  We analyze the ground-state and low-temperature properties of a
one-dimensional Bose gas in a harmonic trapping potential using the numerical
density matrix renormalization group. Calculations cover the whole range from
the Bogoliubov limit of weak interactions to the Tonks-Girardeau limit. Local
quantities such as density and local three-body correlations are calculated and
shown to agree very well with analytic predictions within a local density
approximation. The transition between temperature dominated to quantum
dominated correlation is determined and it is shown that despite the presence
of the harmonic trapping potential first-order correlations display over a
large range the algebraic decay of a harmonic fluid with a Luttinger parameter
determined by the density at the trap center.
