  It is generally accepted that supermassive black holes (SMBHs) grew from
progenitor seeds; but it remains unclear when the seeds formed, how large they
were, and what were their growth rates. This study finds answers to such open
questions in an analysis and interpretation of available mass (MBH) and age (t)
data of SMBHs at z>5.7. Their distribution reveals that a significant number
(36) spread over 300 million years (Myr) of cosmic time have markedly similar
masses (within a factor of 2) even though MBH depends on age. Besides age, MBH
depends upon seed mass Ms, origin time ts, and a variable dubbed accretion
ratio AR. The analysis leads to the conclusion that the 36 SMBHs had
similar-size seeds that formed essentially concurrently and that AR is
inversely proportional to t. Consequently, we get 36 equations with 1 variable
(t) and 3 parameters. Solving these equations using the SANN method, the
optimum values of the parameters define AR as a function of age t, and MBH as a
function of Ms and t. These relationships are tested and applied to thousands
of SMBHs. The results indicate that ts ~100Myr (z=30); that SMBHS > 1E+09M*
(solar masses) require seeds > 1E+03M* but < 5E+04M*; that seeds grew at an
exponentially increasing rate that reached a maximum near z=7 and decreased
thereafter; that seeds may have initially experienced a short period (150 Myr)
of super-Eddington accretion or accretion efficiency much <0.1; that about half
of the growth occurs from z=30-3.5 and the rest thereafter, and that accretion
is the predominant mode of growth with an upper limit of 2.2E+06Ms or
MBH=(6.7+/-2)E+10M*. This study provides quantitative estimates of accretion
rates, constraints for models of seed formation, and insights into the origins
and growth of Sagittarius A*.
