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The real-space renormalization group (RG) treatment of randomtransverse-field Ising spin chains by Fisher ({\it Phys. Rev. B{\bf 51}, 6411(1995)}) has been extended into the strongly ordered and strongly disorderedGriffiths phases and asymptotically exact results are obtained. In thenon-critical region the asymmetry of the renormalization of the couplings andthe transverse fields is related to a non-linear quantum control parameter,$\Delta$, which is a natural measure of the distance from the quantum criticalpoint. $\Delta$, which is found to stay invariant along the RG trajectories andhas been expressed by the initial disorder distributions, stands in thesingularity exponents of different physical quantities (magnetization,susceptibility, specific heat, etc), which are exactly calculated. In this waywe have observed a weak-universality scenario: the Griffiths-McCoysingularities does not depend on the form of the disorder, provided thenon-linear quantum control parameter has the same value. The exact scalingfunction of the magnetization with a small applied magnetic field is calculatedand the critical point magnetization singularity is determined in a simple,direct way.Comment: 11 page

Exact renormalization of the random transverse-field Ising spin chain in the strongly ordered and strongly disordered Griffiths phases