# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Iterative solutions to matrix equations of the form ${A}_{i}X{B}_{i}={F}_{i}$. (English) Zbl 1197.15009
Summary: This paper is concerned with the numerical solutions to the linear matrix equations ${A}_{1}X{B}_{1}={F}_{1}$ and ${A}_{2}X{B}_{2}={F}_{2}$; two iterative algorithms are presented to obtain the solutions. For any initial value, it is proved that the iterative solutions obtained by the proposed algorithms converge to their true values. Finally, simulation examples are given to verify the proposed convergence theorems.
##### MSC:
 15A24 Matrix equations and identities 65F30 Other matrix algorithms
##### References:
 [1] Zheng, B.; Ye, L.; Cvetkovic-Ilic, D. S.: The congruence class of the solutions of some matrix equations, Computers mathematics with applications 57, No. 4, 540-549 (2009) · Zbl 1165.15303 · doi:10.1016/j.camwa.2008.11.010 [2] Xie, L.; Ding, J.; Ding, F.: Gradient based iterative solutions for general linear matrix equations, Computers mathematics with applications 58, No. 7, 1441-1448 (2009) · Zbl 1189.65083 · doi:10.1016/j.camwa.2009.06.047 [3] Zhou, B.; Li, Z. Y.; Duan, G. R.; Wang, Y.: Weighted least squares solutions to general coupled Sylvester matrix equations, Journal of computational and applied mathematics 224, No. 2, 759-776 (2009) · Zbl 1161.65034 · doi:10.1016/j.cam.2008.06.014 [4] Ding, F.; Liu, P. X.; Ding, J.: Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Applied mathematics and computation 197, No. 1, 41-50 (2008) · Zbl 1143.65035 · doi:10.1016/j.amc.2007.07.040 [5] Dehghan, M.; Hajarian, M.: An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation, Applied mathematics and computation 202, No. 2, 571-588 (2008) · Zbl 1154.65023 · doi:10.1016/j.amc.2008.02.035 [6] Dehghan, M.; Hajarian, M.: Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A1X1B1+A2X2B2=C, Mathematical and computer modelling 49, No. 9–10, 1937-1959 (2009) · Zbl 1171.15310 · doi:10.1016/j.mcm.2008.12.014 [7] Qiu, J. Q.; Jiang, W. H.; Shi, Y.; Xi, D. M.: Positive solutions for nonlinear nth-order m-point boundary value problem with the first derivative, International journal of innovative computing, information and control 5, No. 8, 2405-2414 (2009) [8] Zhong, X. Z.; Zhang, T.; Shi, Y.: Oscillation and nonoscillation of neutral difference equation with positive and negative coefficients, International journal of innovative computing, information and control 5, No. 5, 1329-1342 (2009) [9] Yu, C.: Existence and uniqueness of the solution for FM-BEM based on $GMRES\left(m\right)$ algorithm, ICIC express letters 2, No. 1, 89-93 (2008) [10] Liu, J.; Yu, C.; Chen, Y.; Li, Xia: Computational formulations for the fundamental solution and kernel functions of elasto-plastic FM-BEM in spherical coordinate system, ICIC express letters 2, No. 2, 207-212 (2008) [11] Navarra, A.; Odell, P. L.; Young, D. M.: A representation of the general common solution to the matrix equations A1XB1=C1 and A2XB2=C2 with applications, Computers mathematics with applications 41, No. 7–8, 929-935 (2001) [12] Liao, A.; Lei, Y.: Least-squares solution with the minimum-norm for the matrix equation (AXB,GXH)=(C,D), Computers mathematics with applications 50, No. 3–4, 539-549 (2005) · Zbl 1087.65040 · doi:10.1016/j.camwa.2005.02.011 [13] Liu, Y. H.: Ranks of least squares solutions of the matrix equation AXB=C, Computers mathematics with applications 55, No. 6, 1270-1278 (2008) · Zbl 1157.15014 · doi:10.1016/j.camwa.2007.06.023 [14] Yuan, S.; Liao, A.; Lei, Y.: Least squares Hermitian solution of the matrix equation (AXB,CXD)=(E,F) with the least norm over the skew field of quaternions, Mathematical and computer modelling 48, No. 1–2, 91-100 (2008) · Zbl 1145.15303 · doi:10.1016/j.mcm.2007.08.009 [15] Wang, D. Q.; Ding, F.: Extended stochastic gradient identification algorithms for Hammerstein–Wiener ARMAX systems, Computers mathematics with applications 56, No. 12, 3157-3164 (2008) · Zbl 1165.65308 · doi:10.1016/j.camwa.2008.07.015 [16] Ding, J.; Ding, F.: The residual based extended least squares identification method for dual-rate systems, Computers mathematics with applications 56, No. 6, 1479-1487 (2008) · Zbl 1155.93435 · doi:10.1016/j.camwa.2008.02.047 [17] Han, L. L.; Ding, F.: Multi-innovation stochastic gradient algorithms for multi-input multi-output systems, Digital signal processing 19, No. 4, 545-554 (2009) [18] Dehghan, M.; Hajarian, M.: An iterative algorithm for solving a pair of matrix equations AYB=E,CYD=F over generalized centro-symmetric matrices, Computers mathematics with applications 56, No. 12, 3246-3260 (2008) · Zbl 1165.15301 · doi:10.1016/j.camwa.2008.07.031 [19] Cai, J.; Chen, G.: An iterative algorithm for the least squares bisymmetric solutions of the matrix equations A1XB1=C1 and A2XB2=C2, Mathematical and computer modelling 50, No. 7–8, 1237-1244 (2009) · Zbl 1190.65061 · doi:10.1016/j.mcm.2009.07.004 [20] Ding, F.; Chen, T.: Iterative least squares solutions of coupled Sylvester matrix equations, Systems control letters 54, No. 2, 95-107 (2005) · Zbl 1129.65306 · doi:10.1016/j.sysconle.2004.06.008 [21] Ding, F.; Chen, T.: On iterative solutions of general coupled matrix equations, SIAM journal on control and optimization 44, No. 6, 2269-2284 (2006) · Zbl 1115.65035 · doi:10.1137/S0363012904441350 [22] Ding, F.; Chen, T.: Gradient based iterative algorithms for solving a class of matrix equations, IEEE transactions on automatic control 50, No. 8, 1216-1221 (2005)