The Nonlinear Eigenvalue Problem: Part I

Research Matters

February 25, 2009

Nick HighamDirector of Research

School of Mathematics

The Nonlinear Eigenvalue Problem:Part I

Françoise Tisseur

School of MathematicsThe University of Manchester

ESSAM, Kácov, 2018

OutlineThree lectures:

I Part I: Mathematical properties of nonlineareigenproblems (NEPs)

Definition and historical aspectsExamples and applicationsSolution structure

I Part II and part III: Numerical methods for NEPsSolvers based on Newton’s methodSolvers using contour integralsLinear interpolation methods

S. GÜTTEL AND F. TISSEUR, The nonlinear eigenvalue problem.Acta Numerica 26:1–94, 2017.

Françoise Tisseur The nonlinear eigenvalue problem 2 / 31

Nonlinear Eigenvalue Problems (NEPs)Let F : Ω → Cn×n with Ω ⊆ C a nonempty open set.

Nonlinear eigenvalue problem: Find scalars λ andnonzero x , y ∈ Cn satisfying F (λ)x = 0 and y∗F (λ) = 0.

λ is an e’val, x , y are corresponding right/left e’vecs.

E’vals are solutions of p(λ) = det(F (λ)) = 0.

Algebraic multiplicity of λ is multiplicity of λ as root of p.

Familiar examples:I Standard eigenvalue problem: F (λ) = λI − A.

I Generalized eigenvalue problem: F (λ) = λB − A.

I Quadratic eigenvalue problem: F (λ) = λ2M + λD + K .

In practice, elements of F can be polynomial, rational orexponential functions of λ.

Polynomial Eigenvalue Problems (PEPs)

Consider n × n matrix polynomial

P(λ) = λ`A` + · · ·+ λA1 + A0 (monomial form)= φ`(λ)P` + · · ·+ φ1(λ)P1 + φ0(λ)P0 (non-monomial form),

where φi(λ) is a polynomial of degree i .

I Assume P(λ) is regular, i.e., det(P(λ)) 6≡ 0.

I P(λ) has `n e’vals. Finites e’vals are roots ofdet(P(λ)) = 0.

I Zero e’vals when det(A0) = 0 and∞ e’vals whendet(A`) = 0.

Polynomial Eigenvalue Problems (PEPs)

Consider n × n matrix polynomial

P(λ) = λ`A` + · · ·+ λA1 + A0 (monomial form)= φ`(λ)P` + · · ·+ φ1(λ)P1 + φ0(λ)P0 (non-monomial form),

where φi(λ) is a polynomial of degree i .

I Assume P(λ) is regular, i.e., det(P(λ)) 6≡ 0.

I P(λ) has `n e’vals. Finites e’vals are roots ofdet(P(λ)) = 0.

I Zero e’vals when det(A0) = 0 and∞ e’vals whendet(A`) = 0.

Example 1: 2× 2 NEP

F (z) =

[eiz2 11 1

]on Ω = C. Roots of det F (z) = eiz2 − 1

areλk = ±

√2πk , k = 0,±1,±2, . . . ,

with λ0 = 0 of alg. multiplicity 2 and all others simple.

Ω = z ∈ C :− 8 ≤ Re(z) ≤ 8,− 8 ≤ Im(z) ≤ 8.

Note that[ 1−1

]is a left and right e’vec for all of the e’vals!

Remarks

Generally, an NEP can have

no eigenvalues at all, e.g., if F (z) = exp(z),

finitely many eigenvalues, e.g., if F (z) = z3 − 1,

countably many eigenvalues, e.g., if F (z) = cos(z),

a continuum of eigenvalues, e.g., F (z) = 0.

To exclude last case, F is called regular if det F (z) 6≡ 0 onΩ.

Historical Aspects

In the 1930s, Frazer, Duncan & Collar weredeveloping matrix methods for analyzing flutter inaircraft.

Worked in Aerodynamics Division of NPL.

Developed matrix structural analysis.

Wrote Elementary Matrices & Some Applications toDynamics and Differential Equations, 1938.

Olga Taussky, in Frazer’s group at NPL, 1940s.6× 6 quadratic eigenvalue problems from flutter insupersonic aircraft.

Historical Aspects (cont.)

Peter Lancaster, English Electric Co., 1950s solvedquadratic eigenvalue problems of dimension 2 to 20.

I Lancaster, Lambda-Matrices and Vibrating Systems,1966 (Pergamon), 2002 (Dover).

I Gohberg, Lancaster, Rodman, Matrix Polynomials,1982 (Academic Press), 2009 (SIAM).

I Gohberg, Lancaster, Rodman, Indefinite LinearAlgebra and Applications, 2005 (Birkhäuser).

NLEVP Toolbox

Collection of Nonlinear Eigenvalue Problems : T. Betcke,

N. J. Higham, V. Mehrmann, C. Schröder, F. T., TOMS 2013.

I Quadratic, polynomial, rational and other nonlineareigenproblems.

I Provided in the form of a MATLAB Toolbox.

I Problems from real-life applications + specificallyconstructed problems.

http://www.mims.manchester.ac.uk/research/numerical-analysis/nlevp.html

Sample of Quadratic Problems

n ×m quadratic Q(λ) = λ2M + λD + K .

Speaker box (pep,qep,real,symmetric).n = m = 107. Finite element model of a speaker box.‖M‖2 = 1, ‖D‖2 = 5.7× 10−2, ‖K‖2 = 1.0× 107.

Railtrack (pep,qep,t-palindromic,sparse).n = m = 1005. Model of vibration of rail tracks under theexcitation of high speed trains. M = K T , D = DT .

Surveillance (pep,qep,real,nonsquare,nonregular). n = 21, m = 16. From calibration ofsurveillance camera using human body as calibrationtarget.

Sample of Higher Degree Problems

n × n matrix polynomials: P(λ) =∑d

j=0 λjCj .

Orr-Sommerfeld(pep,parameter-dependent). Quarticarising in spatial stability analysis of Orr-Sommerfeld eq.

Plasma drift(pep). Cubic polynomial from modeling of driftinstabilities in the plasma edge inside a Tokamak reactor.

Butterfly (pep,real,T-even, scalable)quartic matrix polynomialwith T-even structure:PT (−λ) = P(λ).

−1.5 −1 −0.5 0 0.5 1 1.5−2

−1.5

−0.5

Eigenvalues

Boundary Value Problem (Solov’ëv 2006)

Differential equations with nonlinear boundary conditions

−u′′(x) = λu(x) on [0,1], u(0) = 0, −u′(1) = φ(λ)u(1).

Finite element discretization yields NEP

F (λ)u = (C1 − λC2 + φ(λ)C3)u = 0

with n × n matrices

C1 = n

2 −1

−1. . . . . .. . . 2 −1

−1 1

, C2 =1

. . . . . .

. . . 4 11 2

,C3 = eneT

n , and en = [0, . . . ,0,1]T .

Sample of Rational/Nonlinear ProblemsLoaded string (rep,real,symmetric,scalable)rational eigenvalue problem describing eigenvibration of aloaded string.

F (λ)v =

(C1 − λC2 +

λ− σC3

)v = 0.

Gun (nep,sparse) nonlinear eigenvalue problemmodeling a radio-frequency gun cavity.

F (λ)v =[K − λM + i(λ− σ2

1)12 W1 + i(λ− σ2

2)12 W2

]v = 0.

I C3, W1 and W2 have low rank.

Time delay(nep,real) from stability analysis of delaydifferential equations (DDEs).

F (λ)v = (λI + A + Be−λ)v = 0,

Re(λ) determines growth of solution.

Sample of Rational/Nonlinear ProblemsLoaded string (rep,real,symmetric,scalable)rational eigenvalue problem describing eigenvibration of aloaded string.

F (λ)v =

(C1 − λC2 +

λ− σC3

)v = 0.

Gun (nep,sparse) nonlinear eigenvalue problemmodeling a radio-frequency gun cavity.

F (λ)v =[K − λM + i(λ− σ2

1)12 W1 + i(λ− σ2

2)12 W2

]v = 0.

I C3, W1 and W2 have low rank.

Time delay(nep,real) from stability analysis of delaydifferential equations (DDEs).

F (λ)v = (λI + A + Be−λ)v = 0,

Re(λ) determines growth of solution.Françoise Tisseur The nonlinear eigenvalue problem 14 / 31

Other Classes of NEPs

Multi-parameter NEPs

F (λ, γ)v = 0.

Arise in stability analysis of parametrized nonlinearwave equations and of delay differential equations.Numerical solution via continuation and solution of anNEP at every iteration.

Eigenvector nonlinearities: find nontrivial solutions of

F (V )V = VD,

where n × n F depends on n × k matrix of e’vecs V ,and k × k diagonal D contains e’vals. [Not consideredin this course.]

Comments

I NEPs underpin many areas of computational scienceand engineering.

I Can be very difficult to solve:

nonlinear,

large problem size,

poor conditioning,

The past ten years has seen numerous breakthroughs inthe development of numerical methods.

Comments

I NEPs underpin many areas of computational scienceand engineering.

I Can be very difficult to solve:

nonlinear,

large problem size,

poor conditioning,

The past ten years has seen numerous breakthroughs inthe development of numerical methods.

Discreteness Result

Assumption: Holomorphic NEPWe assume that F is holomorphic on a domain Ω ⊆ C,denoted by F ∈ H(Ω,Cn×n).

Note that F ∈ H(Ω,Cn×n)⇒ det F (z) ∈ H(Ω,C), hence:

Theorem (Atkinson 1952)Let F ∈ H(Ω,Cn×n) be regular on adomain Ω. Then Λ(F ) is a discrete setwithout accumulation points in Ω.

New Section 1 Page 1

Remark: There may well be accumulation points on ∂Ω,e.g., F (z) = sin(1/z) with roots accumulating at 0 6∈ Ω.

Discreteness Result

Assumption: Holomorphic NEPWe assume that F is holomorphic on a domain Ω ⊆ C,denoted by F ∈ H(Ω,Cn×n).

Note that F ∈ H(Ω,Cn×n)⇒ det F (z) ∈ H(Ω,C), hence:

Theorem (Atkinson 1952)Let F ∈ H(Ω,Cn×n) be regular on adomain Ω. Then Λ(F ) is a discrete setwithout accumulation points in Ω.

New Section 1 Page 1

Remark: There may well be accumulation points on ∂Ω,e.g., F (z) = sin(1/z) with roots accumulating at 0 6∈ Ω.

More Definitions and CommentsLet F ∈ H(Ω,Cn×n).

I Algebraic multiplicity of isolated e’val λ of F is finitebut not necessarily bounded by n.

I Geometric multiplicity of λ := dim(nullF (λ)).

I v0, v1, . . . , vm−1 are generalized e’vecs (Jordan chain)of F with e’val λ if

∑jk=0

F (k)(λ)k! vj−k = 0, 0 ≤ j ≤ m − 1.

v0 is an e’vec of F .

The vi are not necessarily linearly independent.

vj = 0, j 6= 0 can happen!

Maximal length m of Jordan chain starting at v0 ispartial multiplicity.

Factorization: Smith FormTheorem (Smith form)Let F ∈ H(Ω,Cn×n) be regular on a nonempty domain Ω.Let λi (i = 1,2, . . .) be the distinct e’vals of F in Ω withpartial mult mi,1 ≥ mi,2 ≥ · · · ≥ mi,di . Then there exists aglobal Smith form P(z)F (z)Q(z) = D(z) , z ∈ Ω, withunimodular P,Q ∈ H(Ω,Cn×n), andD(z) = diag(δ1(z), δ2(z), . . . , δn(z)), where

δj(z) = hj(z)∏

i=1,2,...

(z − λi)mi,j , j = 1, . . . ,n,

mi,j = 0 if j > di and each hj ∈ H(Ω,C) is free of roots on Ω.

Resolvent is given by

F (z)−1 = Q(z)D(z)−1P(z) =n∑

δj(z)−1qj(z)pj(z)∗.

Keldysh’s Theorem“Locally, near an eigenvalue λ, the resol-vent F (z)−1 of a regular NEP behaveslike that of a linear operator.”

Theorem (Keldysh 1951)

Let λ ∈ Λ(F ) have alg multiplicity m and geom multiplicity d.Then there are n ×m matrices V and W and an m ×mJordan matrix J with eigenvalue λ of partial multiplicitiesm1 ≥ m2 ≥ · · · ≥ md ,

∑di=1 mi = m, such that

F (z)−1 = V (zI − J)−1W ∗ + R(z)

in some open neighborhood U of λ, with R ∈ H(U ,Cn×n).

R. MENNICKEN AND M. MÖLLER. Non-self-adjoint boundaryeigenvalue problems, Elsevier Science, 2003.

Keldysh’s Theorem“Locally, near an eigenvalue λ, the resol-vent F (z)−1 of a regular NEP behaveslike that of a linear operator.”

Theorem (Keldysh 1951)

Let λ ∈ Λ(F ) have alg multiplicity m and geom multiplicity d.Then there are n ×m matrices V and W and an m ×mJordan matrix J with eigenvalue λ of partial multiplicitiesm1 ≥ m2 ≥ · · · ≥ md ,

∑di=1 mi = m, such that

F (z)−1 = V (zI − J)−1W ∗ + R(z)

in some open neighborhood U of λ, with R ∈ H(U ,Cn×n).

R. MENNICKEN AND M. MÖLLER. Non-self-adjoint boundaryeigenvalue problems, Elsevier Science, 2003.

Spectral Portraits z 7→ log10 ‖F (z)−1‖2

F (z) =[eiz2

]. Near λ = 0,±

√2π,±i

√2π, Keldysh’s thm

says thatF (z)−1 ≈ V (zI − J)−1W ∗, J = diag

],±√

2π,±i√

Invariant PairsCan always write F ∈ H(Ω,Cn×n) in “split form”

F (z) = f1(z)C1 + f2(z)C2 + · · ·+ f`(z)C`, ` ≤ n2.

Definition (invariant pair)A pair (V , Λ) ∈ Cn×m × Cm×m is called an invariant pair forF ∈ H(Ω,Cn×n) if

C1Vf1(Λ) + C2Vf2(Λ) + · · ·+ C`Vf`(Λ) = 0. (∗)

I Beyn et al. (2011) introduce (V , Λ) via Cauchy integral.

I Generalizes to NEPs invariant subspaces for A ∈ Cn×n:F (z) = zI − A so that f1(z) = z, f2(z) = 1,C1 = I, C2 = −A.(∗)⇐⇒ AV = VΛ, i.e., V is an invariant subspace for A.

I Generalizes standard/Jordan pairs for matrix poly.

Invariant PairsCan always write F ∈ H(Ω,Cn×n) in “split form”

F (z) = f1(z)C1 + f2(z)C2 + · · ·+ f`(z)C`, ` ≤ n2.

Definition (invariant pair)A pair (V , Λ) ∈ Cn×m × Cm×m is called an invariant pair forF ∈ H(Ω,Cn×n) if

C1Vf1(Λ) + C2Vf2(Λ) + · · ·+ C`Vf`(Λ) = 0. (∗)

I Beyn et al. (2011) introduce (V , Λ) via Cauchy integral.

I Generalizes to NEPs invariant subspaces for A ∈ Cn×n:F (z) = zI − A so that f1(z) = z, f2(z) = 1,C1 = I, C2 = −A.(∗)⇐⇒ AV = VΛ, i.e., V is an invariant subspace for A.

I Generalizes standard/Jordan pairs for matrix poly.Françoise Tisseur The nonlinear eigenvalue problem 22 / 31

Complete Invariant PairInvariant pairs may contain redundant information: if (V , Λ)is an invariant pair so is ([V ,V ],diag(Λ,Λ)).

(V , Λ) ∈ Cn×m × Cm×m is minimal if there is p > 0 s.t.

VΛ...

VΛp−1

Definition (Complete invariant pair)

An invariant pair (V , Λ) of F ∈ H(Ω,Cn×n) is complete if itis minimal and the alg multiplicities of the e’vals of Λ are thesame as the alg multiplicities of corresponding e’vals of F .

See Kressner’s block Newton method (2009) andEfferberger’s deflation strategy (2013).

VΛ...

VΛp−1

VΛ...

VΛp−1

Hermitian NEPs

F (z) is said to be Hermitian if F (z) = F (z)∗ for all z ∈ C.

I Ei’vals are either real or come in pairs (λ, λ).

I If v and w are right and left ei’vecs of F with e’valλ ∈ C, then w and v are right and left ei’vecs for theei’val λ.

I If λ ∈ R with right ei’vec v , then v is a left ei’vec for λ.

Do variational principles for real eigenvalues of HermitianNEPs exist?

Yes but under some assumptions.

Hermitian NEPs

Assumptions (A1)–(A3)

(A1) F : R ⊇ I→ Cn×n is Hermitian and continuouslydifferentiable on open real interval I.

(A2) For every x ∈ Cn \ 0, the real nonlinear equation

x∗F(p(x)

)x = 0 (1)

has at most one real solution p(x) ∈ I.(1) defines implicitly the generalized Rayleigh functionalp on some open subset K(p) ⊆ Cn \ 0. WhenK(p) ≡ Cn \ 0, the NEP F (λ)v = 0 is called overdamped.

(A3) For every x ∈ K(p) and any z ∈ I such that z 6= p(x),(z − p(x)

)(x∗F (z)x

When F is overdamped, (A3) holds if x∗F ′(p(x))x > 0 for allx ∈ Cn \ 0.

Nonlinear Variational Principle

An e’val λ ∈ I of F is a k th eigenvalue if µ = 0 is the k thlargest eigenvalue of the Hermitian matrix F (λ).

Theorem (Hadeler 1968)Assume that F satisfies (A1)–(A3). Then F has at most ne’vals in I. Moreover, if λk is a kth eigenvalue of F , then

λk = minV∈Sk

V∩K(p)6=∅

maxx∈V∩K(p)

p(x) ∈ I,

λk = maxV∈Sk−1

V⊥∩K(p) 6=∅

minx∈V⊥∩K(p)

p(x) ∈ I,

where Sj is the set of all subspaces of Cn of dimension j.

Used to develop safegarded iteration (Werner 1970).Preconditioned CG method (Szyld and Xue 2015).

Nonlinear Variational Principle

An e’val λ ∈ I of F is a k th eigenvalue if µ = 0 is the k thlargest eigenvalue of the Hermitian matrix F (λ).

Theorem (Hadeler 1968)Assume that F satisfies (A1)–(A3). Then F has at most ne’vals in I. Moreover, if λk is a kth eigenvalue of F , then

λk = minV∈Sk

V∩K(p)6=∅

maxx∈V∩K(p)

p(x) ∈ I,

λk = maxV∈Sk−1

V⊥∩K(p) 6=∅

minx∈V⊥∩K(p)

p(x) ∈ I,

where Sj is the set of all subspaces of Cn of dimension j.

Used to develop safegarded iteration (Werner 1970).Preconditioned CG method (Szyld and Xue 2015).

Eigenvalue LocalizationBindel & Hood (2013) provide a Gershgorin’s thm for NEPs.

TheoremLet F (z) = D(z) + E(z) with D,E ∈ H(Ω,Cn×n) and Ddiagonal. Then for any 0 ≤ α ≤ 1,

Λ(F ) ⊂n⋃

Gαj ,

where Gαj is the jth generalized Gershgorin region

Gαj =

z ∈ Ω : |djj(z)| ≤ rj(z)αcj(z)1−α

rj(z) =n∑

|ejk (z)|, cj(z) =n∑

|ekj(z)|.

Sensitivity and Stability

Condition number measures sensitivity of the solutionof a problem to perturbations in the data.

Backward error measures how well the problem hasbeen solved.

error in solution <∼ condition number× backward error.

Sensitivity of Eigenvalues

Assume λ 6= 0 is a simple ei’val with right and left ei’vecs vand w . Define normwise relative condition number of λ:

κ(λ,F )= lim supε→0

|δλ|ε|λ|

:(F (λ + δλ) + ∆F (λ + δλ)

)(v + δv) = 0,

‖∆Cj‖2 ≤ ε‖Cj‖2, j = 1, . . . , `,

where ∆F (z) = f1(z)∆C1 + f2(z)∆C2 + · · ·+ f`(z)∆C`.

Can show that

κ(λ,F ) =

(∑`j=1 ‖Cj‖2|fj(λ)|

)‖v‖2‖w‖2

|λ||w∗F ′(λ)v |.

Sensitivity of Eigenvalues

Assume λ 6= 0 is a simple ei’val with right and left ei’vecs vand w . Define normwise relative condition number of λ:

κ(λ,F )= lim supε→0

|δλ|ε|λ|

:(F (λ + δλ) + ∆F (λ + δλ)

)(v + δv) = 0,

‖∆Cj‖2 ≤ ε‖Cj‖2, j = 1, . . . , `,

where ∆F (z) = f1(z)∆C1 + f2(z)∆C2 + · · ·+ f`(z)∆C`.

Can show that

κ(λ,F ) =

(∑`j=1 ‖Cj‖2|fj(λ)|

)‖v‖2‖w‖2

|λ||w∗F ′(λ)v |.

Example 1 (Cont.)

F (z) = eiz2[

1 00 0

[0 11 1

]=: f1(z)C1 + f2(z)C2.

Can show nonzero e’vals ofF , λk = ±

√2πk , k =

±1,±2, . . ., have small con-dition numbers

κ(λk ,F ) =1 +

√1 + (3 + 51/2)/2

2π|k |

that get smaller as k in-creases in modulus.

Spectral portrait, z 7→ log10 ‖F (z)−1‖2

Backward Error of Approximate Eigenpairs

Backward error of an approximate ei’val λ of F defined by

ηF (λ) = minv∈Cnv 6=0

ηF (λ, v),

ηF (λ, v) = minε : (F (λ) + ∆F (λ))v = 0, ‖∆F‖ ≤ ε

is the backward error of the approximate ei’pair (λ, v).Can show that

ηF (λ, v) =‖F (λ)v‖2

f (λ) ‖v‖2and ηF (λ) =

f (λ) ‖F (λ)−1‖2,

where f (λ) =∑`

j=1 ‖Cj‖2|fj(λ)| if ‖∆F‖ = max1≤j≤`‖∆Cj‖2‖Cj‖2

.Françoise Tisseur The nonlinear eigenvalue problem 31 / 31

References I

F. Atkinson.

A spectral problem for completely continuous operators.

Acta Math. Hungar., 3(1-2):53–60, 1952.

T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, andF. Tisseur.

NLEVP: A collection of nonlinear eigenvalue problems.

ACM Trans. Math. Software, 39(2):7:1–7:28, Feb. 2013.

D. Bindel and A. Hood.

Localization theorems for nonlinear eigenvalue problems.

SIAM J. Matrix Anal. Appl., 34(4):1728–1749, 2013.

References II

R. A. Frazer, W. J. Duncan, and A. R. Collar.

Elementary Matrices and Some Applications to Dynamicsand Differential Equations.

Cambridge University Press, Cambridge, UK, 1938.

xviii+416 pp.

1963 printing.

I. Gohberg, P. Lancaster, and L. Rodman.

Indefinite Linear Algebra and Applications.

Birkhäuser, Basel, Switzerland, 2005.

ISBN 3-7643-7349-0.

xii+357 pp.

References III

I. Gohberg, P. Lancaster, and L. Rodman.

Matrix Polynomials.

Society for Industrial and Applied Mathematics, Philadelphia,PA, USA, 2009.

ISBN 0-898716-81-8.

xxiv+409 pp.

Unabridged republication of book first published by AcademicPress in 1982.

S. Güttel and F. Tisseur.

The nonlinear eigenvalue problem.

In Acta Numerica, volume 26, pages 1–94. CambridgeUniversity Press, 2017.

References IV

K. P. Hadeler.

Variationsprinzipien bei nichtlinearen Eigenwertaufgaben.

Arch. Rational Mech. Anal., 30:297–307, 1968.

D. Kressner.

A block Newton method for nonlinear eigenvalue problems.

Numer. Math., 114(2):355–372, 2009.

References V

P. Lancaster.

Lambda-Matrices and Vibrating Systems.

Pergamon Press, Oxford, 1966.

ISBN 0-486-42546-0.

xiii+196 pp.

Reprinted by Dover, New York, 2002.

R. Mennicken and M. Möller.

Non-Self-Adjoint Boundary Eigenvalue Problems, volume192.

Elsevier Science B. V., Amsterdam, The Netherlands, 2003.

References VI

S. I. Solov′ëv.

Preconditioned iterative methods for a class of nonlineareigenvalue problems.

Linear Algebra Appl., 415:210–229, 2006.

D. B. Szyld and F. Xue.

Local convergence of Newton-like methods for degenerateeigenvalues of nonlinear eigenproblems. I. classicalalgorithms.

Numer. Math., 129(2):353–381, 2015.

References VII

B. Werner.

Das Spektrum von Operatorenscharen mit verallgemeinertenRayleighquotienten.

PhD thesis, Universität Hamburg, Hamburg, Germany, 1970.

The Nonlinear Eigenvalue Problem: Part I

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