  We develop heuristic interpolation methods for the functions $t \mapsto \log
\det \left( \mathbf{A} + t \mathbf{B} \right)$ and $t \mapsto
\operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{p} \right)$ where the
matrices $\mathbf{A}$ and $\mathbf{B}$ are Hermitian and positive (semi)
definite and $p$ and $t$ are real variables. These functions are featured in
many applications in statistics, machine learning, and computational physics.
The presented interpolation functions are based on the modification of sharp
bounds for these functions. We demonstrate the accuracy and performance of the
proposed method with numerical examples, namely, the marginal maximum
likelihood estimation for Gaussian process regression and the estimation of the
regularization parameter of ridge regression with the generalized
cross-validation method.
