Abstract
We say that \(X=[x_{ij}]_{i,j=1}^n\) is symmetric centrosymmetric if x ij = x ji and x n − j + 1,n − i + 1, 1 ≤ i,j ≤ n. In this paper we present an efficient algorithm for minimizing ||AXA T − B|| where ||·|| is the Frobenius norm, A ∈ ℝm×n, B ∈ ℝm×m and X ∈ ℝn×n is symmetric centrosymmetric with a specified central submatrix [x ij ]p ≤ i,j ≤ n − p. Our algorithm produces a suitable X such that AXA T = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bai, Z.J.: The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation. SIAM J. Matrix Anal. Appl. 26, 1100–1114 (2005)
Baruch, M.: Optimization procedure to correct stiffness and flexibility matrices using vibration tests. AIAA J. 16, 1208–1210 (1978)
Berman, A., Nagy, E.J.: Improvement of a large analytical model using test data. AIAA J. 21, 1168–1173 (1983)
Baksalary, J.K.: Nonnegative definite and positive solutions to the matrix equation AXA * = B. Linear Multilinear Algebra 16, 133–139 (1984)
Cantoni, A., Butler, P.: Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl. 13, 275–288 (1976)
Cantoni, A., Butler, P.: Properties of the eigenvectors of persymmetric matrices with applications to communication theory. IEEE Trans. Commun. COM 24(8), 804–809 (1976)
Dai, H., Lancaster, P.: Linear matrix equations from an inverse problem of vibration theory. Linear Algebra Appl. 246, 31–47 (1996)
Datta, L., Morgera, S.: On the reducibility of centrosymmetric matrices—applications in engineering problems. Circuits Syst. Signal Process. 8, 71–96 (1989)
Friswell, M.I., Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics. Kluwer Academic, Dordrecht (1995)
Gersho, A.: Adaptive equalization of highly dispersive channels for data transmission. BSTJ 48, 55–70 (1969)
Hochstadt, H.: On the construction of a Jacobi matrix from mixed given data. Linear Algebra Appl. 28, 113–115 (1979)
Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)
Link, M.: Identification and correction of errors in analytical models using test data-theoretical and practical bounds. In: Proceedings of the Eighth International Modal Analysis Conference, pp. 570–578 (1990)
Luber, W., Lotze, A.: Application of sensitivity methods for error localization in finite element systems. In: Proceedings of the Eighth International Modal Analysis Conference, pp. 598–604 (1990)
O’Callahan, J.C., Chou, C.-M.: Localization of model errors in optimized mass and stiffness matrices using modal test data. Int. J. Anal. Exp. Anal. 4, 8–14 (1989)
Peng, Z.Y., Hu, X.Y., Zhang, L.: The inverse problem of symmetric centrosymmetric matrices with a submatrix constraint. Numer. Linear Algebra Appl. 11, 59–73 (2004)
Peng, Z.Y., Hu, X.Y.: Constructing Jacobi matrix with prescribed ordered defective eigenpairs and a principal submatrix. J. Comput. Appl. Math. 175, 321–333 (2005)
Liao, A.P., Lei, Y.: Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint. Numer. Linear Algebra Appl. 14, 425–444 (2007)
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)
Lancaster, P.: Explicit solutions of linear matrix equation. SIAM Rev. 72, 544–566 (1970)
Rojo, O., Rojo, H.: Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices. Linear Algebra Appl. 392, 211–233 (2004)
Tao, D., Yasuda, M.: A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric centroskew matrices. SIAM J. Matrix Anal. Appl. 23, 885–895 (2002)
Weaver, J.: Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors. Am. Math. Mon. 92, 711–717, (1985)
Wang, Q.W.: Symmetric centrosymmetric and centrosymmetric solutions to systems of real quaternion matrix equations. Comput. Math. Appl. 49, 641–650 (2005)
Wang, R.S.: Functwnal Analysis and Optimization Theory. Beljing Unlv of Aeronautics Astronautics Press, Beijing (2003)
Wei, F.S.: Stiffness matrix correction from incomplete test data. AIAA J. 18, 1274–1275 (1980)
Wei, F.S.: Mass and stiffness interaction effects in analytical model modification. AIAA J. 28, 1686–1688 (1990)
Yuan, Y.X., Dai, H.: Inverse problems for symmetric matrices with a submatrix constraint. Appl. Numer. Math. 57, 646–656 (2007)
Yuan, Y.X., Dai, H.: The nearness problems for symmetric matrix with a submatrix constraint, J. Comput. Appl. Math. 213, 224–231 (2008)
Zhang, O., Zerva, A., Zhang, D.W.: Stiffness matrix adjustment using incomplete measured modes. AIAA J. 35, 917–919 (1997)
Zhao, L.J., Hu, X.Y., Zhang, L.: Least squares solutions to AX = B for symmetric centrosymmetric matrices under a central principal submatrix constraint and the optimal approximation. Linear Algebra Appl. 428, 871–880 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by the National Natural Science Foundation of China (Grant No. 10571047) and Doctorate Foundation of the Ministry of Education of China (Grant No. 20060532014).
Rights and permissions
About this article
Cite this article
Li, JF., Hu, XY. & Zhang, L. The nearness problems for symmetric centrosymmetric with a special submatrix constraint. Numer Algor 55, 39–57 (2010). https://doi.org/10.1007/s11075-009-9356-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-009-9356-2
Keywords
- Symmetric centrosymmetric matrix
- Submatrices constraint
- Iterative method
- Model updating
- Perturbation analysis