Extension and further development of the differential calculus for matrix norms with applications. (Q1398715)

In an earlier work [J. Comput. Appl. Math. 135, 1--21 (2001; Zbl 1025.34004)], the author has discussed differential calculus for the operator norms \(\| \cdot \|_p\), \(p\in \{ 1, 2, \infty\}\), of the fundamental matrix \(\Phi(t)=e^{At}\), \(t\geq0\), where \(A\) is a complex \(n\times n\) matrix. In this paper, he extends the earlier study to \(m\) times continuously differentiable matrix functions \(\chi(t)\), \(t>0\), and other \(p\)-norms \(| \cdot |_p\), \(1<p<\infty\). In particular, formulae for the first two logarithmic derivatives \(D^1_+|\Phi(0)|_p\) and \(D^2_+|\Phi(0)|_p\), \(1<p<\infty\), of the function \(\Phi(t)\) are derived. Further, upper bounds on the discrete evolution \(\Psi(t)\), \(t\geq0\) and on the difference \(R(t)=\Phi(t)-\Psi(t)\), \(t\geq0\), are derived. The results obtained are applied to the computation of the optimal upper bounds on \(\| R(t)\|_{\infty}\), \(\| R(t)\|_2\) and \(| R(t)|_2\).