In mathematics, the logarithmic norm is a real-valued functional on operators, constructed from either a vector norm or an inner product, or directly from the induced operator norm. It quantifies key notions such as positive/negative definiteness in matrix theory, uniformly coercive or monotone vector fields in nonlinear analysis, and strong ellipticity in differential operators on function spaces, subject to specific boundary conditions.
The logarithmic norm has a wide range of applications in matrix theory, stability theory for initial and boundary value problems in differential equations, in nonlinear analysis, applied mathematics, and numerical analysis.
Comprehensive treatments of the theory and its applications can be found in recent monographs,Söderlind, G. (2024). "Logarithmic Norms." Springer Series in Computational Mathematics vol. 63. ISBN 978-3-031-74378-8 https://doi.org/10.1007/978-3-031-74379-5Bullo, F. (2022). "Contraction Theory for Dynamical Systems". 1st ed., (19 Aug. 2022). Kindle Direct Publishing ISBN 979-8836646806 on which this article is based.
History, original definition, and classical notation
The logarithmic norm was introduced in 1958, independently by Germund DahlquistDahlquist, G. (1958). "Stability and error bounds in the numerical integration of ordinary differential equations", Almqvist & Wiksell, Uppsala 1958 and Sergei LozinskiĭLozinskiĭ, S. (1958). "Error estimates for the numerical integration of ordinary differential equations, Part I". (In Russian) Izv. Vyss. Uceb. Zaved. Matematika 6, pp 52-90 (1958) for square matrices and bounded linear operators. The original purpose was to estimate solutions to linear differential equations , to construct sufficient conditions for stability, and to obtain norm bounds of perturbations due to the forcing function .
Let be a square matrix and be the operator norm induced by a given vector norm. The associated logarithmic norm is defined by
where is real and is the identity matrix of the same dimension as . The limit exists on account of the convexity of the matrix norm.
The term logarithmic norm is due to Lozinskiĭ, and refers to the fact that if then
where denotes the upper right Dini derivative with respect to time. In other words, is an upper bound for the (short-term) growth rate of the "logarithmic norm" of . Using logarithmic differentiation, this bound can also be written as the differential inequality
from which it follows that, for ,
This bound is always preferable to the Lipschitz bound , since . Moreover, while , the logarithmic norm may take negative values, e.g. when is negative definite and the Euclidean norm is used. Thus implies that is a contraction, and hence the solutions are stable in forward time.
The ability of the logarithmic norm to distinguish forward and reverse time evolution in dynamical systems is a key property. It may also establish whether a problem is well-posed. This asymmetry is also of importance in many applications. For example, in control theory, there is a fundamental need to discriminate between positive and negative feedback; this too can be characterized by the logarithmic norm.
Depending on the setting, the logarithmic norm is related to Lyapunov stability theory.Bullo, F. (2022). "Contraction Theory for Dynamical Systems". 1st ed., (19 Aug. 2022). Kindle Direct Publishing ISBN 979-8836646806 Other bounds are similar to those obtained using Grönwall's lemma. Thus the state transition matrix associated with a nonautonomous linear differential equation satisfies the upper and lower bounds for ,
More recently, the logarithmic norm has been extended and generalized to nonlinear mapsSöderlind, G. (1986). "Bounds on nonlinear operators in finite-dimensional Banach spaces". Numerical Mathematics 50, pp 27-44. as well as to unbounded operators,Söderlind, G. (2006). "The logarithmic norm. History and modern theory", BIT Numerical Mathematics, 46(3):631-652. covering boundary value problems and elementary applications in elliptic and parabolic partial differential equations.
The logarithmic norm may in some literature appear under less informative names, such as the "matrix measure" (for matrices) and "one-sided Lipschitz constant" (for nonlinear maps),Dekker, K.; Verwer, J.G. (1984). "Stability of Runge-Kutta methods for stiff nonlinear differential equations." New York: North Holland.} although the concept neither relates to measure theory nor implies Lipschitz continuity.
Extended definitions and modern notation
For clarity and generality, the modern notation defines a greatest lower bound (or simply lower) logarithmic norm as well as the classical least upper bound (upper) logarithmic norm by the two left and right limits
Here for all , and the lower and upper logarithmic norms are related by .
Alternatively, if an inner product norm is used, the lower and upper logarithmic norms are defined as the best possible bounds such that for all ,
where denotes the corresponding vector norm. In case of bounded linear operators in a Hilbert space setting, the inner product definition is equivalent to the original definition. But as the inner product definition also admits certain classes of unbounded operators it is more general. In the latter case one of the logarithmic norms may be infinite.
Further variants of use in semigroup theory are based on the limit
which is still equivalent to the classical definition if is a bounded linear operator. In case is a strongly continuous semigroup for , various modifications of this alternative are necessary, since the "vector field" is defined by the infinitesimal generator of the semigroup, i.e.,
The structure of the limit using the matrix exponential is shared by many other pairs of functionals, linking the vector field and the flow of a linear differential equation in a natural manner. For example, it holds that
where and are the (upper) spectral radius and (upper) spectral abscissa, respectively. Both limits exist, although neither the spectral radius nor the determinant is a convex functional. Moreover, for the trace, the left and right limits coincide. Moreover,
In spite of the similarities, there are major differences. Thus
where the important properties, enabling norm bounds, are that the operator norm is submultiplicative, and the logarithmic norm is subadditive. These properties do not hold for the spectral radius and abscissa, which are less powerful tools in the stability analysis of dynamical systems.
Properties and computation of logarithmic norms
For every choice of norm, the basic properties (including some redundancy) of the logarithmic norm of a matrix are the following:
Boundedness Lower and upper bounds Lower and upper bounds Continuity Real translation Weak homogeneity Sublinearity Superlinearity Convexity Concavity Lower and upper exponential bounds Positive definite inverse bound Negative definite inverse bound Spectral bound
As illustrated by properties 14 - 15, the condition () generalizes positive (negative) definiteness to arbitrary norms and non-symmetric matrices, while also quantifying these notions. An indefinite matrix with respect to a given norm satisfies .
For the most common norms, the logarithmic norm of a (complex) matrix can be computed as follows, where represents the matrix element in the row and column, and where denotes the complex conjugate of :
Applications in matrix theory and spectral theory
Let be a (possibly complex) matrix. The symmetric positive definite square root of and the Hermitian part of are denoted by
respectively. Their eigenvalue problems are and , where are the singular values of , and are the logarithmic values of . The singular and logarithmic values are the stationary points of the two quadratic forms
respectively. As for the relations to the eigenvalues of , it holds that
and
together with
and
As the quadratic form is generally complex valued, one can only find extrema of some related real function. For the logarithmic norms one takes the real part. Another option is to take the absolute value, to obtain the numerical radius of , defined by
Because , it follows that for all ,
The numerical radius is a matrix norm (but not an operator norm). However, as it is not defined by a quadratic form, the determination of is more intricate. But to each matrix norm there is a logarithmic norm, and
i.e., the lower and upper logarithmic norms of the numerical radius coincide with the Euclidean logarithmic norms. By the Hausdorff-Toeplitz theorem, the numerical range of a matrix is the set
in the complex plane, given by the intersection of the half-planes
and the numerical radius is
For matrix functions, the use of the von Neumann inequality and spectral sets provides the following classical result, using the Euclidean norm: let a polynomial be a such that . Then .
Using the logarithmic norm, this theorem can be generalized from unit circles to half-planes. Thus, if is a rational function such that
then
This property is of importance in numerical analysis, where implicit Runge-Kutta methods approximate a differential equation by a recursion , where is a rational function, referred to as the method's stability function. The continuous system is stable with a contractive flow if , while the discrete system is stable (a discrete contraction) if . Thus, if the stability function satisfies , a condition known as A-stability, then the Runge-Kutta method produces stable solutions for all negative semidefinite vector fields .
In a similar manner the theory of logarithmic norms can also be extended to unitarily invariant norms, e.g. the Hilbert-Schmidt norm for trace class operators, the Frobenius norm, and more generally the Schatten norms. As logarithmic values are only invariant under unitary similarity transformations, the logarithmic norms are defined subordinate to the matrix norms.
Extensions to nonlinear maps
For real-valued nonlinear maps there are various possibilities to define operator norms. The least upper bound Lipschitz constant is an operator seminorm, which is all that is required. A corresponding lower Lipschitz constant can be included to define
on a simply connected domain . The corresponding lower and upper logarithmic Lipschitz constants are defined by
where both limits exist if is Lipschitz continuous. For inner product norms, one can instead use the special definitions
where one or both of the logarithmic Lipschitz constants may exist even when is not Lipschitz continuous. Although it is difficult to compute actual values of Lipschitz and logarithmic Lipschitz constants in practice, they retain their theoretical importance as useful tools in nonlinear analysis.Söderlind, G. (1986). "Bounds on nonlinear operators in finite-dimensional Banach spaces". Numerical Mathematics 50, pp 27-44.
All properties listed above for linear operators are retained, provided that operations like etc. are allowed on a joint domain of definition. In the special case when is Fréchet differentiable on a convex domain , it further holds that
where is the Jacobian matrix of .
An operator having either or is called uniformly monotone, and the uniform monotonicity theorem states that
generalizing Properties 14 - 15 above, and quantifying the Uniform monotonicity theorem due to Browder and Minty (1963). Here, the Lipschitz constant of the inverse is defined on the image of , while its lower logarithmic Lipschitz constant is defined on the domain of .
Because composition is not distributive over addition for nonlinear maps, the von Neumann type bounds for 'rational functions of operators' have no direct counterparts in the nonlinear case. Nevertheless some results carry over. For example, in a fixed point iteration
converging to a fixed point , it can be shown that the error is bounded by