Fast computation of the Baker-Campbell-Hausdorff and similar series

We provide an efficient C program for the fast computation of the Baker-Campbell-Hausdorff (BCH) and similar Lie series. The Lie series can be represented in the Lyndon basis, in the classical Hall basis, or in the right-normed basis of E.S. Chibrikov. In the Lyndon basis, which proves to be particularly efficient for this purpose, the computation of 111013 coefficients for the BCH series up to terms of degree 20 takes less than half a second on an ordinary personal computer and requires negligible 11MB of memory. Up to terms of degree 30, which is the maximum degree the program can handle, the computation of 74248451 coefficients takes 55 hours but still requires only a modest 5.5GB of memory.

Installation

On a Unix-like system with gcc compiler available just type

$ make

in a directory containing the source code, which causes the shared library libbch.so and the executable bch to be created. To use a different compiler, the Makefile has to be adapted accordingly.

Documentation

A user manual is available at arXiv:2102.06570.

Trying the program without installing it

The bch program has been compiled to WebAssembly. So you can try it in your browser without installing it. For example, to compute the Baker-Campbell-Hausdorff series represented in the Lyndon basis up to terms of degree N=20, just click on the following link:

http://www.harald-hofstaetter.at/BCH/bch.html?verbosity_level=1&N=20&basis=0

Parameters are specified as shown in this link. Their usage is exactly as described in the user manual for the command line version of the bch program.

Note that the actual computation is suprisingly fast. However, in the current version, the generation of the output and its transfer via JavaScript into HTML is not very efficient. This can certainly be improved by replacing the driver program bch.c by one specifically optimized for WebAssembly.

Examples

$ ./bch
+1/1*A+1/1*B+1/2*[A,B]+1/12*[A,[A,B]]+1/12*[[A,B],B]+1/24*[A,[[A,B],B]]-1/720*[A
,[A,[A,[A,B]]]]+1/180*[A,[A,[[A,B],B]]]+1/360*[[A,[A,B]],[A,B]]+1/180*[A,[[[A,B]
,B],B]]+1/120*[[A,B],[[A,B],B]]-1/720*[[[[A,B],B],B],B]

Output in tabular form:

$ ./bch table_output=1
0       1       0       0       1/1
1       1       1       0       1/1
2       2       0       1       1/2
3       3       0       2       1/12
4       3       2       1       1/12
5       4       0       3       0/1
6       4       0       4       1/24
7       4       4       1       0/1
8       5       0       5       -1/720
9       5       0       6       1/180
10      5       3       2       1/360
11      5       0       7       1/180
12      5       2       4       1/120
13      5       7       1       -1/720

Computation up to higher degree:

$ ./bch N=8
+1/1*A+1/1*B+1/2*[A,B]+1/12*[A,[A,B]]+1/12*[[A,B],B]+1/24*[A,[[A,B],B]]-1/720*[A
,[A,[A,[A,B]]]]+1/180*[A,[A,[[A,B],B]]]+1/360*[[A,[A,B]],[A,B]]+1/180*[A,[[[A,B]
,B],B]]+1/120*[[A,B],[[A,B],B]]-1/720*[[[[A,B],B],B],B]-1/1440*[A,[A,[A,[[A,B],B
]]]]+1/720*[A,[[A,[A,B]],[A,B]]]+1/360*[A,[A,[[[A,B],B],B]]]+1/240*[A,[[A,B],[[A
,B],B]]]-1/1440*[A,[[[[A,B],B],B],B]]+1/30240*[A,[A,[A,[A,[A,[A,B]]]]]]-1/5040*[
A,[A,[A,[A,[[A,B],B]]]]]+1/10080*[A,[A,[[A,[A,B]],[A,B]]]]+1/3780*[A,[A,[A,[[[A,
B],B],B]]]]+1/10080*[[A,[A,[A,B]]],[A,[A,B]]]+1/1680*[A,[A,[[A,B],[[A,B],B]]]]+1
/1260*[A,[[A,[[A,B],B]],[A,B]]]+1/3780*[A,[A,[[[[A,B],B],B],B]]]+1/2016*[[A,[A,B
]],[A,[[A,B],B]]]-1/5040*[[[A,[A,B]],[A,B]],[A,B]]+13/15120*[A,[[A,B],[[[A,B],B]
,B]]]+1/10080*[[A,[[A,B],B]],[[A,B],B]]-1/1512*[[A,[[[A,B],B],B]],[A,B]]-1/5040*
[A,[[[[[A,B],B],B],B],B]]+1/1260*[[A,B],[[A,B],[[A,B],B]]]-1/2016*[[A,B],[[[[A,B
],B],B],B]]-1/5040*[[[A,B],B],[[[A,B],B],B]]+1/30240*[[[[[[A,B],B],B],B],B],B]+1
/60480*[A,[A,[A,[A,[A,[[A,B],B]]]]]]-1/15120*[A,[A,[A,[[A,[A,B]],[A,B]]]]]-1/100
80*[A,[A,[A,[A,[[[A,B],B],B]]]]]+1/20160*[A,[[A,[A,[A,B]]],[A,[A,B]]]]-1/20160*[
A,[A,[A,[[A,B],[[A,B],B]]]]]+1/2520*[A,[A,[[A,[[A,B],B]],[A,B]]]]+23/120960*[A,[
A,[A,[[[[A,B],B],B],B]]]]+1/4032*[A,[[A,[A,B]],[A,[[A,B],B]]]]-1/10080*[A,[[[A,[
A,B]],[A,B]],[A,B]]]+13/30240*[A,[A,[[A,B],[[[A,B],B],B]]]]+1/20160*[A,[[A,[[A,B
],B]],[[A,B],B]]]-1/3024*[A,[[A,[[[A,B],B],B]],[A,B]]]-1/10080*[A,[A,[[[[[A,B],B
],B],B],B]]]+1/2520*[A,[[A,B],[[A,B],[[A,B],B]]]]-1/4032*[A,[[A,B],[[[[A,B],B],B
],B]]]-1/10080*[A,[[[A,B],B],[[[A,B],B],B]]]+1/60480*[A,[[[[[[A,B],B],B],B],B],B
]]

Output using right-normed Chibrikov basis:

$ ./bch basis=1
+1/1*A+1/1*B-1/2*[B,A]-1/12*[A,[B,A]]+1/12*[B,[B,A]]+1/24*[B,[A,[B,A]]]+1/720*[A
,[A,[A,[B,A]]]]-1/360*[B,[A,[A,[B,A]]]]+1/120*[A,[B,[A,[B,A]]]]-1/120*[B,[B,[A,[
B,A]]]]+1/360*[A,[B,[B,[B,A]]]]-1/720*[B,[B,[B,[B,A]]]]

Compute symmetric BCH formula:

$ ./bch expression=1
+1/1*A+1/1*B-1/24*[A,[A,B]]+1/12*[[A,B],B]+7/5760*[A,[A,[A,[A,B]]]]-7/1440*[A,[A
,[[A,B],B]]]+1/360*[[A,[A,B]],[A,B]]+1/180*[A,[[[A,B],B],B]]+1/120*[[A,B],[[A,B]
,B]]-1/720*[[[[A,B],B],B],B]

Compute Lie series for more general expressions

$ ./bch "expression=log(exp(3/8*A)*exp(4/5*B)*exp(5/8*A)*exp(1/5*B))"
+1/1*A+1/1*B-1/96*[A,[A,B]]+1/30*[[A,B],B]+1/256*[A,[A,[A,B]]]-3/320*[A,[[A,B],B
]]+1/200*[[[A,B],B],B]+163/368640*[A,[A,[A,[A,B]]]]-43/23040*[A,[A,[[A,B],B]]]+1
9/23040*[[A,[A,B]],[A,B]]+67/28800*[A,[[[A,B],B],B]]+1/384*[[A,B],[[A,B],B]]-13/
18000*[[[[A,B],B],B],B]

$ ./bch "expression=log(exp(A)*exp(-1/24*B)*exp(-2/3*A)*exp(3/4*B)*exp(2/3*A)*ex
p(7/24*B))"
+1/1*A+1/1*B+1/216*[A,[A,[A,B]]]-1/72*[A,[[A,B],B]]+5/2304*[[[A,B],B],B]+1/6480*
[A,[A,[A,[A,B]]]]-1/1620*[A,[A,[[A,B],B]]]+13/6480*[[A,[A,B]],[A,B]]-71/69120*[A
,[[[A,B],B],B]]-37/34560*[[A,B],[[A,B],B]]-53/207360*[[[[A,B],B],B],B]

$ ./bch "expression=log(exp(1/6*B)*exp(1/2*A)*exp(2/3*B+1/72*[B,[A,B]])*exp(1/2*
A)*exp(1/6*B))" 
+1/1*B+1/1*A-41/155520*[B,[B,[B,[B,A]]]]+7/12960*[B,[B,[[B,A],A]]]-1/3240*[[B,[B
,A]],[B,A]]-7/8640*[B,[[[B,A],A],A]]-1/2880*[[B,A],[[B,A],A]]+1/2880*[[[[B,A],A]
,A],A]