On The Computation Of The Multivariable Zassenhaus Formula

The celebrated BCH formula and Zassenhaus formula play a fundamental role in many fields of mathematics and physics.Many researchers concern with the computation of the BCH formula and Zassenhaus formula over the years.Since many problems often involve the exponent of the sum of multiple variables in mathematics and physics,then we try to study the multivariable Zassenhaus formula.The thesis mainly studies the computation of the multivariable Zassenhaus formula with two different methods.And the thesis includes two parts as follows:First,we extend the matrix method of Scholz and Weyrauch to the multivariable Zassenhaus formula,and give a recursion formula of it with matrices.Furthermore,we also give the recursion formula of the symmetric multivariable Zassenhaus formula with matrices by using the same matrix method.Meanwhile,two computing examples are given respectively.From the concrete computations,it can be found that with the increase of the order of the multivariable Zassenhaus exponent,our computations become more complicated.So it's not efficient for us to obtain the commutator representation of the associated exponent.Second,we present a new efficient recursive algorithm.We firstly give the recursion formula of the multivariable Zassenhaus exponent Wk based on the method of Casas,Murua and Nadinic.And we give a concrete procedure to compute Wk(k=1,2,3,4,5).We find that the results of f1,k(k?1)are related to the partition of integer by analyzing these computing results.Then we establish a new combinatorial formula of f1,k(k?1).Moreover,we show that our formula can give a slightly better recursion formula of Wk when k?6 by further computations.These equations clearly show that Wk(k?6)can be expressed as a linear combination of f1,k(k?1)in the end,so we can use the new combinatorial formula of f1,k(k?1)to obtain Wk(k?6).It's easy to find that this method can be used to obtain the explicit commutator representation of the associated exponent Wk quickly for small integer k.Finally,we give the concrete computations of W6 and W7 by using this method.