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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 thematrices $\mathbf{A}$ and $\mathbf{B}$ are Hermitian and positive (semi)definite and $p$ and $t$ are real variables. These functions are featured inmany applications in statistics, machine learning, and computational physics.The presented interpolation functions are based on the modification of sharpbounds for these functions. We demonstrate the accuracy and performance of theproposed method with numerical examples, namely, the marginal maximumlikelihood estimation for Gaussian process regression and the estimation of theregularization parameter of ridge regression with the generalizedcross-validation method.

Interpolating Log-Determinant and Trace of the Powers of Matrix $\mathbf{A} + t \mathbf{B}$