  We present a general and exact formalism for finding the evolution of a
quantum system subject to external telegraph noise. The various qubit
decoherence rates are determined by the eigenvalues of a transfer matrix. The
formalism can be applied to a qubit subject to an arbitrary combination of
dephasing and relaxational telegraph noise, in contrast to existing
non-perturbative methods that treat only one or the other of these limits. We
present 3 applications: 1) We obtain the full qubit dynamics on time scales
short compared with the enviromental correlation times. In the strong coupling
cases this reveals unexpected oscillations and induced magnetization
components; 2) We find in strong coupling case strong violations of the widely
used relation 1/T$_2$ = 1/2T$_1$ + 1/T$_{\phi}$, which is a result of
perturbation theory; 3) We discuss the effects of bang-bang and spin-echo
controls of the qubit dynamics in general settings of the telegraph noises.
%The result shows that these methods are not very effective in %reducing
decoherence arising from a single telegraph noise. Finally, we discuss the
extension of the method to the cases of many telegraph noise sources and
multiple qubits. The method still works when white noise is also present.
