비선형 고유문제

수학에서 비선형 고유문제, 때로는 비선형 고유값 문제는 (일반) 고유값 문제를 고유값에 비선형적으로 의존하는 방정식으로 일반화하는 것이다.구체적으로는 형태 방정식을 가리킨다.

M(\lambda )x=0,

where $x\neq 0$ is a vector, and $M$ is a matrix-valued function of the number $\lambda$ . The number $\lambda$ is known as the (nonlinear) eigenvalue, the vector $x$ as the (nonlinear) eigenvector, and ${\disp$ $레이스타일(\lambda ,x)}$ 을(를) 고유 쌍으로 한다 $(\lambda ,x)$ . $M(\lambda )$ ( $M(\lambda )$ ) ${\displaystyle$ M $(\lambda )}$ 행렬은 고유값 $\lambda$ $\lambda$ 에서 단수형이다 $M(\lambda )$ $\lambda$

정의

수치 선형대수학에서는 일반적으로 다음과 같은 정의를 사용한다.^[1]^[2]^[3]^[4]

$\Omega \subseteq \mathbb {C}$ $\Omega \subseteq \mathbb {C}$ $\Omega \subseteq \mathbb {C}$ ${\$ 을(를) 놓아두고 $M:\Omega \rightarrow \mathbb {C} ^{n\times n}$ : $M:\Omega \rightarrow \mathbb {C} ^{n\times n}$ → $M:\Omega \rightarrow \mathbb {C} ^{n\times n}$ $M:\Omega \rightarrow \mathbb {C} ^{n\times n}$ $M:\Omega \rightarrow \mathbb {C} ^{n\times n}$ n ${\$ $\Oomega \rightarrow \mathb{C} ^{n\time$ n $}}}$ 는 $M:\Omega \rightarrow \mathbb {C} ^{n\times n}$ $스칼라$ 를 행렬에 매핑하는 함수다.A scalar $\lambda \in \mathbb {C}$ is called an eigenvalue, and a nonzero vector $x\in \mathbb {C} ^{n}$ is called a right eigevector if $M(\lambda )x=0$ . Moreover, a nonzero vector $y\in \mathbb {C} ^{n}$ is cal $y^{H}M(\lambda )=0^{H}$ $y^{H}M(\lambda )=0^{H}$ $y^{H}M(\lambda )=0^{H}$ ) $y^{H}M(\lambda )=0^{H}$ = $y^{H}M(\lambda )=0^{H}$ $y^{H}M(\lambda )=0^{H}$ ${\$ 0^{ $H$ 여기서 위첨자 $^{H}$ ${\$ 는 은둔자의 전치사를 나타낸다 $^{H}$ .고유값의 정의는 $\det(M(\lambda ))=0$ ( $\det(M(\lambda ))=0$ ( $\det(M(\lambda ))=0$ ) $\det(M(\lambda ))=0$ = 0 ${\displaystyle \det(M(\lambda )=0$ 여기서 $\det()$ ( $\det()$ ) ${\displaysty \det()}$ 은 결정요인을 나타낸다 $\det()$ .^[1]

$함수$ M $M$ 은(일부 도메인 $\Omega$ ${\displaystyle \lambda$ $})$ 의 홀로모르픽 함수여야 한다 $M$ . $\lambda$ $\Omega$

일반적으로 $M(\lambda )$ ( $M(\lambda )$ ) $M(\lambda )$ 은 선형 지도가 될 수 있지만 $M(\lambda )$ , 가장 일반적으로는 유한 차원, 대게 정사각형의 행렬이다.

정의:문제는 $\det(M(z))\neq 0$ ( $\det(M(z))\neq 0$ ) ${\displaystyle$ $z\in \Oomega$ }과 같은 $z\in \Omega$ $z\in \Omega$ $z\in \Omega$ $z\in \Omega$ ${\displaystyle \$ $det(M(z)\neq$ 0 $}$ 이 있으면 규칙적이라고 하며 $\det(M(z))\neq 0$ 그렇지 않으면 단수라고 한다.^[1]^[4]

정의:고유값 $\lambda$ ${\$ $displaystyle$ $\$ $lambda }$ 은 $k$ (는) $\lambda$ ${\displaystyle$ $k$ $}$ 이 $k$ ( $\det(M(z))$ $)$ $\det(M(z))$ det ${\displaystyle z$ 에 대한 $\det(M(z))$ det( $(z)}$ 의 파생상품이 $k$ ${\displaystystyk$ }일 경우 $k$ 대수 $다중성$ k ${\\\\}$ 을 갖는다고 한다 $\lambda$ . $\$ \ $da }$ 은(는) 0이 아니다 $\lambda$ . $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ ( z $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ ) $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ = $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ 0 $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ {\ $displaystyle \왼쪽)$ 수식. ${\frac {d^}{k}\det(M(z)}{dz^{k}}}\right _{$ z=\ $lambda }\neq$ 0 $}$ $\left.{\frac {d^{\ell }\det(M(z))}{dz^{\ell }}}\right|_{z=\lambda }=0$ d $\left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0$ $\left.{\frac {d^{\ell }\det(M(z))}{dz^{\ell }}}\right|_{z=\lambda }=0$ det $\left.{\frac {d^{\ell }\det(M(z))}{dz^{\ell }}}\right|_{z=\lambda }=0$ ( ) $\left.{\frac {d^{\ell }\det(M(z))}{dz^{\ell }}}\right|_{z=\lambda }=0$ $\left.{\frac {d^{\ell }\det(M(z))}{dz^{\ell }}}\right|_{z=\lambda }=0$ = ${\displaysty \left.$ ${\$ $\ell =0,1,2,\dots ,k-1$ { $\ell =0,1,2,\dots ,k-1$ $^{\$ $\ell =0,1,2,\dots ,k-1$ $}\det(M(z))}{dz^{\\ell }}}\right _{z=\lambda$ }= $0$ $}$ for $\left.{\frac {d^{\ell }\det(M(z))}{dz^{\ell }}}\right|_{z=\lambda }=0$ $\ell =0,1,2,\dots ,k-1$ $\ell =0,1,2,\dots ,k-1$ 0 $\ell =0,1,2,\dots ,k-1$ 1, $\ell =0,1,2,\dots ,k-1$ , $\ell =0,1,2,\dots ,k-1$ k - 1 ${\displaysty \ell$ = $0,1,2,\dots,k-1$ ^[1]^[4]

정의:고유값 $\lambda$ $\lambda$ 의 기하학적 다중성은 $M(\lambda )$ ( $M(\lambda )$ ) ${\displaystyle$ M $(\lambda )}$ 의 nullspace 치수다 $\lambda$ $M(\lambda )$ ^[1]^[4]

특례

비선형 고유문제의 특별한 예는 다음과 같다.

(일반) 고유값 문제: $M(\lambda )=A-\lambda I.$ ( $M(\lambda )=A-\lambda I.$ ) $M(\lambda )=A-\lambda I.$ = $M(\lambda )=A-\lambda I.$ - $M(\lambda )=A-\lambda I.$ $M(\lambda )=A-\lambda I.$ . ${\displaystyle M(\lambda )=A-\lambda$ I $.}$
일반화된 고유값 $M(\lambda )=A-\lambda B.$ : $M(\lambda )=A-\lambda B.$ ( $M(\lambda )=A-\lambda B.$ ) $M(\lambda )=A-\lambda B.$ = A $M(\lambda )=A-\lambda B.$ - $M(\lambda )=A-\lambda B.$ $M(\lambda )=A-\lambda B.$ . ${\displaystyle$ M $(\lambda )=A-\lambda B.$ $}$
2차 고유치 문제: $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ ) $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ = $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ + $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ + $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ $M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.$ . ${\displaystyle$ M $(\lambda )=$ $A_{0}+\lambda A_{1}+\lambda ^{2}$ $A_{2}.}$
다항 고유값 문제: $M(\lambda )=\sum _{i=1}^{m}\lambda ^{i}A_{i}.$ $M(\lambda )=\sum _{i=1}^{m}\lambda ^{i}A_{i}.$ ) $M(\lambda )=\sum _{i=1}^{m}\lambda ^{i}A_{i}.$ = $M(\lambda )=\sum _{i=1}^{m}\lambda ^{i}A_{i}.$ i $M(\lambda )=\sum _{i=1}^{m}\lambda ^{i}A_{i}.$ = 1 $M(\lambda )=\sum _{i=1}^{m}\lambda ^{i}A_{i}.$ $M(\lambda )=\sum _{i=1}^{m}\lambda ^{i}A_{i}.$ $M(\lambda )=\sum _{i=1}^{m}\lambda ^{i}A_{i}.$ ${\displaystyle$ M $(\lambda )=\sum _{i=1}^{m}\lambda ^{i}$ $A_{i}}$
$M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ 인 고유치 문제: M ( $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ ) $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ = $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ i $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ = 1 m 1 $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ i + $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ i = $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ i ( $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ ) ) $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ , ${\displaystyle M(\lambda )=\sum$ _{ $i=1}^{m_{1}}}}.$ $A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}{$ i}(\ $lambda$ ), $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ 여기서 $M(\lambda )=\sum _{i=1}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),$ $r_{i}(\lambda )$ $($ $){\displaysty r_{i}(\lambda$ )는 합리적인 $r_{i}(\lambda )$ 함수다.
$M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ 고유값 문제: M ( $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ ) $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ = $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ - $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ + A $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ + $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ = $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ m $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ - $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ i $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ ${\displaystyle$ M $(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m$ $}A_{i}e^{-\tau _{i}\\lambda }}}$ 여기서 $M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },$ $\tau _{1},\tau _{2},\dots ,\tau _{m}$ 1, $\tau _{1},\tau _{2},\dots ,\tau _{m}$ $\tau _{1},\tau _{2},\dots ,\tau _{m}$ , $\tau _{1},\tau _{2},\dots ,\tau _{m}$ $\tau _{1},\tau _{2},\dots ,\tau _{m}$ ${\$ 은(지연)로 알려진 스칼라가 주어진다 $\tau _{1},\tau _{2},\dots ,\tau _{m}$ .

요르단 체인

정의:( $(\lambda _{0},x_{0})$ $(\lambda _{0},x_{0})$ , x $(\lambda _{0},x_{0})$ ) ${\displaystyle(\lambda _{0},x_{0})$ 을 고유공기로 $(\lambda _{0},x_{0})$ 한다.A tuple of vectors $(x_{0},x_{1},\dots ,x_{r-1})\in \mathbb {C} ^{n}\times \mathbb {C} ^{n}\times \dots \times \mathbb {C} ^{n}$ is called a Jordan chain if

{\displaystyle \sum \{k=0}^{\ell }M^{(k)}(\lambda _{0}x_{\ell -k}=0,})

for

\ell =0,1,\dots ,r-1

, where

M^{(k)}(\lambda _{0})

denotes the

k

th derivative of

M

with respect to

\lambda

and evaluated in

{\displaystyle \lambda =\lambda

_{0}}

.The vectors

x_{0},x_{1},\dots ,x_{r-1}

are called generalized eigenvectors,

r

is called the length of the Jordan chain, and the maximal length a Jordan chain starting with

x_{0}

is called the rank of

{\displaystyle x_

{0}}

.^[1]^[4]

정리:^[1]A tuple of vectors $(x_{0},x_{1},\dots ,x_{r-1})\in \mathbb {C} ^{n}\times \mathbb {C} ^{n}\times \dots \times \mathbb {C} ^{n}$ is a Jordan chain if and only if the function ${\displaystyle M(\lambda )\chi _{\ell }(\lam$ $bda )}$ has a root in $\lambda =\lambda _{0}$ and the root is of multiplicity at least $\ell$ for $\ell =0,1,\dots ,r-1$ , where the vector valued function $\chi _{\ell }(\lambda )$ is defined as

\chi _{\ell \}{k=0}^{\ell }x_{k}(\data -\basedda _{0}}}}}}}

고유벡터 비선형성

고유벡터 비선형성은 관련성이 있지만 때로는 다른 형태의 비선형성을 연구하기도 한다.이 경우 $M$ $M$ 함수는 벡터를 행렬에 매핑하거나, 은둔자 행렬을 은둔자 행렬에 매핑한다 $M$ .^[5]^[6]

참조

^ ^a ^b ^c ^d ^e ^f ^g Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298.
^ Ruhe, Axel (1973). "Algorithms for the Nonlinear Eigenvalue Problem". SIAM Journal on Numerical Analysis. 10 (4): 674–689. doi:10.1137/0710059. ISSN 0036-1429. JSTOR 2156278.
^ Mehrmann, Volker; Voss, Heinrich (2004). "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods". GAMM-Mitteilungen. 27 (2): 121–152. doi:10.1002/gamm.201490007. ISSN 1522-2608.
^ ^a ^b ^c ^d ^e Voss, Heinrich (2014). "Nonlinear eigenvalue problems" (PDF). In Hogben, Leslie (ed.). Handbook of Linear Algebra (2 ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 9781466507289.
^ Jarlebring, Elias; Kvaal, Simen; Michiels, Wim (2014-01-01). "An Inverse Iteration Method for Eigenvalue Problems with Eigenvector Nonlinearities". SIAM Journal on Scientific Computing. 36 (4): A1978–A2001. arXiv:1212.0417. doi:10.1137/130910014. ISSN 1064-8275. S2CID 16959079.
^ Upadhyaya, Parikshit; Jarlebring, Elias; Rubensson, Emanuel H. (2021). "A density matrix approach to the convergence of the self-consistent field iteration". Numerical Algebra, Control & Optimization. 11 (1): 99. doi:10.3934/naco.2020018. ISSN 2155-3297.

추가 읽기

프랑수아즈 티세서르와 카를 메어베르겐, "이차적 고유가치 문제", SIAM Review 43(2), 235–286(2001) (링크)
Gene H. Golub와 Henk A. van der Vorst, "20세기의 고유값 연산", Journal of Computing and Applied Mathics 123, 35–65(2000)
Philipe Guillaume, "비선형 고유문제", Matrix Analysis and Applications 20(3), 575–595(1999년) (링크)에 관한 SIAM 저널.
Cedric Effenberger, "비선형 고유값 문제에 대한 강력한 솔루션 방법", PhD 논문 EPFL(2013) (링크)
Roel Van Beuumen, "비선형 고유값 문제에 대한 Rational Krylov methods for non-linear eigenvalue probles", 박사 논문 KU Leuven(2015) (링크)

[:0-1] ^ ^a ^b ^c ^d ^e ^f ^g Güttel, Stefan; Tisseur, Françoise (2017). "The nonlinear eigenvalue problem" (PDF). Acta Numerica. 26: 1–94. doi:10.1017/S0962492917000034. ISSN 0962-4929. S2CID 46749298.

[2] Ruhe, Axel (1973). "Algorithms for the Nonlinear Eigenvalue Problem". SIAM Journal on Numerical Analysis. 10 (4): 674–689. doi:10.1137/0710059. ISSN 0036-1429. JSTOR 2156278.

[3] Mehrmann, Volker; Voss, Heinrich (2004). "Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods". GAMM-Mitteilungen. 27 (2): 121–152. doi:10.1002/gamm.201490007. ISSN 1522-2608.

[:1-4] Voss, Heinrich (2014). "Nonlinear eigenvalue problems" (PDF). In Hogben, Leslie (ed.). Handbook of Linear Algebra (2 ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 9781466507289.

[5] Jarlebring, Elias; Kvaal, Simen; Michiels, Wim (2014-01-01). "An Inverse Iteration Method for Eigenvalue Problems with Eigenvector Nonlinearities". SIAM Journal on Scientific Computing. 36 (4): A1978–A2001. arXiv:1212.0417. doi:10.1137/130910014. ISSN 1064-8275. S2CID 16959079.

[6] Upadhyaya, Parikshit; Jarlebring, Elias; Rubensson, Emanuel H. (2021). "A density matrix approach to the convergence of the self-consistent field iteration". Numerical Algebra, Control & Optimization. 11 (1): 99. doi:10.3934/naco.2020018. ISSN 2155-3297.

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비선형 고유문제

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정의

특례

요르단 체인

고유벡터 비선형성

참조

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