  We present a new method of exactly calculating neutrino oscillation
probabilities in matter. We leverage the "eigenvector-eigenvalue identity" to
show that, given the eigenvalues, all mixing angles in matter follow
surprisingly simply. The CP violating phase in matter can then be determined
from the Toshev identity. Then, to avoid the cumbersome expressions for the
exact eigenvalues, we have applied previously derived perturbative, approximate
eigenvalues to this scheme and discovered them to be even more precise than
previously realized. We also find that these eigenvalues converge at a rate of
five orders of magnitude per perturbative order which is the square of the
previously realized expectation. Finally, we provide an updated speed versus
accuracy plot for oscillation probabilities in matter, to include the methods
of this paper.
