Improved methods and starting values to solve the matrix equations iteratively
Authors:
Ivan G. Ivanov, Vejdi I. Hasanov and Frank Uhlig
Translated by:
Journal:
Math. Comp. 74 (2005), 263278
MSC (2000):
Primary 65F10
Published electronically:
January 27, 2004
MathSciNet review:
2085410
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The two matrix iterations are known to converge linearly to a positive definite solution of the matrix equations , respectively, for known choices of and under certain restrictions on . The convergence for previously suggested starting matrices is generally very slow. This paper explores different initial choices of in both iterations that depend on the extreme singular values of and lead to much more rapid convergence. Further, the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use Newton's method in part.
 1.
W.
N. Anderson Jr., T.
D. Morley, and G.
E. Trapp, Positive solutions to
𝑋=𝐴𝐵𝑋⁻¹𝐵*, Linear
Algebra Appl. 134 (1990), 53–62. MR 1060009
(91c:47031), http://dx.doi.org/10.1016/00243795(90)90005W
 2.
B.
L. Buzbee, G.
H. Golub, and C.
W. Nielson, On direct methods for solving Poisson’s
equations, SIAM J. Numer. Anal. 7 (1970),
627–656. MR 0287717
(44 #4920)
 3.
S. M. ElSayed, Theorems for the Existence and Computing of Positive Definite Solutions for Two Nonlinear Matrix Equation, Proc. of 25 Spring Conference of the Union of Bulgarian Mathematicians, Kazanlak, 1996, pp.155161, (in Bulgarian).
 4.
Salah
M. ElSayed and André
C. M. Ran, On an iteration method for solving a class of nonlinear
matrix equations, SIAM J. Matrix Anal. Appl. 23
(2001/02), no. 3, 632–645 (electronic). MR 1896810
(2002m:15023), http://dx.doi.org/10.1137/S0895479899345571
 5.
Jacob
C. Engwerda, André
C. M. Ran, and Arie
L. Rijkeboer, Necessary and sufficient conditions for the existence
of a positive definite solution of the matrix equation
𝑋+𝐴*𝑋⁻¹𝐴=𝑄, Linear
Algebra Appl. 186 (1993), 255–275. MR 1217209
(94j:15012), http://dx.doi.org/10.1016/00243795(93)90295Y
 6.
Jacob
C. Engwerda, On the existence of a positive definite solution of
the matrix equation
𝑋+𝐴^{𝖳}𝖷⁻¹𝖠=𝖨,
Linear Algebra Appl. 194 (1993), 91–108. MR 1243822
(94j:15013), http://dx.doi.org/10.1016/00243795(93)901155
 7.
Augusto
Ferrante and Bernard
C. Levy, Hermitian solutions of the equation
𝑋=𝑄+𝑁𝑋⁻¹𝑁*, Linear
Algebra Appl. 247 (1996), 359–373. MR 1412761
(97m:93071), http://dx.doi.org/10.1016/00243795(95)001212
 8.
Gene
H. Golub and Charles
F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins
Series in the Mathematical Sciences, vol. 3, Johns Hopkins University
Press, Baltimore, MD, 1989. MR 1002570
(90d:65055)
 9.
ChunHua
Guo and Peter
Lancaster, Iterative solution of two matrix
equations, Math. Comp. 68
(1999), no. 228, 1589–1603.
MR
1651757 (99m:65061), http://dx.doi.org/10.1090/S0025571899011229
 10.
Alston
S. Householder, The theory of matrices in numerical analysis,
Blaisdell Publishing Co. Ginn and Co. New YorkTorontoLondon, 1964. MR 0175290
(30 #5475)
 11.
Ivan
G. Ivanov and Salah
M. Elsayed, Properties of positive definite solutions of the
equation
𝑋+𝐴*𝑋⁻²𝐴=𝐼, Linear
Algebra Appl. 279 (1998), no. 13, 303–316. MR 1637909
(99c:15019), http://dx.doi.org/10.1016/S00243795(98)000238
 12.
J.
M. Ortega and W.
C. Rheinboldt, Iterative solution of nonlinear equations in several
variables, Academic Press, New York, 1970. MR 0273810
(42 #8686)
 13.
Olga
Taussky, Matrices 𝐶 with 𝐶ⁿ→0, J.
Algebra 1 (1964), 5–10. MR 0161865
(28 #5069)
 14.
J.
H. Wilkinson, The algebraic eigenvalue problem, Clarendon
Press, Oxford, 1965. MR 0184422
(32 #1894)
 15.
Xingzhi
Zhan, Computing the extremal positive definite solutions of a
matrix equation, SIAM J. Sci. Comput. 17 (1996),
no. 5, 1167–1174. MR 1404867
(97g:65074), http://dx.doi.org/10.1137/S1064827594277041
 1.
 W. N. Anderson, T. D. Morley and G. E. Trapp, Positive Solution to , Linear Algebra Appl., 134 (1990), 5362. MR 91c:47031
 2.
 B. L. Buzbee, G. H. Golub and C. W. Nielson, On Direct Methods for Solving Poisson's Equations, SIAM J. Numer. Anal., 7 (1970), 627656. MR 44:4920
 3.
 S. M. ElSayed, Theorems for the Existence and Computing of Positive Definite Solutions for Two Nonlinear Matrix Equation, Proc. of 25 Spring Conference of the Union of Bulgarian Mathematicians, Kazanlak, 1996, pp.155161, (in Bulgarian).
 4.
 S. M. ElSayed and A. C. M. Ran, On an Iteration Method for Solving a Class of Nonlinear Matrix Equations, SIAM J. Matrix Anal. Appl., 23 (2001), 632645. MR 2002m:15023
 5.
 J. C. Engwerda, A. C. M. Ran and A. L. Rijkeboer, Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation , Linear Algebra Appl., 186 (1993), 255275. MR 94j:15012
 6.
 J. C. Engwerda, On the Existence of a Positive Definite Solution of the Matrix Equation , Linear Algebra Appl., 194 (1993), 91108. MR 94j:15013
 7.
 A. Ferrante and B. Levy, Hermitian Solutions of the Equation , Linear Algebra Appl., 247 (1996) 359373. MR 97m:93071
 8.
 C. H. Golub and C. F. Van Loan, Matrix Computations, Second Edition, John Hopkins University Press, Baltimore, 1989. MR 90d:65055
 9.
 C.H. Guo and P. Lancaster, Iterative Solution of Two Matrix Equations, Math. Comp., 68 (1999), 15891603. MR 99m:65061
 10.
 A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964. MR 30:5475
 11.
 I. G. Ivanov and S. M. ElSayed, Properties of Positive Definite Solutions of the Equation , Linear Algebra Appl., 279 (1998), 303316. MR 99c:15019
 12.
 J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970 (in Russian 1975). MR 42:8686
 13.
 O. Taussky, Matrices with , J. Algebra, 1 (1965), 510. MR 28:5069
 14.
 J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, London, 1965. MR 32:1894
 15.
 X. Zhan, Computing the Extremal Positive Definite Solutions of a Matrix Equation, SIAM J. Sci. Computing, 17 (1996), 11671174. MR 97g:65074
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC (2000):
65F10
Retrieve articles in all journals
with MSC (2000):
65F10
Additional Information
Ivan G. Ivanov
Affiliation:
Faculty of Economics and Business Administration, 125 Tzarigradsko chaussee, bl.3, Sofia University, Sofia 1113, Bulgaria
Email:
i_ivanov@feb.unisofia.bg
Vejdi I. Hasanov
Affiliation:
Laboratory of Mathematical Modelling, Shumen University, Shumen 9712, Bulgaria
Email:
v.hasanov@fmi.shubg.net
Frank Uhlig
Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849–5310
Email:
uhligfd@auburn.edu
DOI:
http://dx.doi.org/10.1090/S0025571804016369
PII:
S 00255718(04)016369
Keywords:
Matrix equation,
positive definite solution,
iterative method,
Newton's method
Received by editor(s):
May 29, 2001
Received by editor(s) in revised form:
May 7, 2003
Published electronically:
January 27, 2004
Additional Notes:
This work is partially supported by Shumen University under Grant #3/04.06.2001.
Article copyright:
© Copyright 2004 American Mathematical Society
