  Fast radio bursts (FRBs) are millisecond-duration flashes with unknown
origins. Its formation rate is crucial for unveiling physical origins. However,
the luminosity and formation rate are degenerated when directly fitting the
redshift distribution of FRBs. In contrast to previous forward-fitting methods,
we use the Lynden-Bell's $c^{-}$ method to derive luminosity function and
formation rate of FRBs without any assumptions. Using the non-repeating FRBs
from the first CHIME/FRB catalog, we find a relatively strong luminosity
evolution, and luminosity function can be fitted by a broken power-law model
with a break at $1.33\times10^{41}\ \mathrm{erg}\ \mathrm{s}^{-1}$. The
formation rate declines rapidly as $(1+z)^{-4.9\pm0.3}$ with a local rate
$1.13\times10^4\ \mathrm{Gpc}^{-3}\ \mathrm{yr}^{-1}$. This monotonic decrease
is similar to the rate of short gamma-ray bursts. After comparing it with star
formation rate and stellar mass density, we conclude that the old populations
including neutron stars and black holes, are closely related to the origins of
FRBs. Monte Carlo simulations are used to test our results. The distributions
of mock sample are consistent with the observational data.
