Nonlinear Eigenvalue Problems General Theory, Results and Methods
Heinrich Voss [email protected]
Hamburg University of Technology Institute of Mathematics
TUHH Heinrich Voss Isfahan, July, 2016 1 / 72 On the one hand: Parameter dependent nonlinear (with respect to the state variable) operator equations T (λ, u) = 0 discussing — positivity of solutions — multiplicity of solution — dependence of solutions on the parameter; bifurcation — (change of ) stability of solutions
Nonlinear eigenvalue problem
The term nonlinear eigenvalue problem is not used in a unique way in the literature
TUHH Heinrich Voss Isfahan, July, 2016 2 / 72 Nonlinear eigenvalue problem
The term nonlinear eigenvalue problem is not used in a unique way in the literature
On the one hand: Parameter dependent nonlinear (with respect to the state variable) operator equations T (λ, u) = 0 discussing — positivity of solutions — multiplicity of solution — dependence of solutions on the parameter; bifurcation — (change of ) stability of solutions
TUHH Heinrich Voss Isfahan, July, 2016 2 / 72 Find λ ∈ D and x 6= 0 such that
T (λ)x = 0.
Then λ is called an eigenvalue of T (·), and x a corresponding (right) eigenvector.
In this presentation
For D ⊂ C let T (λ), λ ∈ D be a family of linear operators on a Hilbert space H (more generally a family of closed linear operators on a Banach space).
TUHH Heinrich Voss Isfahan, July, 2016 3 / 72 In this presentation
For D ⊂ C let T (λ), λ ∈ D be a family of linear operators on a Hilbert space H (more generally a family of closed linear operators on a Banach space).
Find λ ∈ D and x 6= 0 such that
T (λ)x = 0.
Then λ is called an eigenvalue of T (·), and x a corresponding (right) eigenvector.
TUHH Heinrich Voss Isfahan, July, 2016 3 / 72 Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 4 / 72 Examples Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 5 / 72 iω t Suppose that the system is exited by a time harmonic force f (t) = f0e 0 .
If C = αK + βM (modal damping) and (xj , µj ), j = 1,..., n denotes an T orthonormal eigensystem of Kx = µMx (i.e. xk Mxj = δjk ) then the periodic response of the system is
n T X xj f0 q(t) = eiω0t x . µ − ω2 + iω (αµ + β) j j=1 j 0 0 j
Usually, one gets good approximations taking into account only a small number of eigenvalues in the vicinity of iω0 (truncated mode superposition).
Examples Example 1: Vibration of structures
Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are
Mq¨(t) + Cq˙ (t) + Kq(t) = f (t)
where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector.
TUHH Heinrich Voss Isfahan, July, 2016 6 / 72 If C = αK + βM (modal damping) and (xj , µj ), j = 1,..., n denotes an T orthonormal eigensystem of Kx = µMx (i.e. xk Mxj = δjk ) then the periodic response of the system is
n T X xj f0 q(t) = eiω0t x . µ − ω2 + iω (αµ + β) j j=1 j 0 0 j
Usually, one gets good approximations taking into account only a small number of eigenvalues in the vicinity of iω0 (truncated mode superposition).
Examples Example 1: Vibration of structures
Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are
Mq¨(t) + Cq˙ (t) + Kq(t) = f (t)
where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector.
iω t Suppose that the system is exited by a time harmonic force f (t) = f0e 0 .
TUHH Heinrich Voss Isfahan, July, 2016 6 / 72 Examples Example 1: Vibration of structures
Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are
Mq¨(t) + Cq˙ (t) + Kq(t) = f (t)
where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector.
iω t Suppose that the system is exited by a time harmonic force f (t) = f0e 0 .
If C = αK + βM (modal damping) and (xj , µj ), j = 1,..., n denotes an T orthonormal eigensystem of Kx = µMx (i.e. xk Mxj = δjk ) then the periodic response of the system is
n T X xj f0 q(t) = eiω0t x . µ − ω2 + iω (αµ + β) j j=1 j 0 0 j
Usually, one gets good approximations taking into account only a small number of eigenvalues in the vicinity of iω0 (truncated mode superposition).
TUHH Heinrich Voss Isfahan, July, 2016 6 / 72 Discretizing by finite elements yields (cf. Hager & Wiberg 2000)
K 2 X 1 T (ω)x := ω M + K − ∆Kj x = 0 1 + bj ω j=1
where M is the consistent mass matrix, K is the stiffness matrix with the instantaneous elastic material parameters used in Hooke’s law, and ∆Kj collects the contributions of damping from elements with relaxation parameter bj .
Examples Example 2: Viscoelastic model of damping
Using a viscoelastic constitutive relation to describe the material behavior in the equations of motion yields a rational eigenvalue problem in the case of free vibrations.
TUHH Heinrich Voss Isfahan, July, 2016 7 / 72 Examples Example 2: Viscoelastic model of damping
Using a viscoelastic constitutive relation to describe the material behavior in the equations of motion yields a rational eigenvalue problem in the case of free vibrations.
Discretizing by finite elements yields (cf. Hager & Wiberg 2000)
K 2 X 1 T (ω)x := ω M + K − ∆Kj x = 0 1 + bj ω j=1
where M is the consistent mass matrix, K is the stiffness matrix with the instantaneous elastic material parameters used in Hooke’s law, and ∆Kj collects the contributions of damping from elements with relaxation parameter bj .
TUHH Heinrich Voss Isfahan, July, 2016 7 / 72 In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels.
The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero.
Therefore, a quantum dot can be thought of as an artificial atom.
Examples Example 3: Electronic structure of quantum dots
Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro– and optoelectronic devices.
TUHH Heinrich Voss Isfahan, July, 2016 8 / 72 The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero.
Therefore, a quantum dot can be thought of as an artificial atom.
Examples Example 3: Electronic structure of quantum dots
Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro– and optoelectronic devices.
In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels.
TUHH Heinrich Voss Isfahan, July, 2016 8 / 72 Examples Example 3: Electronic structure of quantum dots
Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro– and optoelectronic devices.
In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels.
The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero.
Therefore, a quantum dot can be thought of as an artificial atom.
TUHH Heinrich Voss Isfahan, July, 2016 8 / 72 Examples Example 3 ct.
Electron Spectroscopy Group, Fritz-Haber-Institute, Berlin TUHH Heinrich Voss Isfahan, July, 2016 9 / 72 Boundary and interface conditions
Φ = 0 on outer boundary of matrix Ωm
1 ∂Φ 1 ∂Φ BenDaniel–Duke condition = on interface mm ∂n mq ∂n ∂Ωm ∂Ωq
Details in Talk tomorrow
m and V are discontinous across the heterojunction.
Examples Example 3 ct.
Governing equation: Schrodinger¨ equation
2 −∇ · ~ ∇Φ + V (x)Φ = EΦ, x ∈ Ω ∪ Ω 2m(x, E) q m
where ~ is the reduced Planck constant, m(x, E) is the electron effective mass depending on the energy state E, and V (x) is the confinement potential.
TUHH Heinrich Voss Isfahan, July, 2016 10 / 72 Boundary and interface conditions
Φ = 0 on outer boundary of matrix Ωm
1 ∂Φ 1 ∂Φ BenDaniel–Duke condition = on interface mm ∂n mq ∂n ∂Ωm ∂Ωq
Details in Talk tomorrow
Examples Example 3 ct.
Governing equation: Schrodinger¨ equation
2 −∇ · ~ ∇Φ + V (x)Φ = EΦ, x ∈ Ω ∪ Ω 2m(x, E) q m
where ~ is the reduced Planck constant, m(x, E) is the electron effective mass depending on the energy state E, and V (x) is the confinement potential.
m and V are discontinous across the heterojunction.
TUHH Heinrich Voss Isfahan, July, 2016 10 / 72 Details in Talk tomorrow
Examples Example 3 ct.
Governing equation: Schrodinger¨ equation
2 −∇ · ~ ∇Φ + V (x)Φ = EΦ, x ∈ Ω ∪ Ω 2m(x, E) q m
where ~ is the reduced Planck constant, m(x, E) is the electron effective mass depending on the energy state E, and V (x) is the confinement potential.
m and V are discontinous across the heterojunction.
Boundary and interface conditions
Φ = 0 on outer boundary of matrix Ωm
1 ∂Φ 1 ∂Φ BenDaniel–Duke condition = on interface mm ∂n mq ∂n ∂Ωm ∂Ωq
TUHH Heinrich Voss Isfahan, July, 2016 10 / 72 Examples Example 3 ct.
Governing equation: Schrodinger¨ equation
2 −∇ · ~ ∇Φ + V (x)Φ = EΦ, x ∈ Ω ∪ Ω 2m(x, E) q m
where ~ is the reduced Planck constant, m(x, E) is the electron effective mass depending on the energy state E, and V (x) is the confinement potential.
m and V are discontinous across the heterojunction.
Boundary and interface conditions
Φ = 0 on outer boundary of matrix Ωm
1 ∂Φ 1 ∂Φ BenDaniel–Duke condition = on interface mm ∂n mq ∂n ∂Ωm ∂Ωq
Details in Talk tomorrow
TUHH Heinrich Voss Isfahan, July, 2016 10 / 72 Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Recent survey: Tisseur, Meerbergen 2001 2 n×n T (λ) = λ A + λB + C, A, B, C ∈ C
Dynamic analysis of structures Stability analysis of flows in fluid mechanics Signal processing Vibration of spinning structures Vibration of fluid-solid structures (Conca et al. 1992) Lateral buckling analysis (Prandtl 1899) Corner singularities of anisotropic material (Wendland et al. 1992) Constrained least squares problems (Golub 1973) Regularization of total least squares problems (Sima et al. 2004)
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems
k X j n×n T (λ) = λ Aj , Aj ∈ C j=0
Optimal control problems (Mehrmann 1991) Singularities in linear elasticity near vertex of a cone (Kozlov et al. 1992) Dynamic element discretization of linear eigenproblems (V. 1987) Least squares element methods (Rothe 1989) System of coupled Euler-Bernoulli and Timoshenko beams (Balakrishnan et al. 2004) Nonlinear integrated optics (Botchev et al. 2008) Electronic states of quantum dots (Hwang et al. 2005)
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 General nonlinear eigenproblems Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 General nonlinear eigenproblems Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems
NMAT 2 X 1 T (λ) = λ M + K + ∆Km 1 + bmλ m=1 Vibration of structures with viscoelastic damping Vibration of sandwich plates (Soni 1981) Dynamics of plates with concentrated masses (Andreev et al. 1988) Vibration of fluid-solid structures (Conca et al. 1989) Vibration of rails exited by fast trains (Mehrmann, Watkins 2003) Singularities in hydrodynamics near vertex of a cone (Kozlov et al. 1994) Exact condensation of linear eigenvalue problems (Wittrick, Williams 1971) Electronic states of semiconductor heterostructures (Luttinger, Kohn 1954)
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems
p X p T (λ) = K − λM + i λ − σj Wj j=1
Exact dynamic element methods Nonlinear eigenproblems in accelerator design (Igarashi et al. 1995) Vibrations of poroelastic structures (Dazel et al. 2002) Vibro-acoustic behavior of piezoelectric/poroelastic structures (Batifol et al. 2007) Nonlinear integrated optics (Dirichlet-to-Neumann) (Botchev 2008) Stability of acoustic pressure levels in combustion chambers (van Leeuwen 2007)
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Two real parameter complex nonlinear eigenproblem
Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues
p d X s(t) = A s(t) + A s(t − τ ), s(t) = xeλt dt 0 j j j=1 p X −τj λ ⇒ T (λ) = −λI + A0 + e Aj j=1
Stability of time-delay systems (Jarlebring 2008)
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Examples Classes of nonlinear eigenproblems
Quadratic eigenvalue problems Polynomial eigenvalue problems Rational eigenvalue problems General nonlinear eigenproblems Exponential dependence on eigenvalues Two real parameter complex nonlinear eigenproblem
T (ω, τ)x = (iωM + A + e−iωτ B)x = 0
Loss of stability for delay differential equations (Meerbergen, Schroder,¨ Voss 2013)
TUHH Heinrich Voss Isfahan, July, 2016 11 / 72 Basic Properties Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 12 / 72 An eigenvalue hatλ of T (·) has the algebraic multiplicity k, if
d ` d k ( (λ)) = , ` = ,..., − ( (λ)) 6= . ` det T 0 for 0 k 1 and k det T 0 dλ λ=λˆ dλ λ=λˆ
An eigenvalue λˆ is simple, if its algebraic multiplicity is 1.
The geometric multiplicity of an eigenvalue λˆ is the dimension of the kernel of T (λˆ).
An eigenvalue λˆ is called semi–simple, if its algebraic and geometric multiplicities coincide.
Basic Properties Definitions
If T (λˆ)x = 0 has a nontrivial solution xˆ 6= 0, then λˆ is an eigenvalue of T (·) and xˆ is a corresponding (right) eigenvector. Every solution yˆ 6= 0 of the adjoint equation T (λˆ)∗y = 0 is a left eigenvector, and the vector-scalar-vector triplet (yˆ, λ,ˆ xˆ) is called eigentriplet of T (·).
TUHH Heinrich Voss Isfahan, July, 2016 13 / 72 An eigenvalue λˆ is simple, if its algebraic multiplicity is 1.
The geometric multiplicity of an eigenvalue λˆ is the dimension of the kernel of T (λˆ).
An eigenvalue λˆ is called semi–simple, if its algebraic and geometric multiplicities coincide.
Basic Properties Definitions
If T (λˆ)x = 0 has a nontrivial solution xˆ 6= 0, then λˆ is an eigenvalue of T (·) and xˆ is a corresponding (right) eigenvector. Every solution yˆ 6= 0 of the adjoint equation T (λˆ)∗y = 0 is a left eigenvector, and the vector-scalar-vector triplet (yˆ, λ,ˆ xˆ) is called eigentriplet of T (·).
An eigenvalue hatλ of T (·) has the algebraic multiplicity k, if
d ` d k ( (λ)) = , ` = ,..., − ( (λ)) 6= . ` det T 0 for 0 k 1 and k det T 0 dλ λ=λˆ dλ λ=λˆ
TUHH Heinrich Voss Isfahan, July, 2016 13 / 72 The geometric multiplicity of an eigenvalue λˆ is the dimension of the kernel of T (λˆ).
An eigenvalue λˆ is called semi–simple, if its algebraic and geometric multiplicities coincide.
Basic Properties Definitions
If T (λˆ)x = 0 has a nontrivial solution xˆ 6= 0, then λˆ is an eigenvalue of T (·) and xˆ is a corresponding (right) eigenvector. Every solution yˆ 6= 0 of the adjoint equation T (λˆ)∗y = 0 is a left eigenvector, and the vector-scalar-vector triplet (yˆ, λ,ˆ xˆ) is called eigentriplet of T (·).
An eigenvalue hatλ of T (·) has the algebraic multiplicity k, if
d ` d k ( (λ)) = , ` = ,..., − ( (λ)) 6= . ` det T 0 for 0 k 1 and k det T 0 dλ λ=λˆ dλ λ=λˆ
An eigenvalue λˆ is simple, if its algebraic multiplicity is 1.
TUHH Heinrich Voss Isfahan, July, 2016 13 / 72 An eigenvalue λˆ is called semi–simple, if its algebraic and geometric multiplicities coincide.
Basic Properties Definitions
If T (λˆ)x = 0 has a nontrivial solution xˆ 6= 0, then λˆ is an eigenvalue of T (·) and xˆ is a corresponding (right) eigenvector. Every solution yˆ 6= 0 of the adjoint equation T (λˆ)∗y = 0 is a left eigenvector, and the vector-scalar-vector triplet (yˆ, λ,ˆ xˆ) is called eigentriplet of T (·).
An eigenvalue hatλ of T (·) has the algebraic multiplicity k, if
d ` d k ( (λ)) = , ` = ,..., − ( (λ)) 6= . ` det T 0 for 0 k 1 and k det T 0 dλ λ=λˆ dλ λ=λˆ
An eigenvalue λˆ is simple, if its algebraic multiplicity is 1.
The geometric multiplicity of an eigenvalue λˆ is the dimension of the kernel of T (λˆ).
TUHH Heinrich Voss Isfahan, July, 2016 13 / 72 Basic Properties Definitions
If T (λˆ)x = 0 has a nontrivial solution xˆ 6= 0, then λˆ is an eigenvalue of T (·) and xˆ is a corresponding (right) eigenvector. Every solution yˆ 6= 0 of the adjoint equation T (λˆ)∗y = 0 is a left eigenvector, and the vector-scalar-vector triplet (yˆ, λ,ˆ xˆ) is called eigentriplet of T (·).
An eigenvalue hatλ of T (·) has the algebraic multiplicity k, if
d ` d k ( (λ)) = , ` = ,..., − ( (λ)) 6= . ` det T 0 for 0 k 1 and k det T 0 dλ λ=λˆ dλ λ=λˆ
An eigenvalue λˆ is simple, if its algebraic multiplicity is 1.
The geometric multiplicity of an eigenvalue λˆ is the dimension of the kernel of T (λˆ).
An eigenvalue λˆ is called semi–simple, if its algebraic and geometric multiplicities coincide.
TUHH Heinrich Voss Isfahan, July, 2016 13 / 72 Example 1 For the quadratic eigenvalue problem T (λ)x = 0 with
0 1 7 −5 1 0 T (λ) := + λ + λ2 −2 3 10 −8 0 1
the distinct eigenvalues λ = 1 and λ = 2 share the eigenvector [1, 2]T
The following facts known for linear eigenproblems are not true for nonlinear eigenproblem: 1. eigenvectors corresponding to distinct eigenvalues are linearly independent:
Basic Properties Facts
For A ∈ Cn×n and T (λ) := λI − A the terms (left and right) eigenvector, eigenpair, eigentriplet, spectrum, algebraic and geometric multiplicity, and semi–simple have their standard meaning.
TUHH Heinrich Voss Isfahan, July, 2016 14 / 72 Example 1 For the quadratic eigenvalue problem T (λ)x = 0 with
0 1 7 −5 1 0 T (λ) := + λ + λ2 −2 3 10 −8 0 1
the distinct eigenvalues λ = 1 and λ = 2 share the eigenvector [1, 2]T
Basic Properties Facts
For A ∈ Cn×n and T (λ) := λI − A the terms (left and right) eigenvector, eigenpair, eigentriplet, spectrum, algebraic and geometric multiplicity, and semi–simple have their standard meaning.
The following facts known for linear eigenproblems are not true for nonlinear eigenproblem: 1. eigenvectors corresponding to distinct eigenvalues are linearly independent:
TUHH Heinrich Voss Isfahan, July, 2016 14 / 72 Basic Properties Facts
For A ∈ Cn×n and T (λ) := λI − A the terms (left and right) eigenvector, eigenpair, eigentriplet, spectrum, algebraic and geometric multiplicity, and semi–simple have their standard meaning.
The following facts known for linear eigenproblems are not true for nonlinear eigenproblem: 1. eigenvectors corresponding to distinct eigenvalues are linearly independent:
Example 1 For the quadratic eigenvalue problem T (λ)x = 0 with
0 1 7 −5 1 0 T (λ) := + λ + λ2 −2 3 10 −8 0 1
the distinct eigenvalues λ = 1 and λ = 2 share the eigenvector [1, 2]T
TUHH Heinrich Voss Isfahan, July, 2016 14 / 72 Example 2 2 eiλ 1 T (λ)x := x = 0 1 1 √ has a countable set of eigenvalues 2kπ, k ∈ N0.
λˆ = 0 is an algebraically double and geometrically simple eigenvalue with left and right eigenvectors xˆ = yˆ = [1; −1]T , and yˆH T 0(0)xˆ = 0. √ ˆ Every λk = 2kπ is algebraically and geometrically simple with the same eigenvectors xˆ, yˆ as before, and √ H 0 ˆ yˆ T (λk )xˆ = 2 2kπi 6= 0.
Basic Properties Facts
2. Left and right eigenvectors corresponding to distinct eigenvalues are orthogonal.
TUHH Heinrich Voss Isfahan, July, 2016 15 / 72 λˆ = 0 is an algebraically double and geometrically simple eigenvalue with left and right eigenvectors xˆ = yˆ = [1; −1]T , and yˆH T 0(0)xˆ = 0. √ ˆ Every λk = 2kπ is algebraically and geometrically simple with the same eigenvectors xˆ, yˆ as before, and √ H 0 ˆ yˆ T (λk )xˆ = 2 2kπi 6= 0.
Basic Properties Facts
2. Left and right eigenvectors corresponding to distinct eigenvalues are orthogonal.
Example 2 2 eiλ 1 T (λ)x := x = 0 1 1 √ has a countable set of eigenvalues 2kπ, k ∈ N0.
TUHH Heinrich Voss Isfahan, July, 2016 15 / 72 √ ˆ Every λk = 2kπ is algebraically and geometrically simple with the same eigenvectors xˆ, yˆ as before, and √ H 0 ˆ yˆ T (λk )xˆ = 2 2kπi 6= 0.
Basic Properties Facts
2. Left and right eigenvectors corresponding to distinct eigenvalues are orthogonal.
Example 2 2 eiλ 1 T (λ)x := x = 0 1 1 √ has a countable set of eigenvalues 2kπ, k ∈ N0.
λˆ = 0 is an algebraically double and geometrically simple eigenvalue with left and right eigenvectors xˆ = yˆ = [1; −1]T , and yˆH T 0(0)xˆ = 0.
TUHH Heinrich Voss Isfahan, July, 2016 15 / 72 Basic Properties Facts
2. Left and right eigenvectors corresponding to distinct eigenvalues are orthogonal.
Example 2 2 eiλ 1 T (λ)x := x = 0 1 1 √ has a countable set of eigenvalues 2kπ, k ∈ N0.
λˆ = 0 is an algebraically double and geometrically simple eigenvalue with left and right eigenvectors xˆ = yˆ = [1; −1]T , and yˆH T 0(0)xˆ = 0. √ ˆ Every λk = 2kπ is algebraically and geometrically simple with the same eigenvectors xˆ, yˆ as before, and √ H 0 ˆ yˆ T (λk )xˆ = 2 2kπi 6= 0.
TUHH Heinrich Voss Isfahan, July, 2016 15 / 72 Example 3: k T (λ) = [λ ], k ∈ N has the eigenvalue λˆ = 0 with algebraic multiplicity k.
Basic Properties Facts
3. The algebraic multiplicities of all eigenvalues sum up to the dimension of the problem, whereas for nonlinear problems there may exist an infinite number of eigenvalues (cf. Example in item 2.), and an eigenvalue may have any algebraic multiplicity.
TUHH Heinrich Voss Isfahan, July, 2016 16 / 72 Basic Properties Facts
3. The algebraic multiplicities of all eigenvalues sum up to the dimension of the problem, whereas for nonlinear problems there may exist an infinite number of eigenvalues (cf. Example in item 2.), and an eigenvalue may have any algebraic multiplicity.
Example 3: k T (λ) = [λ ], k ∈ N has the eigenvalue λˆ = 0 with algebraic multiplicity k.
TUHH Heinrich Voss Isfahan, July, 2016 16 / 72 1. If λˆ is an algebraically simple eigenvalue of T (·), then λˆ is geometrically simple.
2. Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·). The λˆ is algebraically simple if and only if λˆ is geometrically simple and
yˆH T 0(λ)xˆ 6= 0.
The proofs (cf. Schreiber 2008) require only basic linear algebra, but are quite lengthy.
Basic Properties Facts
The following facts known for linear eigenproblems are also true for nonlinear eigenproblem:
TUHH Heinrich Voss Isfahan, July, 2016 17 / 72 2. Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·). The λˆ is algebraically simple if and only if λˆ is geometrically simple and
yˆH T 0(λ)xˆ 6= 0.
The proofs (cf. Schreiber 2008) require only basic linear algebra, but are quite lengthy.
Basic Properties Facts
The following facts known for linear eigenproblems are also true for nonlinear eigenproblem:
1. If λˆ is an algebraically simple eigenvalue of T (·), then λˆ is geometrically simple.
TUHH Heinrich Voss Isfahan, July, 2016 17 / 72 The proofs (cf. Schreiber 2008) require only basic linear algebra, but are quite lengthy.
Basic Properties Facts
The following facts known for linear eigenproblems are also true for nonlinear eigenproblem:
1. If λˆ is an algebraically simple eigenvalue of T (·), then λˆ is geometrically simple.
2. Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·). The λˆ is algebraically simple if and only if λˆ is geometrically simple and
yˆH T 0(λ)xˆ 6= 0.
TUHH Heinrich Voss Isfahan, July, 2016 17 / 72 Basic Properties Facts
The following facts known for linear eigenproblems are also true for nonlinear eigenproblem:
1. If λˆ is an algebraically simple eigenvalue of T (·), then λˆ is geometrically simple.
2. Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·). The λˆ is algebraically simple if and only if λˆ is geometrically simple and
yˆH T 0(λ)xˆ 6= 0.
The proofs (cf. Schreiber 2008) require only basic linear algebra, but are quite lengthy.
TUHH Heinrich Voss Isfahan, July, 2016 17 / 72 4. If T (·) is complex and x is a right eigenvector, then x is a left eigenvector corresponding to the same eigenvalue.
5. If T (·) is Hermitian, then eigenvalues are real (and left and right eigenvectors corresponding to λ coincide), or they come in pairs, i.e. if (y, λ, x) is an eigentriplet of T (·), then this is also true for (x, λ, y).
The proofs require only basic linear algebra they are not lengthy, but are left as an exercise.
Basic Properties Facts
3. If T (·) is real symmetric and λ is a real eigenvalue, then left and right eigenvectors corresponding to λ coincide.
TUHH Heinrich Voss Isfahan, July, 2016 18 / 72 5. If T (·) is Hermitian, then eigenvalues are real (and left and right eigenvectors corresponding to λ coincide), or they come in pairs, i.e. if (y, λ, x) is an eigentriplet of T (·), then this is also true for (x, λ, y).
The proofs require only basic linear algebra they are not lengthy, but are left as an exercise.
Basic Properties Facts
3. If T (·) is real symmetric and λ is a real eigenvalue, then left and right eigenvectors corresponding to λ coincide.
4. If T (·) is complex and x is a right eigenvector, then x is a left eigenvector corresponding to the same eigenvalue.
TUHH Heinrich Voss Isfahan, July, 2016 18 / 72 The proofs require only basic linear algebra they are not lengthy, but are left as an exercise.
Basic Properties Facts
3. If T (·) is real symmetric and λ is a real eigenvalue, then left and right eigenvectors corresponding to λ coincide.
4. If T (·) is complex and x is a right eigenvector, then x is a left eigenvector corresponding to the same eigenvalue.
5. If T (·) is Hermitian, then eigenvalues are real (and left and right eigenvectors corresponding to λ coincide), or they come in pairs, i.e. if (y, λ, x) is an eigentriplet of T (·), then this is also true for (x, λ, y).
TUHH Heinrich Voss Isfahan, July, 2016 18 / 72 Basic Properties Facts
3. If T (·) is real symmetric and λ is a real eigenvalue, then left and right eigenvectors corresponding to λ coincide.
4. If T (·) is complex and x is a right eigenvector, then x is a left eigenvector corresponding to the same eigenvalue.
5. If T (·) is Hermitian, then eigenvalues are real (and left and right eigenvectors corresponding to λ coincide), or they come in pairs, i.e. if (y, λ, x) is an eigentriplet of T (·), then this is also true for (x, λ, y).
The proofs require only basic linear algebra they are not lengthy, but are left as an exercise.
TUHH Heinrich Voss Isfahan, July, 2016 18 / 72 Theorem (Schreiber 2008) Let (yˆ, λ,ˆ xˆ) be an eigentriplet with simple eigenvalue λˆ. Then for sufficiently small |λ − λˆ| 1 xˆyˆH T (λ)−1 = + O(1). λ − λˆ yˆH T 0(λˆ)xˆ
Representations for the resolvent (A − λI)−1 for an arbitrary matrix A can be derived by several available decompositions, such as the spectral decomposition, Schur decomposition, and singular value decomposition. It turns out that in this case the norm of the inverse shifted operator depends on −1 the eigenvalue gap (minλi ∈σ(A)(λi − λ)) . For the nonlinear case we have
Basic Properties Facts
Theorem (Neumeier 1985) Let T (·) be differentiable and let λˆ be a simple eigenvalue of T (·). Let xˆ be a right eigenvector normalized such that v H x = 1 for some vector v. Then the matrix B := T (λˆ) + T 0(λˆ)xvˆ H is nonsingular.
TUHH Heinrich Voss Isfahan, July, 2016 19 / 72 Theorem (Schreiber 2008) Let (yˆ, λ,ˆ xˆ) be an eigentriplet with simple eigenvalue λˆ. Then for sufficiently small |λ − λˆ| 1 xˆyˆH T (λ)−1 = + O(1). λ − λˆ yˆH T 0(λˆ)xˆ
Basic Properties Facts
Theorem (Neumeier 1985) Let T (·) be differentiable and let λˆ be a simple eigenvalue of T (·). Let xˆ be a right eigenvector normalized such that v H x = 1 for some vector v. Then the matrix B := T (λˆ) + T 0(λˆ)xvˆ H is nonsingular.
Representations for the resolvent (A − λI)−1 for an arbitrary matrix A can be derived by several available decompositions, such as the spectral decomposition, Schur decomposition, and singular value decomposition. It turns out that in this case the norm of the inverse shifted operator depends on −1 the eigenvalue gap (minλi ∈σ(A)(λi − λ)) . For the nonlinear case we have
TUHH Heinrich Voss Isfahan, July, 2016 19 / 72 Basic Properties Facts
Theorem (Neumeier 1985) Let T (·) be differentiable and let λˆ be a simple eigenvalue of T (·). Let xˆ be a right eigenvector normalized such that v H x = 1 for some vector v. Then the matrix B := T (λˆ) + T 0(λˆ)xvˆ H is nonsingular.
Representations for the resolvent (A − λI)−1 for an arbitrary matrix A can be derived by several available decompositions, such as the spectral decomposition, Schur decomposition, and singular value decomposition. It turns out that in this case the norm of the inverse shifted operator depends on −1 the eigenvalue gap (minλi ∈σ(A)(λi − λ)) . For the nonlinear case we have
Theorem (Schreiber 2008) Let (yˆ, λ,ˆ xˆ) be an eigentriplet with simple eigenvalue λˆ. Then for sufficiently small |λ − λˆ| 1 xˆyˆH T (λ)−1 = + O(1). λ − λˆ yˆH T 0(λˆ)xˆ
TUHH Heinrich Voss Isfahan, July, 2016 19 / 72 Rayleigh Functional Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 20 / 72 Let T : C ⊃ D → Cn×n be differentiable in an open set D with Lipschitz continuous derivative. Let (λ,ˆ xˆ) be an eigenpair of T (·), and define neighborhoods B(λ,ˆ τ) := {λ ∈ C : |λ − λˆ| < τ} and n ˆ Kε(xˆ) := {x ∈ C : ∠(span{x}, span{xˆ}) ≤ ε} of λ and xˆ, respectively.
Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·). p : Kε(xˆ) × Kε(yˆ) → B(λ,ˆ τ) is a two-sided Rayleigh functional if the following conditions hold for every x ∈ Kε(xˆ) and every y ∈ Kε(yˆ): (i) p(αx, βy) = p(x, y) for every α, β ∈ C \{0} (ii) y H T (p(x, y))x = 0 (iii) y H T 0(p(x, y))x 6= 0.
Rayleigh Functional Rayleigh Functional
In this section we collect results on the existence and approximation properties of a Rayleigh functional in a vicinity of eigenvectors corresponding to algebraically simple eigenvalues. Rayleigh functionals will play a completely different, but also important roll in the talk on variational characterizations of eigenvalues tomorrow.
TUHH Heinrich Voss Isfahan, July, 2016 21 / 72 Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·). p : Kε(xˆ) × Kε(yˆ) → B(λ,ˆ τ) is a two-sided Rayleigh functional if the following conditions hold for every x ∈ Kε(xˆ) and every y ∈ Kε(yˆ): (i) p(αx, βy) = p(x, y) for every α, β ∈ C \{0} (ii) y H T (p(x, y))x = 0 (iii) y H T 0(p(x, y))x 6= 0.
Rayleigh Functional Rayleigh Functional
In this section we collect results on the existence and approximation properties of a Rayleigh functional in a vicinity of eigenvectors corresponding to algebraically simple eigenvalues. Rayleigh functionals will play a completely different, but also important roll in the talk on variational characterizations of eigenvalues tomorrow.
Let T : C ⊃ D → Cn×n be differentiable in an open set D with Lipschitz continuous derivative. Let (λ,ˆ xˆ) be an eigenpair of T (·), and define neighborhoods B(λ,ˆ τ) := {λ ∈ C : |λ − λˆ| < τ} and n ˆ Kε(xˆ) := {x ∈ C : ∠(span{x}, span{xˆ}) ≤ ε} of λ and xˆ, respectively.
TUHH Heinrich Voss Isfahan, July, 2016 21 / 72 Rayleigh Functional Rayleigh Functional
In this section we collect results on the existence and approximation properties of a Rayleigh functional in a vicinity of eigenvectors corresponding to algebraically simple eigenvalues. Rayleigh functionals will play a completely different, but also important roll in the talk on variational characterizations of eigenvalues tomorrow.
Let T : C ⊃ D → Cn×n be differentiable in an open set D with Lipschitz continuous derivative. Let (λ,ˆ xˆ) be an eigenpair of T (·), and define neighborhoods B(λ,ˆ τ) := {λ ∈ C : |λ − λˆ| < τ} and n ˆ Kε(xˆ) := {x ∈ C : ∠(span{x}, span{xˆ}) ≤ ε} of λ and xˆ, respectively.
Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·). p : Kε(xˆ) × Kε(yˆ) → B(λ,ˆ τ) is a two-sided Rayleigh functional if the following conditions hold for every x ∈ Kε(xˆ) and every y ∈ Kε(yˆ): (i) p(αx, βy) = p(x, y) for every α, β ∈ C \{0} (ii) y H T (p(x, y))x = 0 (iii) y H T 0(p(x, y))x 6= 0.
TUHH Heinrich Voss Isfahan, July, 2016 21 / 72 Theorem 2 (Schwetlick, Schreiber 2012) Under the conditions of Thm. 1 let ξ < π/3 and η < π/3. Then
32 kT (λˆ)k |p(x, y) − λˆ| ≤ kx − xˆkky − yˆk., 3 |yˆH T 0(λˆ)xˆ| where ξ := ∠(span{x}, span{xˆ}).
Rayleigh Functional Rayleigh Functional
Theorem 1 (Schwetlick, Schreiber 2012) Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·) with kxˆk = kyˆk = 1, and assume that yˆH T 0(λˆ)xˆ 6= 0. Then there exist ε > 0 and τ > 0 such that the two-sided Rayleigh functional p(·) is defined in Kε(xˆ) × Kε(yˆ), and it holds that
8 kT (λˆ)k |p(x, y) − λˆ| ≤ tan ξ tan η, 3 |yˆH T 0(λˆ)xˆ|
where ξ := ∠(span{x}, span{xˆ}) and η := ∠(span{y}, span{yˆ}).
TUHH Heinrich Voss Isfahan, July, 2016 22 / 72 Rayleigh Functional Rayleigh Functional
Theorem 1 (Schwetlick, Schreiber 2012) Let (yˆ, λ,ˆ xˆ) be an eigentriplet of T (·) with kxˆk = kyˆk = 1, and assume that yˆH T 0(λˆ)xˆ 6= 0. Then there exist ε > 0 and τ > 0 such that the two-sided Rayleigh functional p(·) is defined in Kε(xˆ) × Kε(yˆ), and it holds that
8 kT (λˆ)k |p(x, y) − λˆ| ≤ tan ξ tan η, 3 |yˆH T 0(λˆ)xˆ|
where ξ := ∠(span{x}, span{xˆ}) and η := ∠(span{y}, span{yˆ}).
Theorem 2 (Schwetlick, Schreiber 2012) Under the conditions of Thm. 1 let ξ < π/3 and η < π/3. Then
32 kT (λˆ)k |p(x, y) − λˆ| ≤ kx − xˆkky − yˆk., 3 |yˆH T 0(λˆ)xˆ| where ξ := ∠(span{x}, span{xˆ}).
TUHH Heinrich Voss Isfahan, July, 2016 22 / 72 Definition p : Kε → B(λ,ˆ τ) is a (one-sided) Rayleigh functional if the following conditions hold: (i) p(αx) = p(x) for every α ∈ C, α 6= 0 H (ii) x T (p(x))x = 0 for every x ∈ Kε(xˆ) H 0 (iii) x T (p(x))x 6= 0 for every x ∈ Kε(xˆ).
Rayleigh Functional Rayleigh Functional
Theorem 3 (Schwetlick, Schreiber 2012) Unter the conditions of Thm. 1 the two–sided Rayleigh functional is stationary at (xˆ, yˆ), i.e. |p(xˆ + s, yˆ + t) − λˆ| = O((ksk + ktk)2).
TUHH Heinrich Voss Isfahan, July, 2016 23 / 72 Rayleigh Functional Rayleigh Functional
Theorem 3 (Schwetlick, Schreiber 2012) Unter the conditions of Thm. 1 the two–sided Rayleigh functional is stationary at (xˆ, yˆ), i.e. |p(xˆ + s, yˆ + t) − λˆ| = O((ksk + ktk)2).
Definition p : Kε → B(λ,ˆ τ) is a (one-sided) Rayleigh functional if the following conditions hold: (i) p(αx) = p(x) for every α ∈ C, α 6= 0 H (ii) x T (p(x))x = 0 for every x ∈ Kε(xˆ) H 0 (iii) x T (p(x))x 6= 0 for every x ∈ Kε(xˆ).
TUHH Heinrich Voss Isfahan, July, 2016 23 / 72 Theorem 4 (Schwetlick, Schreiber 2012) Let (λ,ˆ xˆ) be an eigenpair of T (·) with kxˆk = 1 and xˆH T 0(λˆ)xˆ 6= 0. Then there exist ε > 0 and τ > 0 such that the one-sided Rayleigh functional p(·) is defined in Kε(xˆ), and it holds that 10 kT (λˆ)k |p(x) − λˆ| ≤ tan ξ, 3 |xˆH T 0(λˆ)xˆ|
where ξ := ∠(span{x}, span{xˆ}).
Rayleigh Functional Rayleigh Functional
Theorem 3 (Schwetlick, Schreiber 2012) Let (λ,ˆ xˆ) be an eigenpair of T (·) with kxˆk = 1 and xˆH T 0(λˆ)xˆ 6= 0, and suppose that T (λˆ) = T (λˆ)H . Then there exist ε > 0 and τ > 0 such that the one-sided Rayleigh functional p(·) is defined in Kε(xˆ), and it holds that 8 kT (λˆ)k |p(x) − λˆ| ≤ tan2 ξ, 3 |xˆH T 0(λˆ)xˆ|
where ξ := ∠(span{x}, span{xˆ}).
TUHH Heinrich Voss Isfahan, July, 2016 24 / 72 Rayleigh Functional Rayleigh Functional
Theorem 3 (Schwetlick, Schreiber 2012) Let (λ,ˆ xˆ) be an eigenpair of T (·) with kxˆk = 1 and xˆH T 0(λˆ)xˆ 6= 0, and suppose that T (λˆ) = T (λˆ)H . Then there exist ε > 0 and τ > 0 such that the one-sided Rayleigh functional p(·) is defined in Kε(xˆ), and it holds that 8 kT (λˆ)k |p(x) − λˆ| ≤ tan2 ξ, 3 |xˆH T 0(λˆ)xˆ|
where ξ := ∠(span{x}, span{xˆ}). Theorem 4 (Schwetlick, Schreiber 2012) Let (λ,ˆ xˆ) be an eigenpair of T (·) with kxˆk = 1 and xˆH T 0(λˆ)xˆ 6= 0. Then there exist ε > 0 and τ > 0 such that the one-sided Rayleigh functional p(·) is defined in Kε(xˆ), and it holds that 10 kT (λˆ)k |p(x) − λˆ| ≤ tan ξ, 3 |xˆH T 0(λˆ)xˆ|
where ξ := ∠(span{x}, span{xˆ}). TUHH Heinrich Voss Isfahan, July, 2016 24 / 72 Theorem 5 (Schwetlick, Schreiber 2012) Under the conditions of Thm 1 the generalized Rayleigh quotient pL is defined for all λ ∈ B(λ,ˆ τ) and (x, y) ∈ Kε(xˆ) × Kε(yˆ), and it holds that
ˆ ˆ 2 ˆ 4kT (λ)k 2L |λ − λ| |pL(λ, x, y) − λ| ≤ tan ξ tan η + , |yˆH T 0(λˆ)xˆ| |yˆH T 0(λˆ)xˆ| cos ξ cos η
where L denotes the Lipschitz constant of T 0(·).
Rayleigh Functional Rayleigh Functional
In general, the Rayleigh functional is not easy to evaluate. Applying Newton’s method to the defining equation y H T (p(x))x = 0 one obtains the generalized Rayleigh quotient
y H T (λ)x p : B(λ,ˆ τ) × K (xˆ) × K (yˆ) → B(λ,ˆ τ), p (λ,ˆ x, y) := λ − . L ε ε L y H T 0(λ)x
TUHH Heinrich Voss Isfahan, July, 2016 25 / 72 Rayleigh Functional Rayleigh Functional
In general, the Rayleigh functional is not easy to evaluate. Applying Newton’s method to the defining equation y H T (p(x))x = 0 one obtains the generalized Rayleigh quotient
y H T (λ)x p : B(λ,ˆ τ) × K (xˆ) × K (yˆ) → B(λ,ˆ τ), p (λ,ˆ x, y) := λ − . L ε ε L y H T 0(λ)x
Theorem 5 (Schwetlick, Schreiber 2012) Under the conditions of Thm 1 the generalized Rayleigh quotient pL is defined for all λ ∈ B(λ,ˆ τ) and (x, y) ∈ Kε(xˆ) × Kε(yˆ), and it holds that
ˆ ˆ 2 ˆ 4kT (λ)k 2L |λ − λ| |pL(λ, x, y) − λ| ≤ tan ξ tan η + , |yˆH T 0(λˆ)xˆ| |yˆH T 0(λˆ)xˆ| cos ξ cos η
where L denotes the Lipschitz constant of T 0(·).
TUHH Heinrich Voss Isfahan, July, 2016 25 / 72 Methods for dense nonlinear eigenvalue problems Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 26 / 72 For general nonlinear eigenvalue problems, the classical approach is to formulate the eigenvalue problem as a system of nonlinear equations and to use variations of Newton’s method or the inverse iteration method.
Dense solvers only apply to relatively small eigenproblems (maybe dimensions up to 1000, but this depends on the computer in use). For higher dimensions iterative projection methods like Jacobi–Davidson method or the Nonlinear Arnoldi method are more appropriate. Methods of this type are considered in the lecture on fluid–solid vibrations.
Methods for dense nonlinear eigenvalue problems Dense nonlinear eigenproblems
For polynomial or rational eigenproblems the simplest approach is to use linearization and to apply standard methods for linear eigenproblems.
TUHH Heinrich Voss Isfahan, July, 2016 27 / 72 Dense solvers only apply to relatively small eigenproblems (maybe dimensions up to 1000, but this depends on the computer in use). For higher dimensions iterative projection methods like Jacobi–Davidson method or the Nonlinear Arnoldi method are more appropriate. Methods of this type are considered in the lecture on fluid–solid vibrations.
Methods for dense nonlinear eigenvalue problems Dense nonlinear eigenproblems
For polynomial or rational eigenproblems the simplest approach is to use linearization and to apply standard methods for linear eigenproblems.
For general nonlinear eigenvalue problems, the classical approach is to formulate the eigenvalue problem as a system of nonlinear equations and to use variations of Newton’s method or the inverse iteration method.
TUHH Heinrich Voss Isfahan, July, 2016 27 / 72 Methods for dense nonlinear eigenvalue problems Dense nonlinear eigenproblems
For polynomial or rational eigenproblems the simplest approach is to use linearization and to apply standard methods for linear eigenproblems.
For general nonlinear eigenvalue problems, the classical approach is to formulate the eigenvalue problem as a system of nonlinear equations and to use variations of Newton’s method or the inverse iteration method.
Dense solvers only apply to relatively small eigenproblems (maybe dimensions up to 1000, but this depends on the computer in use). For higher dimensions iterative projection methods like Jacobi–Davidson method or the Nonlinear Arnoldi method are more appropriate. Methods of this type are considered in the lecture on fluid–solid vibrations.
TUHH Heinrich Voss Isfahan, July, 2016 27 / 72 Methods for dense nonlinear eigenvalue problems Linearization Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 28 / 72 mn×mn L(λ) = λX + Y , X, Y ∈ C is a Linearization of P(λ) if
P(λ) O E(λ)L(λ)F(λ) = OI(m−1)n
for some unimodular E(λ) and F(λ) (i.e. det E(λ) = ±1,det F(λ) = ±1).
Obviously, λ is an eigenvalue of p(λ) if and only if λ is an eigenvalue of L(λ).
Methods for dense nonlinear eigenvalue problems Linearization Linearization
Let m X j P(λ) := λ Aj j=0 be a regular (i.e. det P(λ) 6≡ 0) matrix polynomial.
TUHH Heinrich Voss Isfahan, July, 2016 29 / 72 Obviously, λ is an eigenvalue of p(λ) if and only if λ is an eigenvalue of L(λ).
Methods for dense nonlinear eigenvalue problems Linearization Linearization
Let m X j P(λ) := λ Aj j=0 be a regular (i.e. det P(λ) 6≡ 0) matrix polynomial.
mn×mn L(λ) = λX + Y , X, Y ∈ C is a Linearization of P(λ) if
P(λ) O E(λ)L(λ)F(λ) = OI(m−1)n
for some unimodular E(λ) and F(λ) (i.e. det E(λ) = ±1,det F(λ) = ±1).
TUHH Heinrich Voss Isfahan, July, 2016 29 / 72 Methods for dense nonlinear eigenvalue problems Linearization Linearization
Let m X j P(λ) := λ Aj j=0 be a regular (i.e. det P(λ) 6≡ 0) matrix polynomial.
mn×mn L(λ) = λX + Y , X, Y ∈ C is a Linearization of P(λ) if
P(λ) O E(λ)L(λ)F(λ) = OI(m−1)n
for some unimodular E(λ) and F(λ) (i.e. det E(λ) = ±1,det F(λ) = ±1).
Obviously, λ is an eigenvalue of p(λ) if and only if λ is an eigenvalue of L(λ).
TUHH Heinrich Voss Isfahan, July, 2016 29 / 72 x is a right eigenvector of P(λ) if and only if
λm−1x λm−2x . z = . λx x
is a right eigenvector of L(λ)
Left eigenvectors: more complicated.
Methods for dense nonlinear eigenvalue problems Linearization Companion form of linearization
A 0 ... 0 m Am−1 Am−2 ... A0 .. . −I 0 ... 0 0 In . . n L(λ) = λ + . . . . . . .. .. ...... . . . 0 . . 0 ... −I 0 0 ... 0 In n
TUHH Heinrich Voss Isfahan, July, 2016 30 / 72 Left eigenvectors: more complicated.
Methods for dense nonlinear eigenvalue problems Linearization Companion form of linearization
A 0 ... 0 m Am−1 Am−2 ... A0 .. . −I 0 ... 0 0 In . . n L(λ) = λ + . . . . . . .. .. ...... . . . 0 . . 0 ... −I 0 0 ... 0 In n
x is a right eigenvector of P(λ) if and only if
λm−1x λm−2x . z = . λx x
is a right eigenvector of L(λ)
TUHH Heinrich Voss Isfahan, July, 2016 30 / 72 Methods for dense nonlinear eigenvalue problems Linearization Companion form of linearization
A 0 ... 0 m Am−1 Am−2 ... A0 .. . −I 0 ... 0 0 In . . n L(λ) = λ + . . . . . . .. .. ...... . . . 0 . . 0 ... −I 0 0 ... 0 In n
x is a right eigenvector of P(λ) if and only if
λm−1x λm−2x . z = . λx x
is a right eigenvector of L(λ)
Left eigenvectors: more complicated. TUHH Heinrich Voss Isfahan, July, 2016 30 / 72 They show that 2 L1 and L2 are vector spaces of dimension m(m − 1)n + m Almost all pencils in L1 and L2 are linearizations of P(λ).
Quadratic case (m = 2): L(λ) = λX + Y ∈ L1(P) iff v A v A v A X X + Y Y 1 2 1 1 1 0 = 11 12 11 12 v2A2 v2A1 v2A0 X21 X22 + Y21 Y22
Methods for dense nonlinear eigenvalue problems Linearization L1 and L2 linearizations
With Λ := (λm−1, λm−2, . . . , λ, 1)T . Mackey, Mackey, Mehl & Mehrmann (2006) define
m L1(P) := {L(λ): L(λ)(Λ ⊗ In) = v ⊗ P(λ), v ∈ C },
T T m L2(P) := {L(λ) : (Λ ⊗ In)L(λ) = w ⊗ P(λ), w ∈ C }.
TUHH Heinrich Voss Isfahan, July, 2016 31 / 72 Quadratic case (m = 2): L(λ) = λX + Y ∈ L1(P) iff v A v A v A X X + Y Y 1 2 1 1 1 0 = 11 12 11 12 v2A2 v2A1 v2A0 X21 X22 + Y21 Y22
Methods for dense nonlinear eigenvalue problems Linearization L1 and L2 linearizations
With Λ := (λm−1, λm−2, . . . , λ, 1)T . Mackey, Mackey, Mehl & Mehrmann (2006) define
m L1(P) := {L(λ): L(λ)(Λ ⊗ In) = v ⊗ P(λ), v ∈ C },
T T m L2(P) := {L(λ) : (Λ ⊗ In)L(λ) = w ⊗ P(λ), w ∈ C }.
They show that 2 L1 and L2 are vector spaces of dimension m(m − 1)n + m Almost all pencils in L1 and L2 are linearizations of P(λ).
TUHH Heinrich Voss Isfahan, July, 2016 31 / 72 Methods for dense nonlinear eigenvalue problems Linearization L1 and L2 linearizations
With Λ := (λm−1, λm−2, . . . , λ, 1)T . Mackey, Mackey, Mehl & Mehrmann (2006) define
m L1(P) := {L(λ): L(λ)(Λ ⊗ In) = v ⊗ P(λ), v ∈ C },
T T m L2(P) := {L(λ) : (Λ ⊗ In)L(λ) = w ⊗ P(λ), w ∈ C }.
They show that 2 L1 and L2 are vector spaces of dimension m(m − 1)n + m Almost all pencils in L1 and L2 are linearizations of P(λ).
Quadratic case (m = 2): L(λ) = λX + Y ∈ L1(P) iff v A v A v A X X + Y Y 1 2 1 1 1 0 = 11 12 11 12 v2A2 v2A1 v2A0 X21 X22 + Y21 Y22
TUHH Heinrich Voss Isfahan, July, 2016 31 / 72 Note
L(λ)(Λ ⊗ x) = L(λ)(Λ ⊗ In)(1 ⊗ x) = (v ⊗ P(λ))(1 ⊗ x) = v ⊗ P(λ)x.
So (x, λ) is an eigenpair of P(λ) iff (Λ ⊗ x, λ) is an eigenpair of L(λ). Right eigenvectors of P can be recovered from right eigenvectors of linearizations in L1. Left eigenvectors of P can be recovered from right eigenvectors of linearizations in L2.
Methods for dense nonlinear eigenvalue problems Linearization L1 and L2 linearizations ct.
Recall m L1(P) := {L(λ): L(λ)(Λ ⊗ In) = v ⊗ P(λ), v ∈ C },
TUHH Heinrich Voss Isfahan, July, 2016 32 / 72 So (x, λ) is an eigenpair of P(λ) iff (Λ ⊗ x, λ) is an eigenpair of L(λ). Right eigenvectors of P can be recovered from right eigenvectors of linearizations in L1. Left eigenvectors of P can be recovered from right eigenvectors of linearizations in L2.
Methods for dense nonlinear eigenvalue problems Linearization L1 and L2 linearizations ct.
Recall m L1(P) := {L(λ): L(λ)(Λ ⊗ In) = v ⊗ P(λ), v ∈ C },
Note
L(λ)(Λ ⊗ x) = L(λ)(Λ ⊗ In)(1 ⊗ x) = (v ⊗ P(λ))(1 ⊗ x) = v ⊗ P(λ)x.
TUHH Heinrich Voss Isfahan, July, 2016 32 / 72 Methods for dense nonlinear eigenvalue problems Linearization L1 and L2 linearizations ct.
Recall m L1(P) := {L(λ): L(λ)(Λ ⊗ In) = v ⊗ P(λ), v ∈ C },
Note
L(λ)(Λ ⊗ x) = L(λ)(Λ ⊗ In)(1 ⊗ x) = (v ⊗ P(λ))(1 ⊗ x) = v ⊗ P(λ)x.
So (x, λ) is an eigenpair of P(λ) iff (Λ ⊗ x, λ) is an eigenpair of L(λ). Right eigenvectors of P can be recovered from right eigenvectors of linearizations in L1. Left eigenvectors of P can be recovered from right eigenvectors of linearizations in L2.
TUHH Heinrich Voss Isfahan, July, 2016 32 / 72 They show that
L ∈ DL(P) iff w = v in the definitions of L1 and L2 DL(P) is a vector space of dimension m Almost all pencils in DL(P) are linearizations of P
For Q(λ) := λ2A + λB + C, L(λ) ∈ DL(Q) iff v1AV2A v1B − v2A v1C L(λ) = λ + , v ∈ C. v2A v2B − v1C v1C v2C
Methods for dense nonlinear eigenvalue problems Linearization DL(P) linearizations
Mackey, Mackey, Mehl & Mehrmann (2006) define
DL(P) = L1(P) ∩ L2(P).
TUHH Heinrich Voss Isfahan, July, 2016 33 / 72 For Q(λ) := λ2A + λB + C, L(λ) ∈ DL(Q) iff v1AV2A v1B − v2A v1C L(λ) = λ + , v ∈ C. v2A v2B − v1C v1C v2C
Methods for dense nonlinear eigenvalue problems Linearization DL(P) linearizations
Mackey, Mackey, Mehl & Mehrmann (2006) define
DL(P) = L1(P) ∩ L2(P).
They show that
L ∈ DL(P) iff w = v in the definitions of L1 and L2 DL(P) is a vector space of dimension m Almost all pencils in DL(P) are linearizations of P
TUHH Heinrich Voss Isfahan, July, 2016 33 / 72 Methods for dense nonlinear eigenvalue problems Linearization DL(P) linearizations
Mackey, Mackey, Mehl & Mehrmann (2006) define
DL(P) = L1(P) ∩ L2(P).
They show that
L ∈ DL(P) iff w = v in the definitions of L1 and L2 DL(P) is a vector space of dimension m Almost all pencils in DL(P) are linearizations of P
For Q(λ) := λ2A + λB + C, L(λ) ∈ DL(Q) iff v1AV2A v1B − v2A v1C L(λ) = λ + , v ∈ C. v2A v2B − v1C v1C v2C
TUHH Heinrich Voss Isfahan, July, 2016 33 / 72 L(λ) = λX + Y is a strong linearization of P(λ), if it is a linearization of P(λ), and X + λY is a linearization of
m m X j rev P(λ) := λ P(1/λ) = λ Am−j . j=0
The Jordan structure of all finite eigenvalues of P(λ) can be recovered from any linearization of P. For strong linearizations this is also the case for the eigenvalue ∞, i.e. the eigenvalue 0 of rev P(λ).
Methods for dense nonlinear eigenvalue problems Linearization Linearization Theorem (4M, 2006)
Let P(λ) be a regular matrix polynomial, and let L(λ) ∈ L1(P). Then the following statements are equivalent L(λ) is a linearization of P(λ) L(λ) is a regular pencil L(λ) is a strong linearization of P(λ)
TUHH Heinrich Voss Isfahan, July, 2016 34 / 72 The Jordan structure of all finite eigenvalues of P(λ) can be recovered from any linearization of P. For strong linearizations this is also the case for the eigenvalue ∞, i.e. the eigenvalue 0 of rev P(λ).
Methods for dense nonlinear eigenvalue problems Linearization Linearization Theorem (4M, 2006)
Let P(λ) be a regular matrix polynomial, and let L(λ) ∈ L1(P). Then the following statements are equivalent L(λ) is a linearization of P(λ) L(λ) is a regular pencil L(λ) is a strong linearization of P(λ)
L(λ) = λX + Y is a strong linearization of P(λ), if it is a linearization of P(λ), and X + λY is a linearization of
m m X j rev P(λ) := λ P(1/λ) = λ Am−j . j=0
TUHH Heinrich Voss Isfahan, July, 2016 34 / 72 Methods for dense nonlinear eigenvalue problems Linearization Linearization Theorem (4M, 2006)
Let P(λ) be a regular matrix polynomial, and let L(λ) ∈ L1(P). Then the following statements are equivalent L(λ) is a linearization of P(λ) L(λ) is a regular pencil L(λ) is a strong linearization of P(λ)
L(λ) = λX + Y is a strong linearization of P(λ), if it is a linearization of P(λ), and X + λY is a linearization of
m m X j rev P(λ) := λ P(1/λ) = λ Am−j . j=0
The Jordan structure of all finite eigenvalues of P(λ) can be recovered from any linearization of P. For strong linearizations this is also the case for the eigenvalue ∞, i.e. the eigenvalue 0 of rev P(λ).
TUHH Heinrich Voss Isfahan, July, 2016 34 / 72 Methods for dense nonlinear eigenvalue problems Methods based on characteristic equation Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 35 / 72 Applying Newton’s method to this equation, one obtains the iteration 1 λ = λ − k+1 k H H 0 −1 en Q(λk ) T (λk )P(λk )R(λk ) en for approximations to an eigenvalue.
Approximations to left and right eigenvectors can be obtained from
−1 yk = Q(λk )en and xk = P(λk )R(λk ) en.
Methods for dense nonlinear eigenvalue problems Methods based on characteristic equation Characteristic equation
For the characteristic equation
det T (λ) = 0,
it was suggested by Kublanovskaya (1969,1970) to use a QR-decomposition with column pivoting T (λ)P(λ) = Q(λ)R(λ), where P(λ) is a permutation matrix which is chosen such that the diagonal elements rjj (λ) of R(λ) are decreasing in magnitude, i.e. |r11(λ)| ≥ |r22(λ)| ≥ · · · ≥ |rnn(λ)|. Then λ is an eigenvalue if and only if rnn(λ) = 0.
TUHH Heinrich Voss Isfahan, July, 2016 36 / 72 Approximations to left and right eigenvectors can be obtained from
−1 yk = Q(λk )en and xk = P(λk )R(λk ) en.
Methods for dense nonlinear eigenvalue problems Methods based on characteristic equation Characteristic equation
For the characteristic equation
det T (λ) = 0,
it was suggested by Kublanovskaya (1969,1970) to use a QR-decomposition with column pivoting T (λ)P(λ) = Q(λ)R(λ), where P(λ) is a permutation matrix which is chosen such that the diagonal elements rjj (λ) of R(λ) are decreasing in magnitude, i.e. |r11(λ)| ≥ |r22(λ)| ≥ · · · ≥ |rnn(λ)|. Then λ is an eigenvalue if and only if rnn(λ) = 0.
Applying Newton’s method to this equation, one obtains the iteration 1 λ = λ − k+1 k H H 0 −1 en Q(λk ) T (λk )P(λk )R(λk ) en for approximations to an eigenvalue.
TUHH Heinrich Voss Isfahan, July, 2016 36 / 72 Methods for dense nonlinear eigenvalue problems Methods based on characteristic equation Characteristic equation
For the characteristic equation
det T (λ) = 0,
it was suggested by Kublanovskaya (1969,1970) to use a QR-decomposition with column pivoting T (λ)P(λ) = Q(λ)R(λ), where P(λ) is a permutation matrix which is chosen such that the diagonal elements rjj (λ) of R(λ) are decreasing in magnitude, i.e. |r11(λ)| ≥ |r22(λ)| ≥ · · · ≥ |rnn(λ)|. Then λ is an eigenvalue if and only if rnn(λ) = 0.
Applying Newton’s method to this equation, one obtains the iteration 1 λ = λ − k+1 k H H 0 −1 en Q(λk ) T (λk )P(λk )R(λk ) en for approximations to an eigenvalue.
Approximations to left and right eigenvectors can be obtained from
−1 yk = Q(λk )en and xk = P(λk )R(λk ) en.
TUHH Heinrich Voss Isfahan, July, 2016 36 / 72 A similar approach was presented by Yang (1983), via a representation of Newton’s method using the LU-factorization of T (λ).
Other variations of this method can be found in the books of Zurmuhl¨ & Falk (1984, 1985).
However, this relatively simple idea is not efficient, since it computes eigenvalues one at a time and needs several O(n3) factorizations per eigenvalue. It is, however, useful in the context of iterative refinement of computed eigenvalues and eigenvectors.
Methods for dense nonlinear eigenvalue problems Methods based on characteristic equation Characteristic equation ct.
An improved version of Kublanovskaya’s method was suggested by Jain, Singhal & Huseyin (1983), and also quadratic convergence was shown.
TUHH Heinrich Voss Isfahan, July, 2016 37 / 72 Other variations of this method can be found in the books of Zurmuhl¨ & Falk (1984, 1985).
However, this relatively simple idea is not efficient, since it computes eigenvalues one at a time and needs several O(n3) factorizations per eigenvalue. It is, however, useful in the context of iterative refinement of computed eigenvalues and eigenvectors.
Methods for dense nonlinear eigenvalue problems Methods based on characteristic equation Characteristic equation ct.
An improved version of Kublanovskaya’s method was suggested by Jain, Singhal & Huseyin (1983), and also quadratic convergence was shown.
A similar approach was presented by Yang (1983), via a representation of Newton’s method using the LU-factorization of T (λ).
TUHH Heinrich Voss Isfahan, July, 2016 37 / 72 However, this relatively simple idea is not efficient, since it computes eigenvalues one at a time and needs several O(n3) factorizations per eigenvalue. It is, however, useful in the context of iterative refinement of computed eigenvalues and eigenvectors.
Methods for dense nonlinear eigenvalue problems Methods based on characteristic equation Characteristic equation ct.
An improved version of Kublanovskaya’s method was suggested by Jain, Singhal & Huseyin (1983), and also quadratic convergence was shown.
A similar approach was presented by Yang (1983), via a representation of Newton’s method using the LU-factorization of T (λ).
Other variations of this method can be found in the books of Zurmuhl¨ & Falk (1984, 1985).
TUHH Heinrich Voss Isfahan, July, 2016 37 / 72 Methods for dense nonlinear eigenvalue problems Methods based on characteristic equation Characteristic equation ct.
An improved version of Kublanovskaya’s method was suggested by Jain, Singhal & Huseyin (1983), and also quadratic convergence was shown.
A similar approach was presented by Yang (1983), via a representation of Newton’s method using the LU-factorization of T (λ).
Other variations of this method can be found in the books of Zurmuhl¨ & Falk (1984, 1985).
However, this relatively simple idea is not efficient, since it computes eigenvalues one at a time and needs several O(n3) factorizations per eigenvalue. It is, however, useful in the context of iterative refinement of computed eigenvalues and eigenvectors.
TUHH Heinrich Voss Isfahan, July, 2016 37 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 38 / 72 1: start with an approximation λ1 ∈ D to an eigenvalue of T (λ)x = 0 2: for k = 1, 2,... until convergence do 0 3: solve the linear eigenproblem T (λk )x = θT (λk )x 4: choose suitable eigenvalue θ (usually smallest in modulus) 5: set λk+1 = λk − θ 6: end for
Methods for dense nonlinear eigenvalue problems Newton’s method Successive linear approximations
Let λk be an approximation to an eigenvalue of T (λ)x = 0. Linearizing T (λk − θ)x = 0 yields 0 T (λk )x = θT (λk )x. This suggests the method
TUHH Heinrich Voss Isfahan, July, 2016 39 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Successive linear approximations
Let λk be an approximation to an eigenvalue of T (λ)x = 0. Linearizing T (λk − θ)x = 0 yields 0 T (λk )x = θT (λk )x. This suggests the method
1: start with an approximation λ1 ∈ D to an eigenvalue of T (λ)x = 0 2: for k = 1, 2,... until convergence do 0 3: solve the linear eigenproblem T (λk )x = θT (λk )x 4: choose suitable eigenvalue θ (usually smallest in modulus) 5: set λk+1 = λk − θ 6: end for
TUHH Heinrich Voss Isfahan, July, 2016 39 / 72 0 ˆ θ (λ) = 1 yields the quadratic convergence of the method λk+1 = λk − θ(λk ).
Sketch of proof: For
T (λ)x − θT 0(λ)x Φ(x, θ, λ) := `H x − 1
it holds Φ(xˆ, 0, λˆ) = 0, and by the implicit function theorem Φ(x, θ, λ) = 0 defines differentiable functions x : U(λˆ) → Cn and θ : U(λˆ) → C on a neighborhood of λˆ such that Φ(x(λ), θ(λ), λ) = 0.
Methods for dense nonlinear eigenvalue problems Newton’s method Successive linear approximations ct.
THEOREM Let T (λ) be twice continuously differentiable, and let λˆ be an eigenvalue of T (λ)x = 0 such that T 0(λˆ) is nonsingular and 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ). Then the method of successive linear problems converges quadratically to λˆ.
TUHH Heinrich Voss Isfahan, July, 2016 40 / 72 0 ˆ θ (λ) = 1 yields the quadratic convergence of the method λk+1 = λk − θ(λk ).
Methods for dense nonlinear eigenvalue problems Newton’s method Successive linear approximations ct.
THEOREM Let T (λ) be twice continuously differentiable, and let λˆ be an eigenvalue of T (λ)x = 0 such that T 0(λˆ) is nonsingular and 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ). Then the method of successive linear problems converges quadratically to λˆ.
Sketch of proof: For
T (λ)x − θT 0(λ)x Φ(x, θ, λ) := `H x − 1
it holds Φ(xˆ, 0, λˆ) = 0, and by the implicit function theorem Φ(x, θ, λ) = 0 defines differentiable functions x : U(λˆ) → Cn and θ : U(λˆ) → C on a neighborhood of λˆ such that Φ(x(λ), θ(λ), λ) = 0.
TUHH Heinrich Voss Isfahan, July, 2016 40 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Successive linear approximations ct.
THEOREM Let T (λ) be twice continuously differentiable, and let λˆ be an eigenvalue of T (λ)x = 0 such that T 0(λˆ) is nonsingular and 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ). Then the method of successive linear problems converges quadratically to λˆ.
Sketch of proof: For
T (λ)x − θT 0(λ)x Φ(x, θ, λ) := `H x − 1
it holds Φ(xˆ, 0, λˆ) = 0, and by the implicit function theorem Φ(x, θ, λ) = 0 defines differentiable functions x : U(λˆ) → Cn and θ : U(λˆ) → C on a neighborhood of λˆ such that Φ(x(λ), θ(λ), λ) = 0.
0 ˆ θ (λ) = 1 yields the quadratic convergence of the method λk+1 = λk − θ(λk ).
TUHH Heinrich Voss Isfahan, July, 2016 40 / 72 H Assuming that xk is already scaled such that ` xk = 1 the second equation H H reads ` xk+1 = ` xk = 1, and the first one −1 0 xk+1 = −(λk+1 − λk )T (λk ) T (λk )xk .
Multiplying by ` yields `H x λ = λ − k . k+1 k H −1 0 ` T (λk ) T (λk )xk
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration
Applying Newton’s method to the nonlinear system x T (λ)x F = , λ `H x − 1 (` suitable scaling vector) yields
0 T (λk ) T (λk )xk xk+1 − xk T (λk )xk H = − H . ` 0 λk+1 − λk ` xk − 1
TUHH Heinrich Voss Isfahan, July, 2016 41 / 72 Multiplying by ` yields `H x λ = λ − k . k+1 k H −1 0 ` T (λk ) T (λk )xk
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration
Applying Newton’s method to the nonlinear system x T (λ)x F = , λ `H x − 1 (` suitable scaling vector) yields
0 T (λk ) T (λk )xk xk+1 − xk T (λk )xk H = − H . ` 0 λk+1 − λk ` xk − 1
H Assuming that xk is already scaled such that ` xk = 1 the second equation H H reads ` xk+1 = ` xk = 1, and the first one −1 0 xk+1 = −(λk+1 − λk )T (λk ) T (λk )xk .
TUHH Heinrich Voss Isfahan, July, 2016 41 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration
Applying Newton’s method to the nonlinear system x T (λ)x F = , λ `H x − 1 (` suitable scaling vector) yields
0 T (λk ) T (λk )xk xk+1 − xk T (λk )xk H = − H . ` 0 λk+1 − λk ` xk − 1
H Assuming that xk is already scaled such that ` xk = 1 the second equation H H reads ` xk+1 = ` xk = 1, and the first one −1 0 xk+1 = −(λk+1 − λk )T (λk ) T (λk )xk .
Multiplying by ` yields `H x λ = λ − k . k+1 k H −1 0 ` T (λk ) T (λk )xk
TUHH Heinrich Voss Isfahan, July, 2016 41 / 72 THEOREM Let λˆ be an eigenvalue of T (·) such that µ = 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ)y = µy, and let xˆ be a corresponding eigenvector with `H xˆ = 1. Then the inverse iteration converges locally and quadratically to (xˆ, λˆ).
Sketch of proof: Only have to show that the only solution of y T (λˆ) T 0(λˆ)xˆ y F 0(xˆ, λˆ) = = 0 µ `H 0 µ is the trivial one.
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
1: start with an approximation λ1 ∈ D to an eigenvalue of T (λ)x = 0 and an approximation x1 to an appropriate eigenvector 2: for k = 1, 2,... until convergence do 0 3: solve T (λk )xk+1 = T (λk )xk for xk+1 H H 4: set λk+1 = λk − ` xk /` xk+1 5: normalize xk+1 ← xk+1/kxk+1k 6: end for
TUHH Heinrich Voss Isfahan, July, 2016 42 / 72 Sketch of proof: Only have to show that the only solution of y T (λˆ) T 0(λˆ)xˆ y F 0(xˆ, λˆ) = = 0 µ `H 0 µ is the trivial one.
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
1: start with an approximation λ1 ∈ D to an eigenvalue of T (λ)x = 0 and an approximation x1 to an appropriate eigenvector 2: for k = 1, 2,... until convergence do 0 3: solve T (λk )xk+1 = T (λk )xk for xk+1 H H 4: set λk+1 = λk − ` xk /` xk+1 5: normalize xk+1 ← xk+1/kxk+1k 6: end for
THEOREM Let λˆ be an eigenvalue of T (·) such that µ = 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ)y = µy, and let xˆ be a corresponding eigenvector with `H xˆ = 1. Then the inverse iteration converges locally and quadratically to (xˆ, λˆ).
TUHH Heinrich Voss Isfahan, July, 2016 42 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
1: start with an approximation λ1 ∈ D to an eigenvalue of T (λ)x = 0 and an approximation x1 to an appropriate eigenvector 2: for k = 1, 2,... until convergence do 0 3: solve T (λk )xk+1 = T (λk )xk for xk+1 H H 4: set λk+1 = λk − ` xk /` xk+1 5: normalize xk+1 ← xk+1/kxk+1k 6: end for
THEOREM Let λˆ be an eigenvalue of T (·) such that µ = 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ)y = µy, and let xˆ be a corresponding eigenvector with `H xˆ = 1. Then the inverse iteration converges locally and quadratically to (xˆ, λˆ).
Sketch of proof: Only have to show that the only solution of y T (λˆ) T 0(λˆ)xˆ y F 0(xˆ, λˆ) = = 0 µ `H 0 µ is the trivial one. TUHH Heinrich Voss Isfahan, July, 2016 42 / 72 THEOREM (Rothe, V. 1990) Let T (λ) = T (λ)H for every λ ∈ J, and let λˆ ∈ J be an eigenvalue of T (·) such that µ = 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ)y = µy, and let xˆ be a corresponding eigenvector. Then the Rayleigh functional iteration converges locally and cubically to (xˆ, λˆ).
1: start with an approximation λ1 ∈ D to an eigenvalue of T (λ)x = 0 and an approximation x1 to an appropriate eigenvector 2: for k = 1, 2,... until convergence do 0 3: solve T (λk )xk+1 = T (λk )xk for xk+1 4: normalize xk+1 ← xk+1/kxk+1k 5: set λk+1 = p(xk+1) 6: end for
Methods for dense nonlinear eigenvalue problems Newton’s method Rayleigh functional iteration
For linear Hermitian eigenproblems one get’s cubic convergence if the eigenparameter is updated using the Rayleigh quotient. The generalization to nonlinear eigenproblems is the Rayleigh functional iteration
TUHH Heinrich Voss Isfahan, July, 2016 43 / 72 THEOREM (Rothe, V. 1990) Let T (λ) = T (λ)H for every λ ∈ J, and let λˆ ∈ J be an eigenvalue of T (·) such that µ = 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ)y = µy, and let xˆ be a corresponding eigenvector. Then the Rayleigh functional iteration converges locally and cubically to (xˆ, λˆ).
Methods for dense nonlinear eigenvalue problems Newton’s method Rayleigh functional iteration
For linear Hermitian eigenproblems one get’s cubic convergence if the eigenparameter is updated using the Rayleigh quotient. The generalization to nonlinear eigenproblems is the Rayleigh functional iteration
1: start with an approximation λ1 ∈ D to an eigenvalue of T (λ)x = 0 and an approximation x1 to an appropriate eigenvector 2: for k = 1, 2,... until convergence do 0 3: solve T (λk )xk+1 = T (λk )xk for xk+1 4: normalize xk+1 ← xk+1/kxk+1k 5: set λk+1 = p(xk+1) 6: end for
TUHH Heinrich Voss Isfahan, July, 2016 43 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Rayleigh functional iteration
For linear Hermitian eigenproblems one get’s cubic convergence if the eigenparameter is updated using the Rayleigh quotient. The generalization to nonlinear eigenproblems is the Rayleigh functional iteration
1: start with an approximation λ1 ∈ D to an eigenvalue of T (λ)x = 0 and an approximation x1 to an appropriate eigenvector 2: for k = 1, 2,... until convergence do 0 3: solve T (λk )xk+1 = T (λk )xk for xk+1 4: normalize xk+1 ← xk+1/kxk+1k 5: set λk+1 = p(xk+1) 6: end for
THEOREM (Rothe, V. 1990) Let T (λ) = T (λ)H for every λ ∈ J, and let λˆ ∈ J be an eigenvalue of T (·) such that µ = 0 is an algebraically simple eigenvalue of T 0(λˆ)−1T (λˆ)y = µy, and let xˆ be a corresponding eigenvector. Then the Rayleigh functional iteration converges locally and cubically to (xˆ, λˆ).
TUHH Heinrich Voss Isfahan, July, 2016 43 / 72 Then the update for λ becomes
(y k )H T (λ )x k λ = λ − k , k+1 k k H 0 k (y ) T (λk )x which is the Rayleigh functional for general nonlinear eigenproblems proposed in Lancaster (1966), and which can be interpreted as one Newton k H k step for solving the equation fk (λ) := (y ) T (λ)x = 0.
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
For the non-Hermitian nonlinear eigenproblem Ruhe (1973) suggested to use k H k k ` = T (λk ) y for the normalization, where y is an approximation to a left eigenvector.
TUHH Heinrich Voss Isfahan, July, 2016 44 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
For the non-Hermitian nonlinear eigenproblem Ruhe (1973) suggested to use k H k k ` = T (λk ) y for the normalization, where y is an approximation to a left eigenvector.
Then the update for λ becomes
(y k )H T (λ )x k λ = λ − k , k+1 k k H 0 k (y ) T (λk )x which is the Rayleigh functional for general nonlinear eigenproblems proposed in Lancaster (1966), and which can be interpreted as one Newton k H k step for solving the equation fk (λ) := (y ) T (λ)x = 0.
TUHH Heinrich Voss Isfahan, July, 2016 44 / 72 Parlett (1979) proved that the two-sided Rayleigh quotient iteration converges cubically to simple eigenvalues.
Methods for dense nonlinear eigenvalue problems Newton’s method Two-sided Rayleigh quotient iteration
For a highly nonnormal matrix A it is often advantageous to replace the Rayleigh quotient by Ostrowski’s two-sided Rayleigh quotient (1959)
v H Au θ(u, v) = v H u where v and u denotes an approximate left and right eigenvector of A, respectively, and to improve these approximations by solving simultaneously the two linear systems
(A − θI)u˜ = u and (AH − θI)v˜ = v.
TUHH Heinrich Voss Isfahan, July, 2016 45 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Two-sided Rayleigh quotient iteration
For a highly nonnormal matrix A it is often advantageous to replace the Rayleigh quotient by Ostrowski’s two-sided Rayleigh quotient (1959)
v H Au θ(u, v) = v H u where v and u denotes an approximate left and right eigenvector of A, respectively, and to improve these approximations by solving simultaneously the two linear systems
(A − θI)u˜ = u and (AH − θI)v˜ = v.
Parlett (1979) proved that the two-sided Rayleigh quotient iteration converges cubically to simple eigenvalues.
TUHH Heinrich Voss Isfahan, July, 2016 45 / 72 THEOREM (Schreiber 2008) Let λˆ be an algebraically simple eigenvalue of T (·) with left and right eigenvector xˆ and yˆ, respectively, and let T (λ) be holomorphic on a ˆ H H neighborhood of λ. Then with xˆk := xˆ/uk xˆ and yˆk := yˆ/vk yˆ it holds for sufficiently good initial approximations (λ0, u0, v0) that
ˆ ˆ 2 |λk+1 − λ| ≤ C0|λk − λ| kuk − xˆk kkvk − yˆk k 2 kuk+1 − xˆk k ≤ C1kuk − xˆk k kvk − yˆk k 2 kvk+1 − yˆk k ≤ C1kuk − xˆk kkvk − yˆk k
Methods for dense nonlinear eigenvalue problems Newton’s method Two-sided Rayleigh functional iteration
H H Require: Initial triple (λ0, u0, v0) with u0 u0 = v0 v0 = 1 1: for k = 0, 1, 2,... until convergence do 0 2: solve T (λk )xk+1 = T (λk )uk , set uk+1 = xk+1/kxk+1k H 0 H 3: solve T (λk ) yk+1 = T (λk ) vk , set vk+1 = yk+1/kyk+1k H 4: solve vk+1T (λk+1uk+1 = 0 for λk+1 5: end for
TUHH Heinrich Voss Isfahan, July, 2016 46 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Two-sided Rayleigh functional iteration
H H Require: Initial triple (λ0, u0, v0) with u0 u0 = v0 v0 = 1 1: for k = 0, 1, 2,... until convergence do 0 2: solve T (λk )xk+1 = T (λk )uk , set uk+1 = xk+1/kxk+1k H 0 H 3: solve T (λk ) yk+1 = T (λk ) vk , set vk+1 = yk+1/kyk+1k H 4: solve vk+1T (λk+1uk+1 = 0 for λk+1 5: end for
THEOREM (Schreiber 2008) Let λˆ be an algebraically simple eigenvalue of T (·) with left and right eigenvector xˆ and yˆ, respectively, and let T (λ) be holomorphic on a ˆ H H neighborhood of λ. Then with xˆk := xˆ/uk xˆ and yˆk := yˆ/vk yˆ it holds for sufficiently good initial approximations (λ0, u0, v0) that
ˆ ˆ 2 |λk+1 − λ| ≤ C0|λk − λ| kuk − xˆk kkvk − yˆk k 2 kuk+1 − xˆk k ≤ C1kuk − xˆk k kvk − yˆk k 2 kvk+1 − yˆk k ≤ C1kuk − xˆk kkvk − yˆk k
TUHH Heinrich Voss Isfahan, July, 2016 46 / 72 This approach generalizes the method of Kublanovskaya, inverse iteration, and a method proposed in Osborne & Michelson (1964).
It was proved that the rate of convergence is quadratic, and that cubic convergence can be obtained if not only λ, but also x and/or v are updated appropriately, thus unifying the results in Anselone & Rall (1968), Kublanovskaya (1970), Lancaster (2002), Osborne (1964),and Osborne & Michelson (1964).
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
Osborne (1978) considers Newton’s method for the complex function β(λ) defined by T (λ)u = β(λ)x, v H u = κ, where κ is a given constant, and x and v are given vectors.
TUHH Heinrich Voss Isfahan, July, 2016 47 / 72 It was proved that the rate of convergence is quadratic, and that cubic convergence can be obtained if not only λ, but also x and/or v are updated appropriately, thus unifying the results in Anselone & Rall (1968), Kublanovskaya (1970), Lancaster (2002), Osborne (1964),and Osborne & Michelson (1964).
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
Osborne (1978) considers Newton’s method for the complex function β(λ) defined by T (λ)u = β(λ)x, v H u = κ, where κ is a given constant, and x and v are given vectors.
This approach generalizes the method of Kublanovskaya, inverse iteration, and a method proposed in Osborne & Michelson (1964).
TUHH Heinrich Voss Isfahan, July, 2016 47 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
Osborne (1978) considers Newton’s method for the complex function β(λ) defined by T (λ)u = β(λ)x, v H u = κ, where κ is a given constant, and x and v are given vectors.
This approach generalizes the method of Kublanovskaya, inverse iteration, and a method proposed in Osborne & Michelson (1964).
It was proved that the rate of convergence is quadratic, and that cubic convergence can be obtained if not only λ, but also x and/or v are updated appropriately, thus unifying the results in Anselone & Rall (1968), Kublanovskaya (1970), Lancaster (2002), Osborne (1964),and Osborne & Michelson (1964).
TUHH Heinrich Voss Isfahan, July, 2016 47 / 72 The obvious idea then is to use a version of a simplified Newton method, where the shift σ is kept fixed during the iteration, i.e. to use
−1 0 xk+1 = T (σ) T (λk )xk for some fixed shift σ.
However, in general this method does not converge in the nonlinear case.
The iteration converges to an eigenpair of a linear problem
T (σ)x = γT 0(λ˜)x,
from which one cannot recover an eigenpair of the nonlinear problem T (λ)x = 0.
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
The disadvantage of inverse iteration with respect to efficiency is the large number of factorizations that are needed for each of the eigenvalues.
TUHH Heinrich Voss Isfahan, July, 2016 48 / 72 However, in general this method does not converge in the nonlinear case.
The iteration converges to an eigenpair of a linear problem
T (σ)x = γT 0(λ˜)x,
from which one cannot recover an eigenpair of the nonlinear problem T (λ)x = 0.
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
The disadvantage of inverse iteration with respect to efficiency is the large number of factorizations that are needed for each of the eigenvalues.
The obvious idea then is to use a version of a simplified Newton method, where the shift σ is kept fixed during the iteration, i.e. to use
−1 0 xk+1 = T (σ) T (λk )xk for some fixed shift σ.
TUHH Heinrich Voss Isfahan, July, 2016 48 / 72 The iteration converges to an eigenpair of a linear problem
T (σ)x = γT 0(λ˜)x,
from which one cannot recover an eigenpair of the nonlinear problem T (λ)x = 0.
Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
The disadvantage of inverse iteration with respect to efficiency is the large number of factorizations that are needed for each of the eigenvalues.
The obvious idea then is to use a version of a simplified Newton method, where the shift σ is kept fixed during the iteration, i.e. to use
−1 0 xk+1 = T (σ) T (λk )xk for some fixed shift σ.
However, in general this method does not converge in the nonlinear case.
TUHH Heinrich Voss Isfahan, July, 2016 48 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Inverse iteration ct.
The disadvantage of inverse iteration with respect to efficiency is the large number of factorizations that are needed for each of the eigenvalues.
The obvious idea then is to use a version of a simplified Newton method, where the shift σ is kept fixed during the iteration, i.e. to use
−1 0 xk+1 = T (σ) T (λk )xk for some fixed shift σ.
However, in general this method does not converge in the nonlinear case.
The iteration converges to an eigenpair of a linear problem
T (σ)x = γT 0(λ˜)x,
from which one cannot recover an eigenpair of the nonlinear problem T (λ)x = 0.
TUHH Heinrich Voss Isfahan, July, 2016 48 / 72 Neglecting second order terms yields the update
k+1 k −1 k x = x − T (λk ) T (λk+1)x ,
replacing λk by a fixed shift σ yields an update
k+1 k −1 k x = x − T (σ) T (λk+1)x ,
without misconvergence.
Methods for dense nonlinear eigenvalue problems Newton’s method Residual inverse iteration
If T (λ) is twice continuously differentiable,
k k+1 k −1 0 k x − x = x + (λk+1 − λk )T (λk ) T (λk )x −1 0 k = T (λk ) (T (λk ) + (λk+1 − λk )T (λk ))x −1 k 2 = T (λk ) T (λk+1)x + O((λk+1 − λk ) )
TUHH Heinrich Voss Isfahan, July, 2016 49 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Residual inverse iteration
If T (λ) is twice continuously differentiable,
k k+1 k −1 0 k x − x = x + (λk+1 − λk )T (λk ) T (λk )x −1 0 k = T (λk ) (T (λk ) + (λk+1 − λk )T (λk ))x −1 k 2 = T (λk ) T (λk+1)x + O((λk+1 − λk ) )
Neglecting second order terms yields the update
k+1 k −1 k x = x − T (λk ) T (λk+1)x ,
replacing λk by a fixed shift σ yields an update
k+1 k −1 k x = x − T (σ) T (λk+1)x ,
without misconvergence.
TUHH Heinrich Voss Isfahan, July, 2016 49 / 72 THEOREM (Neumaier 1985) Let T (λ) be twice continuously differentiable. Assume that λˆ is a simple eigenvalue of T (λ)x = 0, and let xˆ be a corresponding eigenvector normalized by kxˆk = 1. Then the residual inverse iteration converges for all σ sufficiently close to λˆ, and it holds
kx k+1 − xˆk = O(|σ − λˆ|), and |λ − λˆ| = O(kx k − xˆkt ). kx k − xˆk k+1
Here, t = 1 in the general case, and t = 2 if T (·) is Hermitian.
Methods for dense nonlinear eigenvalue problems Newton’s method Residual inverse iteration ct.
1: start with an approximation x1 ∈ D to an eigenvector of T (λ)x = 0 2: for k = 1, 2,... until convergence do H −1 k 3: solve ` T (σ) T (λk+1)x = 0 for λk+1 k H k (or (x ) T (λk+1)x = 0 if T (·) is Hermitian) k k 4: compute the residual r = T (λk+1)x 5: solve T (σ)d k = r k 6: set x k+1 = x k − d k , x k+1 = x k+1/kx k+1k 7: end for
TUHH Heinrich Voss Isfahan, July, 2016 50 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Residual inverse iteration ct.
1: start with an approximation x1 ∈ D to an eigenvector of T (λ)x = 0 2: for k = 1, 2,... until convergence do H −1 k 3: solve ` T (σ) T (λk+1)x = 0 for λk+1 k H k (or (x ) T (λk+1)x = 0 if T (·) is Hermitian) k k 4: compute the residual r = T (λk+1)x 5: solve T (σ)d k = r k 6: set x k+1 = x k − d k , x k+1 = x k+1/kx k+1k 7: end for
THEOREM (Neumaier 1985) Let T (λ) be twice continuously differentiable. Assume that λˆ is a simple eigenvalue of T (λ)x = 0, and let xˆ be a corresponding eigenvector normalized by kxˆk = 1. Then the residual inverse iteration converges for all σ sufficiently close to λˆ, and it holds
kx k+1 − xˆk = O(|σ − λˆ|), and |λ − λˆ| = O(kx k − xˆkt ). kx k − xˆk k+1
Here, t = 1 in the general case, and t = 2 if T (·) is Hermitian. TUHH Heinrich Voss Isfahan, July, 2016 50 / 72 THEOREM (Schreiber 2008) Let λˆ be an algebraically simple eigenvalue of T (·) with left and right H H eigenvector xˆ and yˆ scaled by wx xˆ = 1, wy yˆ = 1, respectively, and let T (λ) be holomorphic on a neighborhood of λˆ. Then it holds for sufficiently good initial approximations (λ0, u0, v0) that ˆ |λk+1 − λ| = O(|kuk − xˆkkvk − yˆk) ku − xˆk kv − yˆk k+1 = O|(σ − λˆ|) , k+1 = O(|σ − λˆ|) kuk − xˆk kvk − yˆk
Methods for dense nonlinear eigenvalue problems Newton’s method Two-sided residual inverse iteration
Require: (λ0, u0, v0) and normalization vectors wx , wy with H H wx u0 = wy v0 = 1. 1: for k = 0, 1, 2,... until convergence do H k 2: solve vk T (λk+1)u = 0 for λk+1 x y H 3: compute residuals rk = T (λk+1)uk , rk = T (λk+1) vk x 4: solve T (σ)sk = rk for sk H y 5: solve T (σ) tk = rk for tk k 6: set xk+1 = uk − s , uk+1 = xk+1/kxk+1k k 7: set yk+1 = vk − t , vk+1 = yk+1/kyk+1k 8: end for
TUHH Heinrich Voss Isfahan, July, 2016 51 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Two-sided residual inverse iteration
Require: (λ0, u0, v0) and normalization vectors wx , wy with H H wx u0 = wy v0 = 1. 1: for k = 0, 1, 2,... until convergence do H k 2: solve vk T (λk+1)u = 0 for λk+1 x y H 3: compute residuals rk = T (λk+1)uk , rk = T (λk+1) vk x 4: solve T (σ)sk = rk for sk H y 5: solve T (σ) tk = rk for tk k 6: set xk+1 = uk − s , uk+1 = xk+1/kxk+1k k 7: set yk+1 = vk − t , vk+1 = yk+1/kyk+1k 8: end for THEOREM (Schreiber 2008) Let λˆ be an algebraically simple eigenvalue of T (·) with left and right H H eigenvector xˆ and yˆ scaled by wx xˆ = 1, wy yˆ = 1, respectively, and let T (λ) be holomorphic on a neighborhood of λˆ. Then it holds for sufficiently good initial approximations (λ0, u0, v0) that ˆ |λk+1 − λ| = O(|kuk − xˆkkvk − yˆk) ku − xˆk kv − yˆk k+1 = O|(σ − λˆ|) , k+1 = O(|σ − λˆ|) kuk − xˆk kvk − yˆk TUHH Heinrich Voss Isfahan, July, 2016 51 / 72 THEOREM (Schreiber 2008) Under the conditions of the last THEOREM the R-order of convergence of the two-sided residual inverse iteration is at least two, and it holds that ˆ |λk+1 − λ| = O(|kuk − xˆkkvk − yˆk)
kuk+1 − xˆk = O|(kuk − xˆkkuk−1 − xˆkkvk−1 − yˆk)
kvk+1 − yˆk = O(kvk − yˆkkvk−1 − yˆkkuk−1 − xˆk)
Methods for dense nonlinear eigenvalue problems Newton’s method Two-sided residual inverse iteration ct.
If the shift is updated as well one gets
Require: (λ0, u0, v0) and normalization vectors wx , wy with H H wx u0 = wy v0 = 1. 1: for k = 0, 1, 2,... until convergence do H k 2: solve vk T (λk+1)u = 0 for λk+1 x y H 3: compute residuals rk = T (λk+1)uk , rk = T (λk+1) vk x 4: solve T (λk+1)sk = rk for sk H y 5: solve T (λk+1) tk = rk for tk k 6: set xk+1 = uk − s , uk+1 = xk+1/kxk+1k k 7: set yk+1 = vk − t , vk+1 = yk+1/kyk+1k 8: end for
TUHH Heinrich Voss Isfahan, July, 2016 52 / 72 Methods for dense nonlinear eigenvalue problems Newton’s method Two-sided residual inverse iteration ct.
If the shift is updated as well one gets
Require: (λ0, u0, v0) and normalization vectors wx , wy with H H wx u0 = wy v0 = 1. 1: for k = 0, 1, 2,... until convergence do H k 2: solve vk T (λk+1)u = 0 for λk+1 x y H 3: compute residuals rk = T (λk+1)uk , rk = T (λk+1) vk x 4: solve T (λk+1)sk = rk for sk H y 5: solve T (λk+1) tk = rk for tk k 6: set xk+1 = uk − s , uk+1 = xk+1/kxk+1k k 7: set yk+1 = vk − t , vk+1 = yk+1/kyk+1k 8: end for THEOREM (Schreiber 2008) Under the conditions of the last THEOREM the R-order of convergence of the two-sided residual inverse iteration is at least two, and it holds that ˆ |λk+1 − λ| = O(|kuk − xˆkkvk − yˆk)
kuk+1 − xˆk = O|(kuk − xˆkkuk−1 − xˆkkvk−1 − yˆk)
kvk+1 − yˆk = O(kvk − yˆkkvk−1 − yˆkkuk−1 − xˆk)
TUHH Heinrich Voss Isfahan, July, 2016 52 / 72 Invariant pairs Outline
1 Examples
2 Basic Properties
3 Rayleigh Functional
4 Methods for dense nonlinear eigenvalue problems Linearization Methods based on characteristic equation Newton’s method
5 Invariant pairs
TUHH Heinrich Voss Isfahan, July, 2016 53 / 72 This is a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The usual deflation techniques for linear eigenproblems do not work.
In the following we show that the concept of invariant pairs offers a way of representing an eigenvalue problem that is insensitive to this phenomenon.
It is convenient to assume that the eigenvalue problem assumes the following form m X T (λ)x = fj (λ)Aj x = 0 (1) j=1 n×n for holomorphic functions fj : C ⊃ Ω → C and constant matrices Aj ∈ C .
Invariant pairs
One of the most fundamental differences from the linear case is that distinct eigenvalues may have linear dependent eigenvectors or even share the same eigenvalue.
TUHH Heinrich Voss Isfahan, July, 2016 54 / 72 In the following we show that the concept of invariant pairs offers a way of representing an eigenvalue problem that is insensitive to this phenomenon.
It is convenient to assume that the eigenvalue problem assumes the following form m X T (λ)x = fj (λ)Aj x = 0 (1) j=1 n×n for holomorphic functions fj : C ⊃ Ω → C and constant matrices Aj ∈ C .
Invariant pairs
One of the most fundamental differences from the linear case is that distinct eigenvalues may have linear dependent eigenvectors or even share the same eigenvalue.
This is a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The usual deflation techniques for linear eigenproblems do not work.
TUHH Heinrich Voss Isfahan, July, 2016 54 / 72 It is convenient to assume that the eigenvalue problem assumes the following form m X T (λ)x = fj (λ)Aj x = 0 (1) j=1 n×n for holomorphic functions fj : C ⊃ Ω → C and constant matrices Aj ∈ C .
Invariant pairs
One of the most fundamental differences from the linear case is that distinct eigenvalues may have linear dependent eigenvectors or even share the same eigenvalue.
This is a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The usual deflation techniques for linear eigenproblems do not work.
In the following we show that the concept of invariant pairs offers a way of representing an eigenvalue problem that is insensitive to this phenomenon.
TUHH Heinrich Voss Isfahan, July, 2016 54 / 72 Invariant pairs
One of the most fundamental differences from the linear case is that distinct eigenvalues may have linear dependent eigenvectors or even share the same eigenvalue.
This is a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The usual deflation techniques for linear eigenproblems do not work.
In the following we show that the concept of invariant pairs offers a way of representing an eigenvalue problem that is insensitive to this phenomenon.
It is convenient to assume that the eigenvalue problem assumes the following form m X T (λ)x = fj (λ)Aj x = 0 (1) j=1 n×n for holomorphic functions fj : C ⊃ Ω → C and constant matrices Aj ∈ C .
TUHH Heinrich Voss Isfahan, July, 2016 54 / 72 Example 1 For the quadratic eigenvalue problem T (λ)x = 0 with
0 1 7 −5 1 0 T (λ) := + λ + λ2 (3) −2 3 10 −8 0 1
1 with distinct eigenvalues λ = 1 and λ = 2 and shared eigenvector x := 2
1 0 X = x x and S = 0 2
is an invariant pair.
Invariant pairs
Definition Let the eigenvalues of S ∈ Ck×k be contained in Ω ⊂ C and let X ∈ Cn×k . Then (X, S) ∈ Cn×k × Ck×k is called an invariant pair of the nonlinear eigenvalue problem (1) if
m X Aj Xfj (S) = 0. (2) j=1
TUHH Heinrich Voss Isfahan, July, 2016 55 / 72 Invariant pairs
Definition Let the eigenvalues of S ∈ Ck×k be contained in Ω ⊂ C and let X ∈ Cn×k . Then (X, S) ∈ Cn×k × Ck×k is called an invariant pair of the nonlinear eigenvalue problem (1) if
m X Aj Xfj (S) = 0. (2) j=1
Example 1 For the quadratic eigenvalue problem T (λ)x = 0 with
0 1 7 −5 1 0 T (λ) := + λ + λ2 (3) −2 3 10 −8 0 1
1 with distinct eigenvalues λ = 1 and λ = 2 and shared eigenvector x := 2
1 0 X = x x and S = 0 2
is an invariant pair.
TUHH Heinrich Voss Isfahan, July, 2016 55 / 72 Definition A pair (X, S) ∈ Cn×k × Ck×k is called minimal if there is ` ∈ N such that the matrix X XS V (X, S) := ` . . XS`−1 has rank k.
The smallest such ` is called the minimality index of (X, S).
An invariant pair (X, S) is called simple, if (X, S) is minimal and the algebraic multiplicities of the eigenvalues of S are identical to the ones of the corresponding eigenvalues of T (·).
We have already seen that requiring X to have full column rank is not reasonable in the context of nonlinear eigenvalue problems.
Invariant pairs
To avoid the trivial pairs, such as X = 0, an additional property has to be imposed.
TUHH Heinrich Voss Isfahan, July, 2016 56 / 72 The smallest such ` is called the minimality index of (X, S).
An invariant pair (X, S) is called simple, if (X, S) is minimal and the algebraic multiplicities of the eigenvalues of S are identical to the ones of the corresponding eigenvalues of T (·).
Definition A pair (X, S) ∈ Cn×k × Ck×k is called minimal if there is ` ∈ N such that the matrix X XS V (X, S) := ` . . XS`−1 has rank k.
Invariant pairs
To avoid the trivial pairs, such as X = 0, an additional property has to be imposed.
We have already seen that requiring X to have full column rank is not reasonable in the context of nonlinear eigenvalue problems.
TUHH Heinrich Voss Isfahan, July, 2016 56 / 72 The smallest such ` is called the minimality index of (X, S).
An invariant pair (X, S) is called simple, if (X, S) is minimal and the algebraic multiplicities of the eigenvalues of S are identical to the ones of the corresponding eigenvalues of T (·).
Invariant pairs
To avoid the trivial pairs, such as X = 0, an additional property has to be imposed.
We have already seen that requiring X to have full column rank is not reasonable in the context of nonlinear eigenvalue problems.
Definition A pair (X, S) ∈ Cn×k × Ck×k is called minimal if there is ` ∈ N such that the matrix X XS V (X, S) := ` . . XS`−1 has rank k.
TUHH Heinrich Voss Isfahan, July, 2016 56 / 72 An invariant pair (X, S) is called simple, if (X, S) is minimal and the algebraic multiplicities of the eigenvalues of S are identical to the ones of the corresponding eigenvalues of T (·).
Invariant pairs
To avoid the trivial pairs, such as X = 0, an additional property has to be imposed.
We have already seen that requiring X to have full column rank is not reasonable in the context of nonlinear eigenvalue problems.
Definition A pair (X, S) ∈ Cn×k × Ck×k is called minimal if there is ` ∈ N such that the matrix X XS V (X, S) := ` . . XS`−1 has rank k.
The smallest such ` is called the minimality index of (X, S).
TUHH Heinrich Voss Isfahan, July, 2016 56 / 72 Invariant pairs
To avoid the trivial pairs, such as X = 0, an additional property has to be imposed.
We have already seen that requiring X to have full column rank is not reasonable in the context of nonlinear eigenvalue problems.
Definition A pair (X, S) ∈ Cn×k × Ck×k is called minimal if there is ` ∈ N such that the matrix X XS V (X, S) := ` . . XS`−1 has rank k.
The smallest such ` is called the minimality index of (X, S).
An invariant pair (X, S) is called simple, if (X, S) is minimal and the algebraic multiplicities of the eigenvalues of S are identical to the ones of the corresponding eigenvalues of T (·).
TUHH Heinrich Voss Isfahan, July, 2016 56 / 72 1 λˆ = −1 is an eigenvalue with eigenvector xˆ = . 1
The pair (xˆ, λˆ) is a minimal invariant pair with minimality index 1.
It is not simple, because the algebraic multiplicity of λˆ is 2 as an eigenvalue of T (λ) but only 1 as an eigenvalue of S = −1.
Invariant pairs
For the quadratic eigenvalue problem (3)
0 1 7 −5 1 0 T (λ) := + λ + λ2 −2 3 10 −8 0 1
it holds det T (λ) = λ4 − λ3 − 3λ2 + λ + 2.
TUHH Heinrich Voss Isfahan, July, 2016 57 / 72 The pair (xˆ, λˆ) is a minimal invariant pair with minimality index 1.
It is not simple, because the algebraic multiplicity of λˆ is 2 as an eigenvalue of T (λ) but only 1 as an eigenvalue of S = −1.
Invariant pairs
For the quadratic eigenvalue problem (3)
0 1 7 −5 1 0 T (λ) := + λ + λ2 −2 3 10 −8 0 1
it holds det T (λ) = λ4 − λ3 − 3λ2 + λ + 2.
1 λˆ = −1 is an eigenvalue with eigenvector xˆ = . 1
TUHH Heinrich Voss Isfahan, July, 2016 57 / 72 It is not simple, because the algebraic multiplicity of λˆ is 2 as an eigenvalue of T (λ) but only 1 as an eigenvalue of S = −1.
Invariant pairs
For the quadratic eigenvalue problem (3)
0 1 7 −5 1 0 T (λ) := + λ + λ2 −2 3 10 −8 0 1
it holds det T (λ) = λ4 − λ3 − 3λ2 + λ + 2.
1 λˆ = −1 is an eigenvalue with eigenvector xˆ = . 1
The pair (xˆ, λˆ) is a minimal invariant pair with minimality index 1.
TUHH Heinrich Voss Isfahan, July, 2016 57 / 72 Invariant pairs
For the quadratic eigenvalue problem (3)
0 1 7 −5 1 0 T (λ) := + λ + λ2 −2 3 10 −8 0 1
it holds det T (λ) = λ4 − λ3 − 3λ2 + λ + 2.
1 λˆ = −1 is an eigenvalue with eigenvector xˆ = . 1
The pair (xˆ, λˆ) is a minimal invariant pair with minimality index 1.
It is not simple, because the algebraic multiplicity of λˆ is 2 as an eigenvalue of T (λ) but only 1 as an eigenvalue of S = −1.
TUHH Heinrich Voss Isfahan, July, 2016 57 / 72 1 2 1 0 The same is true for the pairs (X , S ) with X = and S = 2 2 2 1 2 2 0 2
and for (X3, S3) with X3 := [X1, X2] and S3 := diag(X1, X2).
Invariant pairs
1 1 −1 1 The pair (X , S ) with X = and S = with 1 1 1 1 1 1 0 −1
1 1 X1 1 1 = X1S1 −1 0 −1 0
is a minimal invariant pair with minimality index 2, which is simple.
TUHH Heinrich Voss Isfahan, July, 2016 58 / 72 and for (X3, S3) with X3 := [X1, X2] and S3 := diag(X1, X2).
Invariant pairs
1 1 −1 1 The pair (X , S ) with X = and S = with 1 1 1 1 1 1 0 −1
1 1 X1 1 1 = X1S1 −1 0 −1 0
is a minimal invariant pair with minimality index 2, which is simple.
1 2 1 0 The same is true for the pairs (X , S ) with X = and S = 2 2 2 1 2 2 0 2
TUHH Heinrich Voss Isfahan, July, 2016 58 / 72 Invariant pairs
1 1 −1 1 The pair (X , S ) with X = and S = with 1 1 1 1 1 1 0 −1
1 1 X1 1 1 = X1S1 −1 0 −1 0
is a minimal invariant pair with minimality index 2, which is simple.
1 2 1 0 The same is true for the pairs (X , S ) with X = and S = 2 2 2 1 2 2 0 2
and for (X3, S3) with X3 := [X1, X2] and S3 := diag(X1, X2).
TUHH Heinrich Voss Isfahan, July, 2016 58 / 72 1. For any invertible matrix Z ∈ Ck×x the pair (XZ, Z −1SZ) is also a minimal invariant pair.
2. The eigenvalues of S are eigenvalues of T (·).
3. The minimality index can not exceed k.
Invariant pairs
Theorem Let (X, S) ∈ Cn×k × Ck×k be a minimal invariant pair of T (·). Then the following statements hold:
TUHH Heinrich Voss Isfahan, July, 2016 59 / 72 2. The eigenvalues of S are eigenvalues of T (·).
3. The minimality index can not exceed k.
Invariant pairs
Theorem Let (X, S) ∈ Cn×k × Ck×k be a minimal invariant pair of T (·). Then the following statements hold:
1. For any invertible matrix Z ∈ Ck×x the pair (XZ, Z −1SZ) is also a minimal invariant pair.
TUHH Heinrich Voss Isfahan, July, 2016 59 / 72 3. The minimality index can not exceed k.
Invariant pairs
Theorem Let (X, S) ∈ Cn×k × Ck×k be a minimal invariant pair of T (·). Then the following statements hold:
1. For any invertible matrix Z ∈ Ck×x the pair (XZ, Z −1SZ) is also a minimal invariant pair.
2. The eigenvalues of S are eigenvalues of T (·).
TUHH Heinrich Voss Isfahan, July, 2016 59 / 72 Invariant pairs
Theorem Let (X, S) ∈ Cn×k × Ck×k be a minimal invariant pair of T (·). Then the following statements hold:
1. For any invertible matrix Z ∈ Ck×x the pair (XZ, Z −1SZ) is also a minimal invariant pair.
2. The eigenvalues of S are eigenvalues of T (·).
3. The minimality index can not exceed k.
TUHH Heinrich Voss Isfahan, July, 2016 59 / 72 Consider the nonlinear matrix operator n×k k×k n×k C × CΩ → C T : Pm (4) (X, S) 7→ j=1 Aj Xfj (S) k×k where CΩ denotes the set of k × k matrices with eigenvalues in Ω. Then an invariant pair (X, S) satisfies T(X, S) = 0, but this relation is not sufficient to characterize (X, S).
To define a scaling condition, choose ` such that the matrix V`(X, S) has rank k, and define the partition W0 W1 −1 W = := V (X, S) V (X, S)H V (X, S) ∈ nk×k . ` ` ` C . W`−1 n×k with Wj ∈ C . Then it holds V(X, S) = 0 for the operator n×k k×k n×k H V : C × CΩ → C , V(X, S) := W V`(X, S) − Ik .
Invariant pairs Simple invariant pairs are well posed objects in the sense of being regular solutions of nonlinear matrix equations.
TUHH Heinrich Voss Isfahan, July, 2016 60 / 72 To define a scaling condition, choose ` such that the matrix V`(X, S) has rank k, and define the partition W0 W1 −1 W = := V (X, S) V (X, S)H V (X, S) ∈ nk×k . ` ` ` C . W`−1 n×k with Wj ∈ C . Then it holds V(X, S) = 0 for the operator n×k k×k n×k H V : C × CΩ → C , V(X, S) := W V`(X, S) − Ik .
Invariant pairs Simple invariant pairs are well posed objects in the sense of being regular solutions of nonlinear matrix equations. Consider the nonlinear matrix operator n×k k×k n×k C × CΩ → C T : Pm (4) (X, S) 7→ j=1 Aj Xfj (S) k×k where CΩ denotes the set of k × k matrices with eigenvalues in Ω. Then an invariant pair (X, S) satisfies T(X, S) = 0, but this relation is not sufficient to characterize (X, S).
TUHH Heinrich Voss Isfahan, July, 2016 60 / 72 Invariant pairs Simple invariant pairs are well posed objects in the sense of being regular solutions of nonlinear matrix equations. Consider the nonlinear matrix operator n×k k×k n×k C × CΩ → C T : Pm (4) (X, S) 7→ j=1 Aj Xfj (S) k×k where CΩ denotes the set of k × k matrices with eigenvalues in Ω. Then an invariant pair (X, S) satisfies T(X, S) = 0, but this relation is not sufficient to characterize (X, S).
To define a scaling condition, choose ` such that the matrix V`(X, S) has rank k, and define the partition W0 W1 −1 W = := V (X, S) V (X, S)H V (X, S) ∈ nk×k . ` ` ` C . W`−1 n×k with Wj ∈ C . Then it holds V(X, S) = 0 for the operator n×k k×k n×k H V : C × CΩ → C , V(X, S) := W V`(X, S) − Ik .
TUHH Heinrich Voss Isfahan, July, 2016 60 / 72 This theorem motivates to apply Newton’s method to the system T(X, S) = 0, V(X, S) = 0 which can be written as −1 (Xp+1, Sp+1) = (Xp, Sp) − L (T(Xp, Sp), V(Xp, Sp)) where L = (DT, DV) is the Jacobian matrix of T(X, S) = 0, V(X, S) = 0.
m X DT(∆X, ∆S) = T(∆X, S) + Aj X[Dfj (S)](∆S), j=1 m H X H j j DV(∆X, ∆S) = W0 ∆X + Wj (∆XS + X[DS ](∆S)). j=1
Invariant pairs Theorem If (X, S) is a minimal invariant pair for the nonlinear eigenvalue problem T (·), then (X, S) is simple if and only if the linear matrix operator
n×k k×k n×k k×k L : C × C → C × C , (∆X, ∆S) 7→ (DT(∆X, ∆S), DV(∆X, ∆S)) is invertible, where DT and DV denotes the Frechet´ derivative of T and V, respectively.
TUHH Heinrich Voss Isfahan, July, 2016 61 / 72 m X DT(∆X, ∆S) = T(∆X, S) + Aj X[Dfj (S)](∆S), j=1 m H X H j j DV(∆X, ∆S) = W0 ∆X + Wj (∆XS + X[DS ](∆S)). j=1
Invariant pairs Theorem If (X, S) is a minimal invariant pair for the nonlinear eigenvalue problem T (·), then (X, S) is simple if and only if the linear matrix operator
n×k k×k n×k k×k L : C × C → C × C , (∆X, ∆S) 7→ (DT(∆X, ∆S), DV(∆X, ∆S)) is invertible, where DT and DV denotes the Frechet´ derivative of T and V, respectively.
This theorem motivates to apply Newton’s method to the system T(X, S) = 0, V(X, S) = 0 which can be written as −1 (Xp+1, Sp+1) = (Xp, Sp) − L (T(Xp, Sp), V(Xp, Sp)) where L = (DT, DV) is the Jacobian matrix of T(X, S) = 0, V(X, S) = 0.
TUHH Heinrich Voss Isfahan, July, 2016 61 / 72 Invariant pairs Theorem If (X, S) is a minimal invariant pair for the nonlinear eigenvalue problem T (·), then (X, S) is simple if and only if the linear matrix operator
n×k k×k n×k k×k L : C × C → C × C , (∆X, ∆S) 7→ (DT(∆X, ∆S), DV(∆X, ∆S)) is invertible, where DT and DV denotes the Frechet´ derivative of T and V, respectively.
This theorem motivates to apply Newton’s method to the system T(X, S) = 0, V(X, S) = 0 which can be written as −1 (Xp+1, Sp+1) = (Xp, Sp) − L (T(Xp, Sp), V(Xp, Sp)) where L = (DT, DV) is the Jacobian matrix of T(X, S) = 0, V(X, S) = 0.
m X DT(∆X, ∆S) = T(∆X, S) + Aj X[Dfj (S)](∆S), j=1 m H X H j j DV(∆X, ∆S) = W0 ∆X + Wj (∆XS + X[DS ](∆S)). j=1
TUHH Heinrich Voss Isfahan, July, 2016 61 / 72 Invariant pairs Newton’s method for computing invariant pairs
n×k k×k Require: Initial pair (X0, S0) ∈ C × C such that H V`(X0, S0) V`(X0, S0) = Ik 1: p ← 0, W ← V`(X0, S0) 2: repeat 3: Res ← T(Xp, Sp) 4: Solve linear matrix equation L(∆X, ∆S) = (Res, O) ˜ ˜ 5: Xp+1 ← Xp − ∆X, Sp+1 ← Sp − ∆S ˜ ˜ 6: Compute compact QR decomposition V`(Xp+1, Sp+1) = WR ˜ −1 ˜ −1 7: Xp+1 ← Xp+1R , Sp+1 ← RSp+1R 8: until convergence
TUHH Heinrich Voss Isfahan, July, 2016 62 / 72 For the stability analysis of the corresponding delay differential equation
x˙ (t) = A0x(t) + A1x(t − τ),
it is of interest to compute eigenvalues with largest real part.
To obtain an initial guess, one approximates T (·) by a polynomial
j X 1 T (λ) ≈ P(λ) = λI − A − A (−λτ)i 0 1 i! i=0
and compute k eigenvalues λ1, . . . , λk with largest real part (e.g. by linearization).
Invariant pairs
Example (by courtesy of Daniel Kressner)
−5 1 −2 1 T (λ) = λI − A − A e−τλ = λI − − e−λ 0 1 2 −6 4 −1
TUHH Heinrich Voss Isfahan, July, 2016 63 / 72 To obtain an initial guess, one approximates T (·) by a polynomial
j X 1 T (λ) ≈ P(λ) = λI − A − A (−λτ)i 0 1 i! i=0
and compute k eigenvalues λ1, . . . , λk with largest real part (e.g. by linearization).
Invariant pairs
Example (by courtesy of Daniel Kressner)
−5 1 −2 1 T (λ) = λI − A − A e−τλ = λI − − e−λ 0 1 2 −6 4 −1
For the stability analysis of the corresponding delay differential equation
x˙ (t) = A0x(t) + A1x(t − τ),
it is of interest to compute eigenvalues with largest real part.
TUHH Heinrich Voss Isfahan, July, 2016 63 / 72 Invariant pairs
Example (by courtesy of Daniel Kressner)
−5 1 −2 1 T (λ) = λI − A − A e−τλ = λI − − e−λ 0 1 2 −6 4 −1
For the stability analysis of the corresponding delay differential equation
x˙ (t) = A0x(t) + A1x(t − τ),
it is of interest to compute eigenvalues with largest real part.
To obtain an initial guess, one approximates T (·) by a polynomial
j X 1 T (λ) ≈ P(λ) = λI − A − A (−λτ)i 0 1 i! i=0
and compute k eigenvalues λ1, . . . , λk with largest real part (e.g. by linearization).
TUHH Heinrich Voss Isfahan, July, 2016 63 / 72 k×k Require: Initial matrix S0 ∈ C n×k Require: Random matrix X0 ∈ C 1: repeat 2: Compute solution Y of the linear equation T(Y , S0) = x0 3: Compute compact QR decomposition V`(Y , S0) = WR −1 −1 4: Update X0 ← YR , S0 ← RS0R 5: until convergence
Invariant pairs Example 1
As initial value one chooses S0 = diag{λ1, . . . , λk } and computes X0 by block–inverse iteration (cf. Kressner 2009).
TUHH Heinrich Voss Isfahan, July, 2016 64 / 72 Invariant pairs Example 1
As initial value one chooses S0 = diag{λ1, . . . , λk } and computes X0 by block–inverse iteration (cf. Kressner 2009).
k×k Require: Initial matrix S0 ∈ C n×k Require: Random matrix X0 ∈ C 1: repeat 2: Compute solution Y of the linear equation T(Y , S0) = x0 3: Compute compact QR decomposition V`(Y , S0) = WR −1 −1 4: Update X0 ← YR , S0 ← RS0R 5: until convergence
TUHH Heinrich Voss Isfahan, July, 2016 64 / 72 Initially, three eigenvalues are well and two eigenvalues are poorly approximated.
The first Fig. shows the location of eigenvalue approximations initially (plus), after 3 iterations (circle), and at convergence after 12 iterations (cross).
The second Fig. shows the residual norm in the course of the Newton iteration.
The condition number of the Jacobian is 9.2e5 at convergence, which could explain the poor transient behavior of Newton’s method.
Invariant pairs Example 1 ct.
The following figures contain the results obtained with the block–Newton method with k = 5 and ` = 3. For the initial approximation j = 4 was chosen.
TUHH Heinrich Voss Isfahan, July, 2016 65 / 72 The first Fig. shows the location of eigenvalue approximations initially (plus), after 3 iterations (circle), and at convergence after 12 iterations (cross).
The second Fig. shows the residual norm in the course of the Newton iteration.
The condition number of the Jacobian is 9.2e5 at convergence, which could explain the poor transient behavior of Newton’s method.
Invariant pairs Example 1 ct.
The following figures contain the results obtained with the block–Newton method with k = 5 and ` = 3. For the initial approximation j = 4 was chosen.
Initially, three eigenvalues are well and two eigenvalues are poorly approximated.
TUHH Heinrich Voss Isfahan, July, 2016 65 / 72 The condition number of the Jacobian is 9.2e5 at convergence, which could explain the poor transient behavior of Newton’s method.
Invariant pairs Example 1 ct.
The following figures contain the results obtained with the block–Newton method with k = 5 and ` = 3. For the initial approximation j = 4 was chosen.
Initially, three eigenvalues are well and two eigenvalues are poorly approximated.
The first Fig. shows the location of eigenvalue approximations initially (plus), after 3 iterations (circle), and at convergence after 12 iterations (cross).
The second Fig. shows the residual norm in the course of the Newton iteration.
TUHH Heinrich Voss Isfahan, July, 2016 65 / 72 Invariant pairs Example 1 ct.
The following figures contain the results obtained with the block–Newton method with k = 5 and ` = 3. For the initial approximation j = 4 was chosen.
Initially, three eigenvalues are well and two eigenvalues are poorly approximated.
The first Fig. shows the location of eigenvalue approximations initially (plus), after 3 iterations (circle), and at convergence after 12 iterations (cross).
The second Fig. shows the residual norm in the course of the Newton iteration.
The condition number of the Jacobian is 9.2e5 at convergence, which could explain the poor transient behavior of Newton’s method.
TUHH Heinrich Voss Isfahan, July, 2016 65 / 72 Invariant pairs Eigenvalue approximations
6
4
2
0
−2
−4
−6 −2.5 −2 −1.5 −1 −0.5 0 0.5
Location of eigenvalue approximations initially (plus), after 3 iterations (circle), and at convergence after 12 iterations (cross). TUHH Heinrich Voss Isfahan, July, 2016 66 / 72 Invariant pairs Residual
105
100
10−5
10−10
10−15 0 2 4 6 8 10
Residual norm in the course of the Newton iteration.
TUHH Heinrich Voss Isfahan, July, 2016 67 / 72 This time, j = 2 in the polynomial approximation, and k = 4, ` = 2.
The condition number of the Jacobian is 2.2e6 at convergence.
Invariant pairs Example 2
Example (by courtesy of Daniel Kressner) The experiments from the last example are repeated for the matrices
−0.8498 0.1479 44.37 0.28 0 0 A0 = 0.003756 −0.2805 −229.2 , A1 = 0 −0.28 0 , −0.1754 0.02296 −0.3608 0 0 0
and τ = 1, which has – according to Green & Wagenknecht (2006) – applications in the stability analysis of a semiconductor laser subject to external feedback.
TUHH Heinrich Voss Isfahan, July, 2016 68 / 72 The condition number of the Jacobian is 2.2e6 at convergence.
Invariant pairs Example 2
Example (by courtesy of Daniel Kressner) The experiments from the last example are repeated for the matrices
−0.8498 0.1479 44.37 0.28 0 0 A0 = 0.003756 −0.2805 −229.2 , A1 = 0 −0.28 0 , −0.1754 0.02296 −0.3608 0 0 0
and τ = 1, which has – according to Green & Wagenknecht (2006) – applications in the stability analysis of a semiconductor laser subject to external feedback.
This time, j = 2 in the polynomial approximation, and k = 4, ` = 2.
TUHH Heinrich Voss Isfahan, July, 2016 68 / 72 Invariant pairs Example 2
Example (by courtesy of Daniel Kressner) The experiments from the last example are repeated for the matrices
−0.8498 0.1479 44.37 0.28 0 0 A0 = 0.003756 −0.2805 −229.2 , A1 = 0 −0.28 0 , −0.1754 0.02296 −0.3608 0 0 0
and τ = 1, which has – according to Green & Wagenknecht (2006) – applications in the stability analysis of a semiconductor laser subject to external feedback.
This time, j = 2 in the polynomial approximation, and k = 4, ` = 2.
The condition number of the Jacobian is 2.2e6 at convergence.
TUHH Heinrich Voss Isfahan, July, 2016 68 / 72 Invariant pairs Eigenvalue approximations
4
2
0
−2
−4 −6 −5 −4 −3 −2 −1 0
Location of eigenvalue approximations initially (plus), after 3 iterations (circle), and at convergence after 12 iterations (cross). TUHH Heinrich Voss Isfahan, July, 2016 69 / 72 Invariant pairs Residual
105
100
10−5
10−10
10−15 0 2 4 6 8 10
Residual norm in the course of the Newton iteration.
TUHH Heinrich Voss Isfahan, July, 2016 70 / 72 Invariant pairs References
T. Betcke & D. Kressner: Perturbation, computation and refinement of invariant pairs for matrix polynomials. Linear Algebra Appl. 435, 514-536 (2011) I. Gohberg, P. Lancaster & L. Rodman: Matrix Polynomials. Academic Press, New York 1982 P. Hager & N.E. Wiberg: The rational Krylov algorithm for nonlinear eigenvalue problems. pp. 379-402 in B.H.V. Topping (ed.): Computational Mechanics for the Twenty-First Century. Saxe–Coburg Publications, Edinburgh 2000 D. Kressner: A block Newton method for nonlinear eigenvalue problems. Numer. Math. 114, 355-372 (2009) P. Lancaster: Lambda–matrices and Vibrating Systems. Dover Publications 2002 D.S. Mackey, N. Mackey, C. Mehl & V. Mehrmann: Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl. 28, 971-1004 (2006) A. Neumaier: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 22, 914-923 (1985) M.R. Osborne: Inverse iteration, Newton’s method and nonlinear eigenvalue problems. pp. 21-53 in The Contribution of Dr. J.H. Wilkinson to Numerical Analysis, Inst. Math. Appl., London 1978
TUHH Heinrich Voss Isfahan, July, 2016 71 / 72 Invariant pairs References ct.
A. Ruhe: Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 10, 674-689 (1973) K. Schreiber: Nonlinear Eigenvalue Problems: Newton-type Methods and Nonlinear Rayleigh Functionals. VDM Verlag Dr. Muller¨ 2008 H. Schwetlick & K. Schreiber: Nonlinear Rayleigh functionals. Linear Algebra Appl. 436, 3991 – 4016 (2012) H. Voss: Numerical methods for sparse nonlinear eigenproblems. pp. 133-160 in I. Marek (ed.): Proceedings of the XV-th Summer School on Software and Algorithms of Numerical Mathematics. Hejnice 2003 H. Voss: Nonlinear Eigenvalue Problems. Chapter 60 in L. Hogben (ed.): Handbook of Linear Algebra. CRC Press, Boca Raton 2014
TUHH Heinrich Voss Isfahan, July, 2016 72 / 72