y)\).
Assuming \(d_ i(T_ i(x),T_ i(\tilde x))\leq \sum^ N_{j=1}k_{ij}d_ j(x_ j,\tilde x_ j)\), the authors are interested in the convergence of the iterative procedure \(x^ k=T(x^{k-1})\), \(k=1,2,\dots\) and they prove the following result:
Let \(K\) be the real matrix with the entries \(k_{ij}\). If the spectral radius \(\rho(K)<1\), then the authors construct a metric \(\hat d\) on \(X\) such that \(T\) is a uniform contraction on the whole space \((X,\hat d)\).
An application to a system consisting of an initial value problem and a system of algebraic equations closes the paper.
Reviewer: E.Bohl (Konstanz)

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65H10 Numerical computation of solutions to systems of equations
47J25 Iterative procedures involving nonlinear operators
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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References:

[1] G. Frobenius: Über Matrizen aus positiven Elementen. S. -B. Preuss. Akad. Wiss, Berlin 1908, 471-476, 1909, 514-518. · JFM 39.0213.03
[2] G. Frobenius: Über Matrizen aus nichtnegativen Elementen. S. -B. Preuss. Akad. Wiss. Berlin 1912, 456-477. · JFM 43.0204.09
[3] E. Lelarasmee A. E. Ruehli A. L. Sangiovanni- Vincentelli: The waveform relaxation method for the time domain analysis of large scale integrated circuits. IEEE Trans. CAD 1, (1982), 131-145. · Zbl 05448742
[4] A. R. Newton. A. L. Sangiovanni-Vincentelli: Relaxation based electrical simulation. IEEE Trans ED 30 (1983), 1184-1207. · Zbl 0526.65008
[5] J. M. Ortega W. C. Rheinboldt: Iterative Solutions of Nonlinear Equations in Several Variables. New York: Academic Press, 1970. · Zbl 0241.65046
[6] O. Perron: Zur Theorie der Matrizen. Math. Ann. 64 (1907), 248-263, · JFM 38.0202.01
[7] K. R. Schneider: A remark on the waveform relaxation method. Int. J. Circuit Theory Appl. 18 (1990).
[8] R. S. Varga: Matrix iterative analysis. Prentice-Hall, Englewood Cliffs, N. J. 1962. · Zbl 0133.08602
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