Abstract
In actual practice, iteration methods applied to the solution of finite systems of equations yield inconclusive results as to the existence or nonexistence of solutions and the accuracy of any approximate solutions obtained. On the other hand, construction of interval extensions of ordinary iteration operators permits one to carry out interval iteration computationally, with results which can give rigorous guarantees of existence or nonexistence of solutions, and error bounds for approximate solutions. Examples are given of the solution of a nonlinear system of equations and the calculation of eigenvalues and eigenvectors of a matrix by interval iteration. Several ways to obtain lower and upper bounds for eigenvalues are given.
Similar content being viewed by others
References
G. Alefeld,Intervallanalytische Methoden bei nichtlinearen Gleichungen, Jahrb. überblicke Math. 1979 (1979), 63–78.
P. M. Anselone and L. B. Rall,The solution of characteristic value-vector problems by Newton's method, Numer. Math. 11 (1968), 38–45.
O. Caprani and K. Madsen,Mean value forms in interval analysis, Computing 25 (1980), 147–154.
A. Cuyt and P. Van der Cruyssen,Abstract Padé-approximants for the solution of a system of nonlinear equations, Rept. No. 80-17, Department of Mathematics, Univ. Antwerp, 1980.
R. Krawczyk,Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing 4 (1969), 187–201.
D. Kuba and L. B. Rall,A UNIVAC 1108 program for obtaining rigorous error estimates for approximate solutions of systems of equations, MRC Tech. Summary Rept. No. 1168, Univ. of Wisconsin-Madison, 1972.
U. Kulisch and W. Miranker,Computer Arithmetic in Theory and Practice, Academic Press, New York, 1981.
U. Kulisch and H.-W. Wippermann,PASCAL-SC: Pascal for Scientific Computation, Inst. for Appl. Math. Univ. Karlsruhe, 1980.
R. E. Moore,Interval Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1966.
R. E. Moore,A test for existence of solutions to nonlinear systems, SIAM J. Numer. Anal. 14 (1977), 611–615.
R. E. Moore,Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics, 2, Soc. for Ind. Appl. Math., Philadelphia, 1979.
R. E. Moore,Interval methods for nonlinear systems, Computing, Suppl. 2 (1980), 113–120.
R. E. Moore,New results on nonlinear systems, [16], pp. 165–180 (1980).
R. E. Moore and S. T. Jones,Safe starting regions for iterative methods, SIAM J. Numer. Anal. 14 (1977), 1051–1065.
K. Nickel,Stability and convergence of monotonic algorithms, J. Math. Anal. Appl. 54 (1976), 157–172.
K. Nickel (Ed.),Interval Mathematics 1980, Academic Press, New York, 1980.
L. B. Rall,Quadratic equations in Banach spaces, Rend. Circ. Mat. Palermo (2) 10 (1961), 314–332.
L. B. Rall,Newton's method for the characteristic value problem Ax = λBx, J. Soc. Indust. Appl. Math. 9 (1961), 288–293, Errata 10 (1962), 228.
L. B. Rall,A quadratically convergent iteration method for computing zeros of operators satisfying autonomous differential equations, Math. Comp. 30 (1976), 112–114.
L. B. Rall,Computational Solution of Nonlinear Operator Equations, Wiley, New York, 1969. Reprinted by Krieger, Huntington, N.Y., 1979.
L. B. Rall,A comparison of the existence theorems of Kantorovich and Moore, SIAM J. Number. Anal. 17 (1980), 148–161.
L. B. Rall,Applications of software for automatic differentiation in numerical computation, Computing, Suppl. 2 (1980), 141–156.
L. B. Rall,An interval arithmetic package for the HP33-E, Freib. Intervall-Ber. 80/7 (1980), 21–23.
L. B. Rall,A theory of interval iteration, MRC Tech. Summary Rept. No. 2196, Univ. of Wisconsin-Madison, 1981.
J. M. Yohe,Implementing nonstandard arithmetics, SIAM Rev. 21 (1979), 34–56.
J. M. Yohe,Portable software for interval arithmetic, Computing, Suppl. 2 (1980), 211–229.
Author information
Authors and Affiliations
Additional information
Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.
Rights and permissions
About this article
Cite this article
Rall, L.B. Solution of finite systems of equations by interval iteration. BIT 22, 233–251 (1982). https://doi.org/10.1007/BF01944479
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01944479