The weighted logarithmic matrix norm and bounds of the matrix exponential. (English) Zbl 1060.15024

Hu, Guang-Da; Liu, Mingzhu

Linear Algebra Appl. 390, 145-154 (2004).

Given a symmetric positive definite matrix \(H\), the authors consider the “weighted” inner product \[ (x,y)_{(H)}:=y^THx. \] Then they consider the corresponding \(H\)-weighted vector and matrix norms, namely \[ \| x\| _{(H)}:=\sqrt{x^THx},\;\| A\| _{(H)}:=\max_{x\in R^n,x\neq0}\,\frac{\| A\| _{(H)}}{\| x\| _{(H)}}, \] and the “weighted logarithmic norm” (which is not a norm!) \[ \mu_{(H)}[A]:=\max_{x\neq0}\frac{(Ax,x)_{(H)}}{\| x\| _{(H)}^2}. \] The paper presents some results for \(\mu_{(H)}\) regarding bounds of the matrix exponential \(e^{At}\), under the requirement that the weight \(H\) satisfies the Lyapunov equation \(A^TH+HA=-2I\). In the last section, the authors present numerical evidence that their bounds improve the bounds that can be obtained through the usual norms, in the case of stable matrices (that is, matrices where the real part of each of the eigenvalues is negative).

Reviewer: Martín Argerami (Regina)

MSC:

15A60	Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A24	Matrix equations and identities
15A63	Quadratic and bilinear forms, inner products

Keywords:

weighted logarithmic matrix norm; Lyapunov matrix equation; matrix exponential; weighted inner product

References:

[1]	Bellen, A.; Guglielmi, N.; Ruehli, A. E., Methods for linear systems of circuit delay differential equations of neutral type, IEEE Trans. Circuits Systems I, 46, 212-216 (1999) · Zbl 0952.94015
[2]	Brunner, H., A note on modified optimal linear methods, Math. Comp, 26, 625-631 (1972) · Zbl 0262.65044
[3]	Dekker, K.; Verwer, J. G., Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations (1984), North-Holland: North-Holland Amsterdam · Zbl 0571.65057
[4]	Desoer, C. A.; Vidyasagar, M., Feedback Systems: Input-Output Properties (1975), Academic Press: Academic Press New York · Zbl 0327.93009
[5]	Fang, Y.; Loparo, K. A.; Feng, X., New estimates for solutions of Lyapunov equations, IEEE. Trans. Automat. Control, 42, 408-411 (1997) · Zbl 0866.93048
[6]	Hu, G.-D.; Cahlon, B., Estimations on numerically stable step-size for neutral delay differential systems with multiple delays, J. Comput. Appl. Math, 102, 221-234 (1999) · Zbl 0948.65080
[7]	Hu, G.-D.; Hu, G. D., A relation between the weighted logarithmic norm of matrix and Lyapunov equation, BIT, 40, 506-510 (2000)
[8]	Kågström, B., Bounds and perturbation bounds for the matrix exponential, BIT, 17, 39-57 (1977) · Zbl 0356.65034
[9]	Lancaster, P., The Theory of Matrices with Applications (1985), Academic Press, Inc: Academic Press, Inc Orlando · Zbl 0558.15001
[10]	Ström, T., On logarithmic norms, SIAM J. Numer. Anal, 12, 741-753 (1975) · Zbl 0321.15012