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              ∈ℝ 
(Strictly it is wrong. For example  ∘       ∘         ≠ ). But I
assumed the uniqueness of

 

 composition for convenience of notation.)

 

   
  



 

         ∈ℤ 


      ∈ℤ∈ℕ  ⇒
 

    lim 

 




 


 

 

  →∞

 ∈ℝ  ∞   

     lim    
→ ∞

*Definition of iexp(iterative exponential), ilog(iterative logarithm)
   is a transformation of (real valued)function such that
′     ⇒       ∘ .
   is a inverse transformation of   .
    
   
         lim 


→ 
          (Similarly to Matrix exponential, logarithm case, it
hold only for some  .)

 ×      ,     ×   (where ∈ℝ )
Examples)

 
 
  
    ≠

      


  


          ≠ 
 







    




          ≠








    
        




  



      
   
   

*Series form of iexp, ilog (I don't know necessity and sufficiency
condition for convergence.)



         ′   ″ ′    ″′  ″′ ′  









∇  (where ∇    ×′










∇   ∇   ∇ ∇    ∀∈ℕ )



(Derived from           ′     ″     ″′         ∘   )




         ∇      



   ∇   ∆  ?


(where ∆    ∇   ∆     ∆    ∆  ∆      ∀∈ℕ )

     
   
 ∀∈ℕ



 

  


*Compositional Calculus
    ℝ × ℝ→ℝ       ∀∈ℝ∀∈ℝ
        
  ∃      




        ∘     ∘  ∘    ∘  
        ∘    ∘  ∘    ∘  
 


 

∘        
      lim  


 
→ ∞
 




     lim 
→ ∞

     lim

→∞


 


  

   

    ∘ 
∘  




 

 

   

 lim
→∞

 



   

  
   


  
  
     ×       ∘          






       lim   
→ ∞

 
 

     
 






∆
lim   





∆ → 

                ∆        ∈      



        lim    
→∞

 
 

   
 




∆   

lim     

∆  →





 

     

 lim     
→ ∞

 
 

     
 






∆  
lim       




∆  → 





    

       ,        ,
      
        , 
  
        ∘  ,
                
  


 

 

 


 







  


 





  
  ∘    ,


  


  





 








 







*Lie Product formula
        ≡ lim 


 


∘


 
  



→ ∞

*Baker-Campbell-Hausdorff formula


   ∘                   



*Adjoint formula(?)
    ′ ×   ∘     ∆   ∘    ,        ′  ′  ∆    ∇  
Be careful!    ∆    ∇   It is different from lie bracket of vector fields.




                       





(where        



  ∀∈ℕ 





            )



* ′    
Let  is continuous on     and differentiable on    . (    satisfies the
same

condition

and

    ∘          ∘    

  ℝ→ℝ   ℝ→ℝ).)
Let  ' is continuous on     and y'' exists on    .

        ′          ∘     

(where

         ∘            ∘     
∵    ∘          ∘     ∘          ∘     ∘      


 


         



 

∘             ∘     







If  is invertible,     lim     .
→∞

*Generalized Magnus expansion(derived only 1, 2 term)
If we use Baker-Campbell-Hausdorff formula.
          ∘  ∘     ∘   

                                                       









 

  

  



 


    
            
  

       
      
 





 

Let ∈ℚ  .



 

 

 
      



 lim 





   



 




       

→∞   

    







→∞




 

  

      
      





 
lim     
 


 
  



     

  









We can guess it holds for ∈ℝ.
Also we can guess the following(Magnus expansion holds for all lie group.).



 




        





  
  

  






  

  





 
  



    





  

              







  

We want to find approximate solution of following ordinary differential equation.
′        
Let  ≈ .
    ′                   ∘     
(Composite by  variable.)
Similarly,
      ∘            ∘     ∘     ,

         ∘                 ∘     

           (Derived from series form of    .)
Hence          ∘       .
         ∘                ∘          ,

  

     ∘     .

Let   .
Then        ∘    ×  


 

 







 , lim   lim   
lim   lim 


→ ∞
→∞
→ ∞ 
→ ∞



lim    lim    ∘     ×    
→ ∞

→ ∞




      

  



  



 
  




      ∘ 

             ∘   



  

(I used that      for some . I don't know necessity and sufficiency
condition for convergence of generalized Magnus expansion.)

Examples)
①′     

    
 ∘ 


Exact solution :   

        
        
 
    

        




      
  )




(where   


 
 
 
  
1 term :





  

   




       

  





        


      

2 term :







  




       

  

 



  




     

  

 




            




            


1 term error : 0.102976 (when      )
1+2 term error : 0.00158228 (when      )

In this graph,  means .

②′      









Exact solution :               ∘   





              



       


 

          )
(where   


  

 

1 term :





  

   

 
          


  







 
 
                 
 



2 term :








    

  

 
  




 

 


                      
  
 

1 term error : 0.0704295 (when      )
1+2 term error : 0.00116553 (when      )

In this graph,  means   .

③ (Although I didn't explained Magnus expansion for system of ordinary differential
equation.

I

guess


    









′′    

many

of

same

formulas

hold.

For

example


  .)

   


        

Exact solution : 
  ′   ′  ′  ′  
 



(where         are Airy functions.)
1 term :





  

   

  

    










2 term : (    ∇    ∇   )







 
  



  



   
  
   

  
     
 

 



       








  



  

1 term error(matrix norm) : 0.0920343(when   )
1+2 term error(matrix norm) : 0.0120930(when   )



References)
Eri Jabotinsky : ANALYTIC ITERATION(1963)
Henryk Trappmann and Dmitrii Kouznetsov : 5+ methods for real analytic
tetration(2010)
Shota Kojima : Convergence of Infinite Composition of Entire Functions(2010)
Michael Grossman & Robert Katz : Non-Newtonian Calculus(1972)
Warren D. Smith : Quaternions, octonions, and now, 16-ons and 2n-ons; New kinds
of numbers.(2004)
S. Blanes , F. Casas , J.A. Oteo and J. Ros : The Magnus expansion and some of
its applications(2008)
Ibrahim M. Alabdulmohsin : Theorems and Methods on Partial Functional Iteration
(2009)
Brian Hall : Lie Groups, Lie Algebras, and Representations: An Elementary
Introduction(2004)
Erwin Kreyszig : Advanced Engineering Mathematics 9th Edition(2006)
http://en.wikipedia.org/wiki/Multiplicative_calculus
http://en.wikipedia.org/wiki/Magnus_expansion
http://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula
http://en.wikipedia.org/wiki/Matrix_exponential


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