zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
New matrix iterative methods for constraint solutions of the matrix equation $AXB=C$. (English) Zbl 1206.65145
Summary: Two new matrix iterative methods are presented to solve the matrix equation $AXB=C$, the minimum residual problem $\min_{X\in\cal S}\|AXB-C\|$ and the matrix nearness problem $\min_{X\in S_E}\|X-X^*\|$, where $\cal S$ is the set of constraint matrices, such as symmetric, symmetric $R$-symmetric and $(R,S)$-symmetric, and $S_E$ is the solution set of above matrix equation or minimum residual problem. These matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in [{\it Y.-B. Deng} et al., Numer. Linear Algebra Appl. 13, No. 10, 801--823 (2006; Zbl 1174.65382); {\it G.-X. Huang} et al., J. Comput. Appl. Math. 212, No. 2, 231--244 (2008; Zbl 1146.65036); the author, Appl. Math. Comput. 170, No. 1, 711--723 (2005; Zbl 1081.65039); and {\it Y. Lei} and {\it A. Liao}, Appl. Math. Comput. 188, No. 1, 499--513 (2007; Zbl 1131.65038)]. Paige’s algorithm is used as the frame method for deriving these matrix iterative methods. Numerical examples are used to illustrate the efficiency of these new methods.

65F30Other matrix algorithms
Full Text: DOI
[1] Trench, William F.: Hermitian, Hermitian R-symmetric, and Hermitian R-skew symmetric procrustes problems, Linear algebra appl. 387, 83-98 (2004) · Zbl 1121.15016
[2] Peng, Z. Y.; Hu, X. Y.: The reflexive and anti-reflexive solutions of the matrix equation AX=B, Linear algebra appl. 375, 147-155 (2003) · Zbl 1050.15016 · doi:10.1016/S0024-3795(03)00607-4
[3] Trench, William F.: Minimization problem for (R,S)-symmetric and (R,S)-skew symmetric matrices, Linear algebra appl. 389, 23-31 (2004) · Zbl 1059.15019 · doi:10.1016/j.laa.2004.03.035
[4] Dai, H.: On the symmetric solutions of linear matrix equations, Linear algebra appl. 131, 1-7 (1990) · Zbl 0712.15009 · doi:10.1016/0024-3795(90)90370-R
[5] Dai, H.: On the symmetric solutions of linear matrix equations, Linear algebra appl. 39, 61-72 (1981)
[6] Chu, K. E.: Symmetric solutions of linear matrix equations by matrix decompositions, Linear algebra appl. 119, 35-50 (1989) · Zbl 0688.15003 · doi:10.1016/0024-3795(89)90067-0
[7] Liao, A. P.; Lei, Y.: Optimal approximate solution of the matrix equation AXB=C over symmetric matrices, J. comput. Math. 25, 543-552 (2007) · Zbl 1140.65323
[8] Liao, A. P.; Bai, Z. Z.: Least-squares solution of AXB=D over symmetric positive semidefinite matrices X, J. comput. Math. 21, 175-182 (2003) · Zbl 1029.65042
[9] Liao, A. P.; Bai, Z. Z.: Least-squares solutions of the matrix equation a\astxa=D in bisymmetric matrix set, Math. numer. Sinica 24, 9-20 (2002)
[10] Liao, A. P.; Bai, Z. Z.: The constrained solutions of two matrix equations, Acta math. Sinica (English ser.) 18, 671-678 (2002) · Zbl 1028.15011
[11] Peng, Z. Y.: The centro-symmetric solutions of linear matrix equation AXB=C and its optimal approximation, J. eng. Math. 6, 60-64 (2003)
[12] Wang, Q. W.; Yang, C. L.: The re-nonnegative definite solutions to the matrix equation AXB=C, Comment. math. Univ. carolinae 39, 7-13 (1998) · Zbl 0937.15008
[13] Deng, Y. B.; Bai, Z. Z.; Gao, Y. H.: Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. linear algebra appl. 13, 801-823 (2006) · Zbl 1174.65382 · doi:10.1002/nla.496
[14] Peng, X. Y.; Hu, X. Y.; Zhang, L.: An iteration method for the symmetric solutions and the optomal approximation solution of matrix equation AXB=C, Appl. math. Comput. 160, 763-777 (2005) · Zbl 1068.65056 · doi:10.1016/j.amc.2003.11.030
[15] Huang, G. X.; Yin, F.; Guo, K.: An iterative method for the skew-symmetic solution and the optimal approximate solution of the matrix equation AXB=C, J. comput. Appl. math. 212, 231-244 (2008) · Zbl 1146.65036 · doi:10.1016/j.cam.2006.12.005
[16] Peng, Z. Y.: An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C, Appl. math. Comput. 170, 711-723 (2005) · Zbl 1081.65039 · doi:10.1016/j.amc.2004.12.032
[17] Lei, Y.; Liao, A. P.: A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. math. Comput. 188, 499-513 (2007) · Zbl 1131.65038 · doi:10.1016/j.amc.2006.10.011
[18] Paige, C. C.: Bidiagonalization of matrices and solution of linear equation, SIAM J. Numer. anal. 11, 197-209 (1974) · Zbl 0244.65023 · doi:10.1137/0711019
[19] Golub, G. H.; Kahan, W.: Calculating the singular values and pseudoinverse of a matrix, SIAM J. Numer. anal. 2, 205-224 (1965) · Zbl 0194.18201