Loudon, Tyson2021-08-162021-08-162021-04https://hdl.handle.net/11299/223151University of Minnesota Ph.D. dissertation.April 2021. Major: Mathematics. Advisor: Douglas Arnold. 1 computer file (PDF); vii, 181 pages.The Lanczos process is a method to approximate quadratic forms (f(A)v, v) for a symmetric matrix A, vector v, and smooth function f. In this thesis we focus on utilizing the Lanczos process to approximate joint spectral quantities determined by the spectra of two distinct operators. Joint spectral quantities are relevant in many areas of physics. The application we focus on is modeling the optical properties of semiconductors. Joint spectral quantities, when computed exactly, require complete knowledge of the spectra of two operators. In practical situations the computational cost of such a calculation is outside the realm of possibility. Instead, we focus on implementing an inexpensive and accurate method to approximate joint spectral quantities which relies on the Lanczos algorithm.enDensity of statesJoint density of statesJoint spectral quantityLanczos algorithmOptical absorptionThesis or Dissertation