The spectrum of delay-differential equations: numerical methods - KTH

Notation
R real numbers
R+ positive real numbers
R− negative real numbers
Z integer numbers
Q rational numbers
C complex numbers
C+ open right half plane
C− open left half plane
D, ∂D open unit disc (D := {z ∈ C : |z| < 1}), unit circle
∂C boundary of the set C
clos C closure of the set C, clos C = ∂C ∪ C
i imaginary unit, i2 = −1
Re z real part of the complex number z
Im z imaginary part of the complex number z
l(z1, z2) straight line connecting z1, z2 ∈ C not including endpoints
Arg z principal branch argument of the complex number z
¯z complex conjugate of the complex number z
A∗ complex conjugate transpose of the matrix A
σ(A), σ(Σ) spectrum of the matrix A or system Σ
σ(G) solutions of s ∈ σ(G(s)) where G : C → Cn×n σ(A, B) solutions of the generalized eigenvalue problem det(A − λB) = 0
σmin(A) the smallest singular value of the matrix A
� · � Eucledian or spectral norm
κ(A) condition number of matrix A
rσ(A)
atan
spectral radius of matrix A
� � a
b four quadrant inverse of tangent, atan � � a
b = Arg (b + ai)
sgn(z) sign of complex number z, sgn(z) = z/|z| if z �= 0
O(g(x)) big O, f(x) = O(g(x)) ⇔ ∃M > 0, x0 > 0 : |f(x)| < M|g(x)| ∀ x > x0
ɛmach machine precision, where not explicitly stated: ɛmach ≈ 2.2 · 10−16 ⊗, ⊕ Kronecker product, Kronecker sum
Wk kth branch of the Lambert W function
dm(S1, S2) maxmin distance between sets S1 and S2
dH(S1, S2) Hausdorff distance between sets S1 and S2
C([a, b]) A continuous function on the segment t ∈ [−a, b]