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We apply polynomial approximation methods -- known in the numerical PDEscontext as spectral methods -- to approximate the vector-valued function thatsatisfies a linear system of equations where the matrix and the right hand sidedepend on a parameter. We derive both an interpolatory pseudospectral methodand a residual-minimizing Galerkin method, and we show how each can beinterpreted as solving a truncated infinite system of equations; the differencebetween the two methods lies in where the truncation occurs. Using classicaltheory, we derive asymptotic error estimates related to the region ofanalyticity of the solution, and we present a practical residual errorestimate. We verify the results with two numerical examples.

Spectral Methods for Parameterized Matrix Equations