  In this work, we report an application of Hylleraas-$B$-spline basis set to
the nonrelativistic Bethe logarithm calculation of helium. The Bethe logarithm
for $n\ ^1S$, $n$ up to 10, states of helium are calculated with a precision of
7-9 significant digits in two gauges, which greatly improves the accuracy of
the traditional $B$-spline basis set. In addition, to deal with the numerical
linear correlation problem in Bethe logarithm calculation, we developed a
multiple-precision generalized symmetric eigenvalue problem solver (MGSEPS).
This program may be very useful to precision calculations.
