Eikonal Blog

2010.01.26

Magnus on matrix exponentials

Definition: The Magnus’ bracket is defined as \{y,x^n\}_M :\equiv (- ad(x))^n y.

Properties:

  • e^{-ad(x)} y = \sum_{n=0}^\infty \{y, x^n\}_M
  • for a power series P(x)=\sum_{n=0}^\infty p_l x^l one has \{y, P(x)\}_M = \sum_{n=0}^\infty p_l \{y, x^n\}_M
  • Hausdorf:
    • e^{-x} (y D_x) e^x = \left\{y,\frac{e^x -1}{x}\right\}_M,
    • (y D_x e^{-x}) e^x = \left\{y,\frac{1-e^{-x}}{x}\right\}_M,

    where the Hausdorff polarization operator (y D_x) is defined as following:

    (y D_x) x^n :\equiv \sum_{k=0}^{n-1} x^k y x^{n-k-1}.

  • Lemma: For two power series P(x) and Q(x) s/t P(x) Q(x) = 1 one has the following equivalence:
      \{y,P(x)\}_M=u \Leftrightarrow y = \{u, Q(x)\}_M.
  • \{\{y,x^n\}_M,x^m\}_M = \{y,x^{n+m}\}_M.

For e^{z} :\equiv e^{x}e^{y} one has:

  • z=\ln(1+u) = u - \frac12 u^2 + \frac13 u^3 + \cdots

    where
    u:\equiv e^x e^y - 1 = x + y + \frac12 x^2 + xy + \frac12 y^2 + \cdots
  • z = x + y + \frac12 [x,y] + \frac1{12} \{x,y^2\}_M + \frac1{12} \{y,x^2\}_M -\frac1{24} [\{x,y^2\}_M,x] - \frac1{720} \{x,y^4\}_M - \frac1{720} \{y,x^4\}_M + \frac1{180} [[\Delta_1,x],y] - \frac1{180}\{\Delta_1,y^2\}_M - \frac1{120} [\Delta_1,\Delta] - \frac1{360} [\Delta_2,\Delta] + \cdots

    where

    \Delta=[x,y],
    \Delta_1=[\Delta,x],
    \Delta_2=[\Delta,y].

The Magnus’ formula for ordered exponentials

And this is really beautiful thing: for the ordered exponential U(t|A):\equiv \left(e^{\int_0^t dt_1 A(t_1)}\right)_+ one can observe its logarithm (the Magnus function of A) \Omega(t|A) :\equiv \ln U(t|A) (i.e. e^{\Omega}=U) and get following nonlinear ODE for it:


\frac{d\Omega(t)}{dt} = \{A, \frac{\Omega}{1-e^{-\Omega}}\}_M = \sum_{n=0}^\infty \beta_n \{A,\Omega^n\}_M = A + \frac12 [A,\Omega] + \frac1{12}\{A,\Omega^2\}_M + \cdots

where \beta_{2n1+1}=0 (\forall n\in{\Bbb N}_0), \beta_{2n} = (-)^{n-1} \frac{B_{2n}}{(2n)!} and B_n‘s are ordinary Bernoulli’s numbers.


Source: T1264 = W. Magnus “On the exponential solution of differential equations for a linear operator”; Communications of Pure and Applied mathematics, Vol VII, 6490673 (1954). Note: This paper is probably mother of the whole this area of matrix exponentials.


Related here: Magnus’ Bracket Operator – https://eikonal.wordpress.com/2010/02/15/magnus-bracket-operator/

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A Theorem of Ellis and Pinsky

Filed under: mathematics — Tags: — sandokan65 @ 13:12

Theorem (Ellis & Pinsky): Let $\latex A$, B be real symmetric n\times n matrices, and assume that B is negative semi-definite. Then as \epsilon \rightarrow +0:

  • e^{t(A+i\frac1{\epsilon}B)} e^{-i\frac{t}{\epsilon}B} = e^{t A_0} + {\cal O}(\epsilon),

where A_0 :\equiv \sum_{\lambda\in\sigma(B)} P_\lambda A P_\lambda, here P_\lambda being the orthogonal projector on the eigenspace of B associated with the eigenvalue \lambda.

Source: T1270 = S.L.Campbell “On the limit of a product of matrix exponentials”; Linear and Multilinear Algebra; 1978, Vol 6, p.p. 55-59.

Expansions of the exponentials of the sums of matrices

Richmond’s formula #1: e^{t(A+B)} = e^{tA}e^{tB} + \sum_{r=0}^\infty E_r t^r, where:

  • E_{r+1} = \frac1{r+1}\left((A+B)E_r+[B,F_r]\right) \ with E_0=E_1=0,
  • F_{r+1} = \frac1{r+1}(A F_r + F_r B) \ with F_0=1.

Richmond’s formula #2: e^{t(A+B)} = \frac12(e^{tA}e^{tB}+e^{tB}e^{tA}) + \sum_{r=0}^\infty E'_r t^r, where:

  • E'_{f+1} = \frac1{r+1}((A+B)E'_r+\frac12[B,F_r]+\frac12[A,F^{*}_r]) \ with E'_0=E'_1=E'_2=0, and where F_r‘s are the same as in the Richmond’s formula #1.
  • Following bounds are valid: ||E'_r|| < \frac1{(r-1)!} (||A||+||B||)^r.

Note that the exponential generating function {\cal F}(z):\equiv \sum_{r=0}^\infty \frac{z^r}{r!} F_r satisfies the second order ODE: \partial_z (z\partial_z {\cal F}(z)) = A{\cal F}(z)+{\cal F}(z)B.

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Source: T1269 = A.N.Richmond “Expansion for the exponential of a sum of matrices”; Int. Journal of Control (issue and year unknown).

Exponential splittings

Filed under: mathematics — Tags: — sandokan65 @ 12:06

First order splitting:

  • e^{t\sum_{i=1}^{n} A_i } = \prod_{i=1}^n e^{t A_i} + {\cal O}(t^2).

Second order splittings:

  • The Strang’s Splitting:
    e^{t\sum_{i=1}^{n} A_i } = e^{\frac{t}2 A_1} e^{\frac{t}2 A_2} \cdots e^{\frac{t}2 A_{n-1}} e^{t A_n} e^{\frac{t}2 A_{n-1}} \cdots e^{\frac{t}2 A_2} e^{\frac{t}2 A_1} + {\cal O}(t^3).
  • The parallel Splitting:
    e^{t\sum_{i=1}^{n} A_i } = \frac12(e^{t A_1}\cdots e^{t A_n} + e^{t A_n}\cdots e^{t A_1})  + {\cal O}(t^3).

Source: T1268 = Qin Sheng “Global Error Estimates for Exponential Splitting”; IMA Journal of Numerical Analysis (1993) 14, 27-56.

Error estimates for exponential splittings

Filed under: mathematics — Tags: , — sandokan65 @ 11:58

Def: \varphi(t):\equiv \frac{||e^{t(A+E}-e^{tA}||}{||e^{tA}||}.

Estimate:
\varphi(t) \le t ||E|| e^{t(\mu(A)-\eta(A)+||E||)}
where:

  • \mu(A):\equiv \max\{\Re(\lambda)|\lambda \in \Sigma_{\frac12(A+A^\dagger)}\} (i.e. the largest eigenvalue of the matrix \frac12(A+A^\dagger), i.e. “the logarithmic norm” of A),
  • \eta(A):\equiv \max\{\Re(\lambda)|\lambda \in \Sigma_A\},
  • \mu(A) \ge \eta(A).

Source: T1268 = Qin Sheng “Global Error Estimates for Exponential Splitting”; IMA Journal of Numerical Analysis (1993) 14, 27-56.

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