Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Logarithmic Norms for Matrix Pencils
SIAM Journal on Matrix Analysis and Applications
Second Logarithmic Derivative of a Complex Matrix in the Chebyshev Norm
SIAM Journal on Matrix Analysis and Applications
Differential calculus for some p-norms of the fundamental matrix with applications
Journal of Computational and Applied Mathematics
How Close Can the Logarithmic Norm of a Matrix Pencil Come to the Spectral Abscissa?
SIAM Journal on Matrix Analysis and Applications
Differential calculus for p-norms of complex-valued vector functions with applications
Journal of Computational and Applied Mathematics
Extension and further development of the differential calculus for matrix norms with applications
Journal of Computational and Applied Mathematics
Higher Order Logarithmic Derivatives of Matrices in the Spectral Norm
SIAM Journal on Matrix Analysis and Applications
| Hi-index | 7.29 |
The new idea is to study the stability behavior of the solution x=x(t) of the initial value problem x@?=Ax,t=t"0,x(t"0)=x"0, with A@?C^n^x^n, in a weighted (semi-) norm @?.@?"R where R is taken as an appropriate solution of the matrix eigenvalue problemRA+A^*R=@rR, rather than as the solution of the algebraic Lyapunov matrix equationRA+A^*R=-S with given positive (semi-) definite matrix S. Substantially better results are obtained by the new method. For example, if A is diagonalizable and all eigenvalues @l"i(A) have negative real parts, i.e., Re@l"i(A)0(t-~), which is much more than the old result, which only states that lim"t"-"~x(t)=0. Especially, the semi-norms @?.@?"R"""i have a decoupling and filter effect on x(t). Further, new two-sided bounds (depending on x"0) for the asymptotic behavior can be derived. The best constants in the bounds are obtained by the differential calculus of norms. Applications are made to free linear dynamical systems, and computations underpin the theoretical findings.