OPERATIONS ON TYPE DAMAP

Map Concatenation

 

DA Concatenation

TPSA Concatenation

Click before for explanations
MoM M=M*M M=M .o. M Explanations/Explanations
toM t=t * M t=t .o. M <-- See program
r(lnv)=Mor(:)   r=M*r(:) Explanations
r(6)=matrix(6,6) o r(6)   r=matrix*r Explanations
r(lnv)=tree o r(:)      

Tree is a fast trackable map

 r=tree*r(:) Constant part not present in a matrix! Explanations
r(lnv)=g o r(:)    Generating function Tracking

 

 Constant part always ignored with generating function Explanations

Norm of a Map

One can check the norm of a map, particularly useful in iterative procedures: NORM=FULL_ABS(M)

Power of a Map

A damap can be raised to a power M=M**N . If N is negative, then the DA inverse is computed. Example, click here.

Partial Inversion of a Map

A map can be partially inverted. This is very useful in the computation of a generating function. Click here for explanations and example.

Vector Fields Action on Taylor Series

To understand vector fields of the type and the Poisson bracket kind, it is useful to put the cart in front of the ox for a while. So click on the following examples first:

M2=Texp()M1 and M2=Texp(:f:)M1
Introducing Vector Fields and Poisson Bracket Fields using COSY-Infinity style technique to create a FODO cell.
  1. Making maps (type damap) with vectors fields (type vecfield), click here
  2. Making maps (type damap) with Poisson bracket fields (type pbfield), click here

 

Various Lie Representation of the Map ( besides Normal Form)

One Lie Exponent M = exp(F·grad)Id    F = Σ Fk  Explanations
Dragt-Finn M = exp(ONo)...exp(O2) L T Id
Reverse Dragt-Finn M = Lexp(O2)...exp(ONo) T Id                Ok=Fk · grad

Exponentiation of the Vector Fields of the above representations using Texp

F%ifac FPP Calls
One Lie Exponent 0 M = exp(F · grad)Id                     Ok=Fk · grad     => M=Texp(F,Id) Vector fields are characterized by a ifac 
Dragt-Finn 1 M = exp(ONo)...exp(O2) Id         Ok=Fk · grad      => M=Texp(F,Id)
Reverse Dragt-Finn -1 M = exp(O2)...exp(ONo) Id         Ok=Fk · grad => M=Texp(F,Id)

Transforming Vector Fields by Map Directly: Differential Algebraic Operation Only

 :f: M  :f: M-1  --> M*f Explanations
M M-1  --> M*F

 

Available DA Concatenation

* M Df rDf O
M M M M M
Df M no no no
rDf M no no no
O M no no no

 

Simple Numerical Operations on DAMAPs
M3=M1+M2 M3=M1-M2 M2=r*M1 or M1* r   where r = real(8), real or integer

Action of Vector fields on Taylor Series and Damaps

Action on Taylor Action on Damaps
 :f:  :f: t-->    f*t  :f: M-->    f*M
t -->  F*t M -->  F*M

Example: Programming a subroutine that reproduces Texp  using the above operations, click here for explanations and program.

 

Overloading of the (=) sign to pass from one representation to the other

=

 M Df rDf O N  g t f F tr fr Fr r(:) r(:,:) t(:) p p(:) cp
M yes yes yes yes yes yes yes yes yes yes
Df yes no no no no no no
rDf yes no no no no no no
O yes no no no no no no
N yes no no no no no no
g yes no no no no no no
t yes yes yes           yes  
f yes yes yes   yes            
F yes yes     yes          
tr yes     no              
fr no yes     yes            
Fr   yes     yes          
r(:) yes           f90        
r(:,:) yes no no no no no           f90      
t(:) yes                      
p yes                    
p(:) yes                      
cp                      

M,df,fd,Nf, O, t(:), matrix(:,:), r(:) (peekmap),  :f:=t, t=:f:,  f=F F=f