# A convergent nonlinear splitting via orthogonal projection

Aplikace matematiky (1984)

• Volume: 29, Issue: 4, page 250-257
• ISSN: 0862-7940

top

## Abstract

top
We study the convergence of the iterations in a Hilbert space $V,{x}_{k+1}=W\left(P\right){x}_{k},W\left(P\right)z=w=T\left(Pw+\left(I-P\right)z\right)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W\left(P\right)$ is continuous and the Lipschitz constant $∥\left(I-P\right)W\left(P\right)∥<1$. If an operator $W\left({P}_{1}\right)$ satisfies these assumptions and ${P}_{2}$ is an orthogonal projection such that ${P}_{1}{P}_{2}={P}_{2}{P}_{1}={P}_{1}$, then the operator $W\left({P}_{2}\right)$ is defined and continuous in $V$ and satisfies $∥\left(I-{P}_{2}\right)W\left({P}_{2}\right)∥\le ∥\left(I-{P}_{1}\right)W\left({P}_{1}\right)∥$.

## How to cite

top

Mandel, Jan. "A convergent nonlinear splitting via orthogonal projection." Aplikace matematiky 29.4 (1984): 250-257. <http://eudml.org/doc/15355>.

@article{Mandel1984,
abstract = {We study the convergence of the iterations in a Hilbert space $V,x_\{k+1\}=W(P)x_k, W(P)z=w=T(Pw+(I-P)z)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W(P)$ is continuous and the Lipschitz constant $\left\Vert (I-P)W(P)\right\Vert <1$. If an operator $W(P_1)$ satisfies these assumptions and $P_2$ is an orthogonal projection such that $P_1P_2=P_2P_1=P_1$, then the operator $W(P_2)$ is defined and continuous in $V$ and satisfies $\left\Vert (I-P_2)W(P_2)\right\Vert \le \left\Vert (I-P_1)W(P_1)\right\Vert$.},
author = {Mandel, Jan},
journal = {Aplikace matematiky},
keywords = {convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point; convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point},
language = {eng},
number = {4},
pages = {250-257},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A convergent nonlinear splitting via orthogonal projection},
url = {http://eudml.org/doc/15355},
volume = {29},
year = {1984},
}

TY - JOUR
AU - Mandel, Jan
TI - A convergent nonlinear splitting via orthogonal projection
JO - Aplikace matematiky
PY - 1984
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 29
IS - 4
SP - 250
EP - 257
AB - We study the convergence of the iterations in a Hilbert space $V,x_{k+1}=W(P)x_k, W(P)z=w=T(Pw+(I-P)z)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W(P)$ is continuous and the Lipschitz constant $\left\Vert (I-P)W(P)\right\Vert <1$. If an operator $W(P_1)$ satisfies these assumptions and $P_2$ is an orthogonal projection such that $P_1P_2=P_2P_1=P_1$, then the operator $W(P_2)$ is defined and continuous in $V$ and satisfies $\left\Vert (I-P_2)W(P_2)\right\Vert \le \left\Vert (I-P_1)W(P_1)\right\Vert$.
LA - eng
KW - convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point; convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point
UR - http://eudml.org/doc/15355
ER -

## References

top
1. M. Fiedler V. Pták, On aggregation in matrix theory and its applications to numerical inverting of large matrices, Bull. Acad. Pol. Sci. Math. Astr. Phys. 11 (1963) 757-759. (1963) MR0166911
2. N. S. Kurpeľ, Projection-iterative Methods of Solution of Operator Equations, (Russian). Naukova Dumka, Kiev 1968. (1968) MR0254691
3. D. P. Looze N. R. Sandell, Jr., 10.1007/BF00934678, J. Optim. Theory Appl. 34 (1981) 371-382. (1981) MR0628203DOI10.1007/BF00934678
4. A. Ju. Lučka, Projection-iterative Methods of Solution of Differential and Integral Equations, (Russian). Naukova Dumka, Kiev 1980. (1980) MR0598991
5. A. Ju. Lučka, Convergence criteria of the projection-iterative method for nonlinear equations, (Russian). Preprint 82.24, Institute of Mathematics AN Ukrain. SSR, Kiev 1982. (1982)
6. J. Mandel, Convergence of some two-level iterative methods, (Czech). PhD Thesis, Charles University, Prague 1982. (1982)
7. J. Mandel, On some two-level iterative methods, In: Defect Correction - Theory and Applications (K. Böhmer, H. J. Stetter. editors), Computing Supplementum Vol. 5, Springer-Verlag, Wien, to appear. Zbl0552.65049MR0782691
8. J. Mandel B. Sekerka, A local convergence proof for the iterative aggregation method, Linear Algebra Appl. 51 (1983), 163-172. (1983) MR0699731
9. O. Pokorná I. Prágerová, Approximate matrix invertion by aggregation, In: Numerical Methods of Approximation Theory, Vol. 6 (L. Collatz, G. Meinhardus, H. Werner, editors), Birghäuser Verlag, Basel-Boston-Stuttgart 1982. (1982)
10. A. E. Taylor, Introduction to Functional Analysis, J. Wiley Publ., New York 1958. (1958) Zbl0081.10202MR0098966
11. R. S. Varga, Matrix Iterative Analysis, Prentice Hall Inc., Englewood Cliffs, New Jersey 1962. (1962) MR0158502
12. T. Wazewski, Sur une procédé de prouver la convergence des approximations successives sans utilisation des séries de comparaison, Bull. Acad. Pol. Sci. Math. Astr. Phys. 8 (1960) 47-52. (1960) MR0126109

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.