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, on which this article is based.

History, original definition, and classical notation

The logarithmic norm was introduced in 1958, independently by Germund Dahlquist and Sergei Lozinskiĭ for square matrices.

is defined by

is the identity matrix. The limit exists on account of the convexity of the matrix norm.

The term logarithmic norm then

denotes the upper right Dini derivative. Using logarithmic differentiation, this bound can also be written as the differential inequality

,

, the logarithmic norm may take negative values 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. Other bounds are similar to those obtained using Grönwall's lemma associated with a nonautonomous linear differential equation satisfies the upper and lower bounds,

More recently, the logarithmic norm has been extended and generalized to nonlinear maps as well as to unbounded operators, 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), 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 as well as the classical least upper bound (upper) logarithmic norm by the two left and right limits

.

Alternatively, if an inner product norm,

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

is a strongly continuous semigroup, 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

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,

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:

.

:

Applications in matrix theory and spectral theory

are denoted by

stationary points of the two quadratic forms

, it holds that

and

together with

and

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, defined by

,

The numerical radius is a matrix norm 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 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.

is a rational function

This property is of importance in numerical analysis, where implicit Runge-Kutta methods is a rational function, referred to as the method's stability function, a condition known as A-stability.

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

. The corresponding lower and upper logarithmic Lipschitz constants are defined by

is Lipschitz continuous. For inner product norms, one can instead use the special definitions

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.

, it further holds that

is the Jacobian matrix.

is called uniformly monotone, and the uniform monotonicity theorem states that

generalizing Properties 14 - 15 above, and quantifying the Uniform monotonicity theorem.

, it can be shown that the error is bounded by

where the difference between the two most recent iterates is directly computable, and where any one of the three bounds

. Using the first (best) expression directly, a sharper result holds, but only for inner product norms, viz.,

. The bound relies on using the polarization identity, in the linear as well as the nonlinear case, and therefore only holds for the Euclidean norm.

. Both are A-stable Runge-Kutta methods is a linear map, both methods are contractive (see above). In the nonlinear case, however, the trapezoidal rule yields the recursion

while the implicit midpoint method is

, i.e.,

Thus A-stability is not sufficient. Instead a qualified condition, known as B-stability, is required for contractivity in the Euclidean norm; the results again rely on the polarization identity. Of the two methods above, only the implicit midpoint method is B-stable, but there are other, advanced B-stable Runge--Kutta methods of high order of convergence.

For nonlinear maps boundedness and continuity, together with an upper logarithmic norm

Likewise, in inner product spaces one may define lower and upper logarithmic norms by

Differential inequalities

, where

From the differential inequality

the following general bound is obtained using logarithmic Lipschitz constants,

covers the effect of forcing functions. In the latter case one can estimate

it also holds on infinite intervals, showing that the problem is well posed, i.e.,

In case of a stiff differential equation it holds that

where the expression on the left represents the maximal growth in reverse time, while the right expression shows the maximal growth in forward time. Hence the condition

is referred to as the stiffness indicator.

, the criterion can be given a local interpretation.

Note that

are the largest and smallest logarithmic values. Because it is relatively expensive to compute the logarithmic norms, an alternative of similar utility is to compute the average of all logarithmic values. The scaled trace is defined by

. Moreover, it equals the scaled divergence,

preserves phase volume (often referred to as area preserving is large and negative, implying that phase volume is compressed, and quantifies step size restrictions along the trajectory for explicit methods.

Elliptic differential operators

is called elliptic. In particular, the logarithmic norm quantifies the concept of strong ellipticity, with the inner product

, integration by parts we obtain

where the last step is the Poincaré inequality is the best possible constant. Thus

reduces ellipticity compared with the case of Dirichlet conditions.

and homogeneous Dirichlet conditions, Property 14 above offers the solution bound

is unique and depends continuously on the data, implying stability, where the inverse of the lower logarithmic norm is the stability constant.

Similar results hold for finite element, which becomes somewhat larger (since the minimizer in the Poincaré inequality cannot be represented). The theory can be extended to Sobolev norms and weak solutions, e.g. allowing "point loads" (Dirac functions in the right hand side), as is common in Galerkin methods, using different norms in the solution and data and spaces as needed.

If instead a finite difference method is positive definite and therefore invertible.

The importance of strong ellipticity in selfadjoint differential operators can be illustrated by Euler buckling represents the deflection from the column's center line, and satisfies the eigenvalue problem

is the cross-section moment of inertia of the column. The logarithmic norm of the associated operator is

is the unique solution. The zero solution is then no longer unique, in agreement with the Fredholm alternative, as the problem also admits nonzero (buckled) solutions satisfying the homogeneous boundary conditions.

, where it quantifies the growth or decay rates of solutions. This is primarily of use in dissipative systems. The difference between two solutions decays if

in which case the problem is well posed for sufficient dissipation, such as in a reaction-diffusion equation.

may depend in a nontrivial way on the type of boundary conditions and the balance of convection and diffusion rates (the Péclet number), leading to questions of well posedness. For problems in several space dimensions, it is usually difficult to calculate the logarithmic norm, as it depends on the geometry of the computational domain where the boundary conditions are imposed.

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