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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
Generalized Magnus Expansion.pdf (PDF, 204.27 KB)
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