Introduction to functional analysis, 2nd ed.
Introduction to functional analysis, 2nd ed.
The algebraic eigenvalue problem
The algebraic eigenvalue problem
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
Journal of Computational and Applied Mathematics
International Journal of Computer Mathematics
| Hi-index | 7.29 |
In this paper, the differential calculus for the operator norms || ċ ||p, p ∈ {1,2, ∞}, of the fundamental matrix or evolution φ(t) = e At, t ≥ 0, of a complex n × n matrix A, introduced by the author in a former paper, is extended to m times continuously differentiable matrix functions χ(t), t ≥ 0, and developed further for other p-norms | ċ |p 1 p t) are obtained. In addition, for this function Φ(t), formulae for the first two logarithmic derivatives D+1|Φ(0)|p and D+2|Φ(0)|p, 1 p t), t ≥ 0 (that is, a matrix power function) and on the difference (or remainder) R(t) = Φ(t) - Ψ(t), t ≥ 0, are derived. The discrete evolution occurs when a step-by-step method is employed to approximate the exact solution of the initial-value problem x(t) = Ax(t), x(0)= x0, which here models a vibration problem. The results are applied to the computation of the optimal upper bounds on ||R(t)||∞, ||R(t)||2, and |R(t)|2.