How far is a function from its linearisation? Typically, one would assume that is twice continuously differentiable and use the following second-order version of the mean value theorem:
This is typically used in the context of linearisation of nonlinear dynamical systems as in [Sec. 220.127.116.11, 1]. The requirement that is twice continuously differentiable, can, however, be reduced to being continuously differentiable with Lipschitz gradient.For a function we denote its Jacobian as . The inner product of two vectors is denoted as . The product of matrix with a vector will be also denoted as .
Let be a continuously differentiable function with L-Lipschitz Jacobian, that is, for all it is
for some . We will show that
Indeed, by the fundamental theorem of calculus
Note that in the above the integral is meant component-wise, so it is a matrix itself.
For convenience, define . Then
Such bounds can also be derived – for real-valued functions – when second order information is available. Read this post for details. In that case, the right-hand side involves a cubic term .