在数学 和物理学 中，马格努斯展开 （英语：Magnus expansion ）为线性算子的一阶齐次线性微分方程 的解提供了指数表示，得名于数学家威廉·马格努斯 。特别地，这种方法提供了变系数n阶线性常微分方程 组的基矩阵。指数是无穷级数，其项涉及多重积分和嵌套换元。
给定n × n A (t )
Y
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t
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=
A
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Y
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t
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Y
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{\displaystyle Y'(t)=A(t)Y(t),\quad Y(t_{0})=Y_{0}}
其中Y (t )
n = 1时，解为
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=
exp
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∫
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0
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A
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d
s
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Y
0
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{\displaystyle Y(t)=\exp \left(\int _{t_{0}}^{t}A(s)\,ds\right)Y_{0}.}
若A (t )t , t 1 、t 2 仍满足A (t 1 ) A (t 2 ) = A (t 2 ) A (t 1 )n > 1情形。A t
马格努斯提出的解矩阵初值问题的方法是，用某个n阶方阵函数Ω(t , t 0 )
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exp
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{\displaystyle Y(t)=\exp {\big (}\Omega (t,t_{0}){\big )}\,Y_{0},}
稍后可将其构造为级数 展开式：
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1
∞
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{\displaystyle \Omega (t)=\sum _{k=1}^{\infty }\Omega _{k}(t),}
为简单起见习惯将Ω(t , t 0 ) Ω(t ) t 0 = 0.
马格努斯意识到，由于d / dt e Ω ) e −Ω = A (t )Ω Ω 伴随自同态 联系起来
Ω
′
=
ad
Ω
exp
(
ad
Ω
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−
1
A
,
{\displaystyle \Omega '={\frac {\operatorname {ad} _{\Omega }}{\exp(\operatorname {ad} _{\Omega })-1}}A,}
并以“BCH展开 的连续类似物”递归求解Ω
上式构成了矩阵线性初值问题求解的马格努斯展开式 或马格努斯级数 。前4项：
Ω
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)
,
{\displaystyle {\begin{aligned}\Omega _{1}(t)&=\int _{0}^{t}A(t_{1})\,dt_{1},\\\Omega _{2}(t)&={\frac {1}{2}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\,[A(t_{1}),A(t_{2})],\\\Omega _{3}(t)&={\frac {1}{6}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}\,{\Bigl (}{\big [}A(t_{1}),[A(t_{2}),A(t_{3})]{\big ]}+{\big [}A(t_{3}),[A(t_{2}),A(t_{1})]{\big ]}{\Bigr )},\\\Omega _{4}(t)&={\frac {1}{12}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}\int _{0}^{t_{3}}dt_{4}\,\left({\Big [}{\big [}[A_{1},A_{2}],A_{3}{\big ]},A_{4}{\Big ]}\right.\\&\qquad +{\Big [}A_{1},{\big [}[A_{2},A_{3}],A_{4}{\big ]}{\Big ]}+{\Big [}A_{1},{\big [}A_{2},[A_{3},A_{4}]{\big ]}{\Big ]}+\left.{\Big [}A_{2},{\big [}A_{3},[A_{4},A_{1}]{\big ]}{\Big ]}\right),\end{aligned}}}
其中[A , B ] ≡ A B − B A A 、B 的矩阵交换子 。
这些方程可解释如下：Ω1 (t ) n 李群 ），则要对指数进行修正。马格努斯级数的剩余部分系统地提供了修正：Ω 李群 的李代数 中。
在应用中，很少能对马格努斯级数精确求和，而要截断才能得到近似解。马格努斯方法的主要优势在于，中截级数通常和精确解具有相同的重要性质，这异于传统摄动理论 。例如，经典力学 中，时间演化 的辛几何 特征在每阶近似都得到保留。同样，量子力学 时间演化算子的幺正性 也得到保留（例如，与解决同一问题的戴森级数 相反）。
从数学角度看，收敛问题如下：给定矩阵A (t )Ω(t )
t ∈ [0,T )收敛 的充分条件是
∫
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{\displaystyle \int _{0}^{T}\|A(s)\|_{2}\,ds<\pi ,}
其中
‖
⋅
‖
2
{\displaystyle \|\cdot \|_{2}}
矩阵范数 。这个结果是通用的，因为可构造特定矩阵，t > T
生成马格努斯展开式中所有项的递归过程利用了下面的递归定义的矩阵 S n (k )
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{\displaystyle S_{n}^{(j)}=\sum _{m=1}^{n-j}\left[\Omega _{m},S_{n-m}^{(j-1)}\right],\quad 2\leq j\leq n-1,}
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{\displaystyle S_{n}^{(1)}=\left[\Omega _{n-1},A\right],\quad S_{n}^{(n-1)}=\operatorname {ad} _{\Omega _{1}}^{n-1}(A),}
然后得到
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{\displaystyle \Omega _{1}=\int _{0}^{t}A(\tau )\,d\tau ,}
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{\displaystyle \Omega _{n}=\sum _{j=1}^{n-1}{\frac {B_{j}}{j!}}\int _{0}^{t}S_{n}^{(j)}(\tau )\,d\tau ,\quad n\geq 2.}
此处adk Ω 是迭代交换子的简写（参见伴随自同态 ）：
ad
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A
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{\displaystyle \operatorname {ad} _{\Omega }^{0}A=A,\quad \operatorname {ad} _{\Omega }^{k+1}A=[\Omega ,\operatorname {ad} _{\Omega }^{k}A],}
其中B j 伯努利数 ，而B 1 = −1/2
最后，明确算得这一递归后，就可将Ωn t ) A 的n-1个嵌套换元的n重积分的线性组合：
Ω
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{\displaystyle \Omega _{n}(t)=\sum _{j=1}^{n-1}{\frac {B_{j}}{j!}}\sum _{k_{1}+\cdots +k_{j}=n-1 \atop k_{1}\geq 1,\ldots ,k_{j}\geq 1}\int _{0}^{t}\operatorname {ad} _{\Omega _{k_{1}}(\tau )}\operatorname {ad} _{\Omega _{k_{2}}(\tau )}\cdots \operatorname {ad} _{\Omega _{k_{j}}(\tau )}A(\tau )\,d\tau ,\quad n\geq 2,}
随着n
要推广到随机常微分方程，令
(
W
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)
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∈
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{\textstyle \left(W_{t}\right)_{t\in [0,T]}}
R
q
{\textstyle \mathbb {R} ^{q}}
布朗运动 ，
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>
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{\textstyle q\in \mathbb {N} _{>0}}
概率空间
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{\textstyle \left(\Omega ,{\mathcal {F}},\mathbb {P} \right)}
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{\displaystyle dX_{t}=B_{t}X_{t}dt+A_{t}^{(j)}X_{t}dW_{t}^{j},\quad X_{0}=I_{d},\qquad d\in \mathbb {N} _{>0},}
其中
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{\textstyle B_{\cdot },A_{\cdot }^{(1)},\dots ,A_{\cdot }^{(j)}}
d
×
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{\textstyle d\times d}
随机过程 ，
I
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{\textstyle I_{d}}
单位矩阵 。参考确定情形，并依随机情形做修改[ 1]
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{\textstyle Y_{t}^{(1)}=Y_{t}^{(1,0)}+Y_{t}^{(0,1)}}
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{\textstyle Y_{t}^{(2)}=Y_{t}^{(2,0)}+Y_{t}^{(1,1)}+Y_{t}^{(0,2)}}
i j
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{\displaystyle {\begin{aligned}Y_{t}^{(0,0)}&=0,\\Y_{t}^{(1,0)}&=\int _{0}^{t}A_{s}^{(j)}\,dW_{s}^{j},\\Y_{t}^{(0,1)}&=\int _{0}^{t}B_{s}\,ds,\\Y_{t}^{(2,0)}&=-{\frac {1}{2}}\int _{0}^{t}{\big (}A_{s}^{(j)}{\big )}^{2}\,ds+{\frac {1}{2}}\int _{0}^{t}{\Big [}A_{s}^{(j)},\int _{0}^{s}A_{r}^{(i)}\,dW_{r}^{i}{\Big ]}dW_{s}^{j},\\Y_{t}^{(1,1)}&={\frac {1}{2}}\int _{0}^{t}{\Big [}B_{s},\int _{0}^{s}A_{r}^{(j)}\,dW_{r}{\Big ]}\,ds+{\frac {1}{2}}\int _{0}^{t}{\Big [}A_{s}^{(j)},\int _{0}^{s}B_{r}\,dr{\Big ]}\,dW_{s}^{j},\\Y_{t}^{(0,2)}&={\frac {1}{2}}\int _{0}^{t}{\Big [}B_{s},\int _{0}^{s}B_{r}\,dr{\Big ]}\,ds.\end{aligned}}}
随机情形下，收敛将受制于停止时间
τ
{\textstyle \tau }
[ 2]
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{\textstyle X=(X_{t})_{t\in [0,T]}}
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{\textstyle \tau }
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{\displaystyle X_{t}=e^{Y_{t}},\qquad 0\leq t<\tau ;}
以下表示有
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{\textstyle \mathbb {P} }
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{\displaystyle Y_{t}=\sum _{n=0}^{\infty }Y_{t}^{(n)},\qquad 0\leq t<\tau ,}
其中
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{\textstyle Y^{(n)}}
存在正常数C
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{\textstyle \|A^{(1)}\|_{T},\dots ,\|A^{(q)}\|_{T},\|B\|_{T},T,d}
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{\textstyle \|A_{\cdot }\|_{T}=\|\|A_{t}\|_{F}\|_{L^{\infty }(\Omega \times [0,T])}}
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{\displaystyle \mathbb {P} (\tau \leq t)\leq Ct,\qquad t\in [0,T].}
随机马格努斯展开的推广形式：
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{\displaystyle Y_{t}=\sum _{n=0}^{\infty }Y_{t}^{(n)}\quad {\text{with}}\quad Y_{t}^{(n)}:=\sum _{r=0}^{n}Y_{t}^{(r,n-r)},}
其中通用项
Y
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{\textstyle Y^{(r,n-r)}}
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{\displaystyle Y_{t}^{(r,n-r)}=\int _{0}^{t}\mu _{s}^{r,n-r}ds+\int _{0}^{t}\sigma _{s}^{r,n-r,j}dW_{s}^{j},\qquad n\in \mathbb {N} _{0},\ r=0,\dots ,n,}
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{\textstyle \sigma ^{r,n-r,j},\mu ^{r,n-r}}
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{\displaystyle {\begin{aligned}\sigma _{s}^{r,n-r,j}&:=\sum _{i=0}^{n-1}{\frac {\beta _{i}}{i!}}S_{s}^{r-1,n-r,i}{\big (}A^{(j)}{\big )},\\\mu _{s}^{r,n-r}&:=\sum _{i=0}^{n-1}{\frac {\beta _{i}}{i!}}S_{s}^{r,n-r-1,i}(B)-{\frac {1}{2}}\sum _{j=1}^{q}\sum _{i=0}^{n-2}{\frac {\beta _{i}}{i!}}\sum _{q_{1}=2}^{r}\sum _{q_{2}=0}^{n-r}S^{r-q_{1},n-r-q_{2},i}{\big (}Q^{q_{1},q_{2},j}{\big )},\end{aligned}}}
其中
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{\displaystyle {\begin{aligned}Q_{s}^{q_{1},q_{2},j}:=\sum _{i_{1}=2}^{q_{1}}\sum _{i_{2}=0}^{q_{2}}\sum _{h_{1}=1}^{i_{1}-1}\sum _{h_{2}=0}^{i_{2}}&\sum _{p_{1}=0}^{q_{1}-i_{1}}\sum _{{p_{2}}=0}^{q_{2}-i_{2}}\ \sum _{m_{1}=0}^{p_{1}+p_{2}}\ \sum _{{m_{2}}=0}^{q_{1}-i_{1}-p_{1}+q_{2}-i_{2}-p_{2}}\\&{\Bigg (}{{\frac {S_{s}^{p_{1},p_{2},m_{1}}{\big (}\sigma _{s}^{h_{1},h_{2},j}{\big )}}{({m_{1}}+1)!}}{\frac {S_{s}^{q_{1}-i_{1}-p_{1},q_{2}-i_{2}-p_{2},m_{2}}{\big (}\sigma _{s}^{i_{1}-h_{1},i_{2}-h_{2},j}{\big )}}{({m_{2}}+1)!}}}\\&\qquad \qquad +{\frac {{\big [}S_{s}^{p_{1},p_{2},m_{1}}{\big (}\sigma _{s}^{i_{1}-h_{1},i_{2}-h_{2},j}{\big )},S_{s}^{q_{1}-i_{1}-p_{1},q_{2}-i_{2}-p_{2},m_{2}}{\big (}\sigma _{s}^{h_{1},h_{2},j}{\big )}{\big ]}}{({m_{1}}+{m_{2}}+2)({m_{1}}+1)!{m_{2}}!}}{\Bigg )},\end{aligned}}}
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{\displaystyle {\begin{aligned}S_{s}^{r-1,n-r,0}(A)&:={\begin{cases}A&{\text{if }}r=n=1,\\0&{\text{otherwise}},\end{cases}}\\S_{s}^{r-1,n-r,i}(A)&:=\sum _{\begin{array}{c}(j_{1},k_{1}),\dots ,(j_{i},k_{i})\in \mathbb {N} _{0}^{2}\\j_{1}+\cdots +j_{i}=r-1\\k_{1}+\cdots +k_{i}=n-r\end{array}}{\big [}Y_{s}^{(j_{1},k_{1})},{\big [}\dots ,{\big [}Y_{s}^{(j_{i},k_{i})},A_{s}{\big ]}\dots {\big ]}{\big ]}\\&=\sum _{\begin{array}{c}(j_{1},k_{1}),\dots ,(j_{i},k_{i})\in \mathbb {N} _{0}^{2}\\j_{1}+\cdots +j_{i}=r-1\\k_{1}+\cdots k_{i}=n-r\end{array}}\operatorname {ad} _{Y_{s}^{(j_{1},k_{1})}}\circ \cdots \circ \operatorname {ad} _{Y_{s}^{(j_{i},k_{i})}}(A_{s}),\qquad i\in \mathbb {N} .\end{aligned}}}
Magnus, W. On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 1954, VII (4): 649–673. doi:10.1002/cpa.3160070404 Blanes, S.; Casas, F.; Oteo, J.A.; Ros, J. Magnus and Fer expansions for matrix differential equations: The convergence problem. J. Phys. A: Math. Gen. 1998, 31 (1): 259–268. Bibcode:1998JPhA...31..259B doi:10.1088/0305-4470/31/1/023 Iserles, A.; Nørsett, S. P. On the solution of linear differential equations in Lie groups. Phil. Trans. R. Soc. Lond. A. 1999, 357 (1754): 983–1019. Bibcode:1999RSPTA.357..983I CiteSeerX 10.1.1.15.4614 S2CID 90949835 doi:10.1098/rsta.1999.0362 Blanes, S.; Casas, F.; Oteo, J.A.; Ros, J. The Magnus expansion and some of its applications. Phys. Rep. 2009, 470 (5–6): 151–238. Bibcode:2009PhR...470..151B S2CID 115177329 arXiv:0810.5488 doi:10.1016/j.physrep.2008.11.001 Kamm, K.; Pagliarani, S.; Pascucci, A. On the Stochastic Magnus Expansion and Its Application to SPDEs. Journal of Scientific Computing. 2021, 89 (3): 56. S2CID 211259118 arXiv:2001.01098 doi:10.1007/s10915-021-01633-6
![{\displaystyle {\begin{aligned}\Omega _{1}(t)&=\int _{0}^{t}A(t_{1})\,dt_{1},\\\Omega _{2}(t)&={\frac {1}{2}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\,[A(t_{1}),A(t_{2})],\\\Omega _{3}(t)&={\frac {1}{6}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}\,{\Bigl (}{\big [}A(t_{1}),[A(t_{2}),A(t_{3})]{\big ]}+{\big [}A(t_{3}),[A(t_{2}),A(t_{1})]{\big ]}{\Bigr )},\\\Omega _{4}(t)&={\frac {1}{12}}\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\int _{0}^{t_{2}}dt_{3}\int _{0}^{t_{3}}dt_{4}\,\left({\Big [}{\big [}[A_{1},A_{2}],A_{3}{\big ]},A_{4}{\Big ]}\right.\\&\qquad +{\Big [}A_{1},{\big [}[A_{2},A_{3}],A_{4}{\big ]}{\Big ]}+{\Big [}A_{1},{\big [}A_{2},[A_{3},A_{4}]{\big ]}{\Big ]}+\left.{\Big [}A_{2},{\big [}A_{3},[A_{4},A_{1}]{\big ]}{\Big ]}\right),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d98fd6d46f84435fb731a73803d8446226f27580)
![{\displaystyle S_{n}^{(j)}=\sum _{m=1}^{n-j}\left[\Omega _{m},S_{n-m}^{(j-1)}\right],\quad 2\leq j\leq n-1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81703c24da029c6762a365928f7467ddc49d6777)
![{\displaystyle S_{n}^{(1)}=\left[\Omega _{n-1},A\right],\quad S_{n}^{(n-1)}=\operatorname {ad} _{\Omega _{1}}^{n-1}(A),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d35b59f701c4d7052c38bdb2b6dbceecb7b8b3bc)
![{\displaystyle \operatorname {ad} _{\Omega }^{0}A=A,\quad \operatorname {ad} _{\Omega }^{k+1}A=[\Omega ,\operatorname {ad} _{\Omega }^{k}A],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3579110e0417de1ffa276dda567ffaf7499ed3c6)
![{\textstyle \left(W_{t}\right)_{t\in [0,T]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c398893e410dc670983bfbbad4d2d322df05e191)
![{\displaystyle {\begin{aligned}Y_{t}^{(0,0)}&=0,\\Y_{t}^{(1,0)}&=\int _{0}^{t}A_{s}^{(j)}\,dW_{s}^{j},\\Y_{t}^{(0,1)}&=\int _{0}^{t}B_{s}\,ds,\\Y_{t}^{(2,0)}&=-{\frac {1}{2}}\int _{0}^{t}{\big (}A_{s}^{(j)}{\big )}^{2}\,ds+{\frac {1}{2}}\int _{0}^{t}{\Big [}A_{s}^{(j)},\int _{0}^{s}A_{r}^{(i)}\,dW_{r}^{i}{\Big ]}dW_{s}^{j},\\Y_{t}^{(1,1)}&={\frac {1}{2}}\int _{0}^{t}{\Big [}B_{s},\int _{0}^{s}A_{r}^{(j)}\,dW_{r}{\Big ]}\,ds+{\frac {1}{2}}\int _{0}^{t}{\Big [}A_{s}^{(j)},\int _{0}^{s}B_{r}\,dr{\Big ]}\,dW_{s}^{j},\\Y_{t}^{(0,2)}&={\frac {1}{2}}\int _{0}^{t}{\Big [}B_{s},\int _{0}^{s}B_{r}\,dr{\Big ]}\,ds.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed5c55f67f226450b8d429fe79b917589101af1)
![{\textstyle X=(X_{t})_{t\in [0,T]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eff1b6cd34a6413e006f83160616dc98472dab6)
![{\textstyle \|A_{\cdot }\|_{T}=\|\|A_{t}\|_{F}\|_{L^{\infty }(\Omega \times [0,T])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c21da16bf89b8a325a71ad502fb3ee0bc458f409)
![{\displaystyle \mathbb {P} (\tau \leq t)\leq Ct,\qquad t\in [0,T].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28da67281b17b15920260b463065d7e5b17ea431)
![{\displaystyle {\begin{aligned}Q_{s}^{q_{1},q_{2},j}:=\sum _{i_{1}=2}^{q_{1}}\sum _{i_{2}=0}^{q_{2}}\sum _{h_{1}=1}^{i_{1}-1}\sum _{h_{2}=0}^{i_{2}}&\sum _{p_{1}=0}^{q_{1}-i_{1}}\sum _{{p_{2}}=0}^{q_{2}-i_{2}}\ \sum _{m_{1}=0}^{p_{1}+p_{2}}\ \sum _{{m_{2}}=0}^{q_{1}-i_{1}-p_{1}+q_{2}-i_{2}-p_{2}}\\&{\Bigg (}{{\frac {S_{s}^{p_{1},p_{2},m_{1}}{\big (}\sigma _{s}^{h_{1},h_{2},j}{\big )}}{({m_{1}}+1)!}}{\frac {S_{s}^{q_{1}-i_{1}-p_{1},q_{2}-i_{2}-p_{2},m_{2}}{\big (}\sigma _{s}^{i_{1}-h_{1},i_{2}-h_{2},j}{\big )}}{({m_{2}}+1)!}}}\\&\qquad \qquad +{\frac {{\big [}S_{s}^{p_{1},p_{2},m_{1}}{\big (}\sigma _{s}^{i_{1}-h_{1},i_{2}-h_{2},j}{\big )},S_{s}^{q_{1}-i_{1}-p_{1},q_{2}-i_{2}-p_{2},m_{2}}{\big (}\sigma _{s}^{h_{1},h_{2},j}{\big )}{\big ]}}{({m_{1}}+{m_{2}}+2)({m_{1}}+1)!{m_{2}}!}}{\Bigg )},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1007971f2e29febe1a5af9f266bad2f0617ae6d8)
![{\displaystyle {\begin{aligned}S_{s}^{r-1,n-r,0}(A)&:={\begin{cases}A&{\text{if }}r=n=1,\\0&{\text{otherwise}},\end{cases}}\\S_{s}^{r-1,n-r,i}(A)&:=\sum _{\begin{array}{c}(j_{1},k_{1}),\dots ,(j_{i},k_{i})\in \mathbb {N} _{0}^{2}\\j_{1}+\cdots +j_{i}=r-1\\k_{1}+\cdots +k_{i}=n-r\end{array}}{\big [}Y_{s}^{(j_{1},k_{1})},{\big [}\dots ,{\big [}Y_{s}^{(j_{i},k_{i})},A_{s}{\big ]}\dots {\big ]}{\big ]}\\&=\sum _{\begin{array}{c}(j_{1},k_{1}),\dots ,(j_{i},k_{i})\in \mathbb {N} _{0}^{2}\\j_{1}+\cdots +j_{i}=r-1\\k_{1}+\cdots k_{i}=n-r\end{array}}\operatorname {ad} _{Y_{s}^{(j_{1},k_{1})}}\circ \cdots \circ \operatorname {ad} _{Y_{s}^{(j_{i},k_{i})}}(A_{s}),\qquad i\in \mathbb {N} .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30ca63471e2d7cd8ac3417c7f54a3e2df56bdfcc)
. S2CID 90949835. doi:10.1098/rsta.1999.0362.