Higher order alternate directions methods
Let \(f\) be a continuous function from a neighborhood of real \(t = 0\) to the vector space \(\mathcal M\) of square matrices, \(f(t,a;b) = \text{exp} (a_ 1 tA) \cdot \text{exp} (b_ 1 tB) \cdot \dots \cdot \text{exp} (a_ k tA) \cdot \text{exp} (b_ k tB)\), \(A \in {\mathcal M}\), \(B \in {\mathcal M}\), \(a_ j\), \(b_ j\) real parameters. Alternate direction methods are based on the construction of a function \(f\) with \(f(t/n)^ n - \text{exp} (t (A+B)) = O(n^{-p})\). \(k = 1\), \(a_ 1 = b_ 1 = 1\) is the well-known Lie formula with the order \(p = 1\). \textit{G. Strang} [Arch. Rat. Mech. Anal. 12, 392-402 (1963; 113, 323), SIAM J. Numer. Anal. 5, 506-517 (1968; Zbl 0184.385)] has given a formula with order \(p = 2 : k = 2\), \(a_ 1 = a_ 2 = 1/2\), \(b_ 1 = 1\), \(b_ 2 = 0\). But \textit{Q. Sheng} [IMA J. Numer. Anal. 9, No. 2, 199- 212 (1989; Zbl 0676.65116)] has pointed out that (in the non-commutative case) no \(k\) exists with \(a_ j \geq 0\), \(b_ j \geq 0\) and order \(p \geq 3\). Hence, the search of more accurate product formulas is negative. However, there exist formulas of third order with (partially) negative coefficients, e.g. \(k = 3\) with \(a_ 1 = 1\), \(a_ 2 = -a_ 3 = -2/3\), \(b_ 1 = -1/24\), \(b_ 2 = 3/4\), \(b_ 3 = 7/24\). The author studies linear combinations of product formulas which yield order \(p = 4\), e.g. the result of \textit{S. Z. Burstein, A. A. Mirin} [J. Comput. Phys. 5, 547- 571 (1970; Zbl 0223.65053)] with the use of \(2/3 f(t, 1/2, 1/2; 1, 0) + 2/3 f(t,1, 0; 1/2, 1/2) - 1/6 f(t, 1, 0; 1, 0) - 1/6 f(t,0, 1; 0, 1)\) (this means \(k = 2\)), or a new result with \(k = 3\) obtained by Richardson's extrapolation: \(4/3 f(t, 1/4, 1/2, 1/4; 1/2, 1/2, 0) - 1/3 f(t, 1/2, 1/2, 0; 1, 0, 0).\) There are only few numerical experiments. But the method (based on a Campbell-Baker-Dynkin-Hausdorff formal series) is a promising one, of course, it requires care and, probably, the use of computer algebra.
Campbell-Baker-Dynkin-Hausdorff formal series
scientific article; zbMATH DE number 764276
scientific article