Map Concatenation
DA Concatenation |
TPSA Concatenation |
Click before for explanations |
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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 |
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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.
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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
Action of Vector fields on Taylor Series and Damaps
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