在数学 和物理学 中，马格努斯展开 （英語：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，得名於數學家威廉·馬格努斯 。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 )
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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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{\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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{\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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{\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
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ad
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exp
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A
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{\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：
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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
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{\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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{\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：
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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
(
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{\textstyle \left(W_{t}\right)_{t\in [0,T]}}
R
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{\textstyle \mathbb {R} ^{q}}
布朗运动 ，
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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)}}
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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
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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 ,}
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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
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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.