Methods for the approximation of the matrix exponential in a Lie-algebraic setting (Q2725335)

The authors derive \(p\)th-order approximation formulas for exponentials representing elements of a Lie group which may be written in canonical coordinates of the second kind (SKC). Given a Lie group \(G\subseteq \text{GL}(n)\) with associated Lie algebra \(g\subseteq \text{gl}(n)\) if \(B= \sum^n_{i=1} \beta_i V_i\) where \(V_i\) is a basis for \(g\) then for some \(g_i(t)\), NEWLINE\[NEWLINE\exp(tB)= \prod^n_{i=1} g_i(t) V_i.\tag{i}NEWLINE\]NEWLINE The authors approximate \(\exp(tB)\) by approximating \(g_i(t)\) by Taylor polynomials NEWLINE\[NEWLINE\alpha_i(t)= \sum^p_{j=0} (j!)^{-1} g^{(j)}_i(0) t^j.NEWLINE\]NEWLINE Explicit formulas for determining \(g^{(j)}_i(0)\) are obtained by differentiating both sides of (i) with respect to \(t\). The \(g^{(j)}_i(0)\) are computable if the basis matrices are sufficiently sparse, \(p\leq 4\). They are determined explicitly for the orthogonal group, the special linear group, and the Lorenz group choosing appropriate bases. A time symmetric approximation to \(\exp(tB)\) is given by NEWLINE\[NEWLINE\prod^n_{i=1} \alpha^{(2k)}_i(t) V_i \prod^{n-1}_{i=1}\alpha^{(2k)}_{n-i}(t) V_{n-i},\tag{ii}NEWLINE\]NEWLINE where \(\alpha^{(2k)}_i(t)\) are odd polynomials with \(k\) terms. A recursive method to obtain \(\alpha^{(2j)}_i\) from \(\alpha^{(2j- 2)}_i\) is described which leads to a \((2k+2)\) order approximation to \(\exp(tB)\). The method is worked out explicitly for \(k=1\) for the special orthogonal group \(\text{SR}(n)\) and requires \(11.5 n^3\) flops.NEWLINENEWLINENEWLINEResults of a computation to solve a \((30,30)\) matrix differential equation for an isospectral flow are presented using the \(k=1\) algorithm. The cost was found to be comparable to the cost of using a Cayley map.