  We derive and implement a new way to find lower bounds on the smallest
limiting trace-to-degree ratio of totally positive algebraic integers and
improve the previously best-known bound to 1.798249. Our method adds new
constraints to Smyth's linear programming method to decrease the variables
required in the new problem of interest. This allows for faster convergence
recovering Schur's bound in the simplest case and Siegel's bound in the second
simplest case of our new family of bounds.
  We also prove the existence of a unique optimal solution to our newly phrased
problem and express the optimal solution in terms of polynomials. Lastly, we
solve this new problem numerically with a gradient descent algorithm to attain
the new bound 1.798249.
