Journal of Applied Mathematics

Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

Xiaomin Duan, Huafei Sun, and Xinyu Zhao

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A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equation Q = X + i = 1 m A i T X A i . In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method.

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J. Appl. Math., Volume 2014 (2014), Article ID 507175, 7 pages.

First available in Project Euclid: 26 March 2014

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Duan, Xiaomin; Sun, Huafei; Zhao, Xinyu. Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations. J. Appl. Math. 2014 (2014), Article ID 507175, 7 pages. doi:10.1155/2014/507175.

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  • M. Berzig, “Solving a class of matrix equations via the Bhaskar-Lakshmikantham coupled fixed point theorem,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1638–1643, 2012.
  • J. C. Engwerda, “On the existence of a positive definite solution of the matrix equation $X+{A}^{T}{X}^{-1}A=I$,” Linear Algebra and Its Applications, vol. 194, pp. 91–108, 1993.
  • W. L. Green and E. W. Kamen, “Stabilizability of linear systems over a commutative normed algebra with applications to spatially-distributed and parameter-dependent systems,” SIAM Journal on Control and Optimization, vol. 23, no. 1, pp. 1–18, 1985.
  • W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,” Reports on Mathematical Physics, vol. 8, no. 2, pp. 159–170, 1975.
  • C.-Y. Chiang, E. K.-W. Chu, and W.-W. Lin, “On the $\bigstar $-Sylvester equation $AX\pm {X}^{\bigstar }{B}^{\bigstar }=C$,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8393–8407, 2012.
  • Z. Gajic and M. T. J. Qureshi, Lyapunov Matrix Equation in System Stability and Control, vol. 195, Academic Press, London, UK, 1995.
  • A. C. M. Ran and M. C. B. Reurings, “The symmetric linear matrix equation,” Electronic Journal of Linear Algebra, vol. 9, pp. 93–107, 2002.
  • M. C. B. Reurings, Symmetric matrix equations [Ph.D. thesis], Vrije Universiteit, Amsterdam, The Netherlands, 2003.
  • A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004.
  • Y. Su and G. Chen, “Iterative methods for solving linear matrix equation and linear matrix system,” International Journal of Computer Mathematics, vol. 87, no. 4, pp. 763–774, 2010.
  • X. Duan, H. Sun, L. Peng, and X. Zhao, “A natural gradient descent algorithm for the solution of discrete algebraic Lyapunov equations based on the geodesic distance,” Applied Mathematics and Computation, vol. 219, no. 19, pp. 9899–9905, 2013.
  • X. Duan, H. Sun, and Z. Zhang, “A natural gradient descent algorithm for the solution of Lyapunov equations based on the geodesic distance,” Journal of Computational Mathematics, vol. 219, no. 19, pp. 9899–9905, 2013.
  • M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1, Publish or Perish, Berkeley, Calif, USA, 3rd edition, 1999.
  • S. Lang, Fundamentals of Differential Geometry, vol. 191, Springer, 1999.
  • F. Barbaresco, “Interactions between symmetric cones and information geometrics: Bruhat-tits and Siegel spaces models for high resolution autoregressive Doppler imagery,” in Emerging Trends in Visual Computing, vol. 5416 of Lecture Notes in Computer Science, pp. 124–163, 2009.
  • M. Moakher, “A differential geometric approach to the geometric mean of symmetric positive-definite matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 3, pp. 735–747, 2005.
  • M. Moakher, “On the averaging of symmetric positive-definite tensors,” Journal of Elasticity, vol. 82, no. 3, pp. 273–296, 2006.
  • A. Schwartzman, Random ellipsoids and false discovery rates: statistics for diffusion tensor imaging data [Ph.D. thesis], Stanford University, 2006.
  • J. D. Lawson and Y. Lim, “The geometric mean, matrices, metrics, and more,” The American Mathematical Monthly, vol. 108, no. 9, pp. 797–812, 2001.
  • R. Bhatia, Positive definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, USA, 2007.
  • C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, vol. 297 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, Germany, 1994.
  • M. P. D. Carmo, Riemannian Geometry, Springer, 1992.
  • D. G. Luenberger, “The gradient projection method along geodesics,” Management Science, vol. 18, pp. 620–631, 1972. \endinput