  Numerical tools for computation of $\wp$-functions, also known as Kleinian,
or multiply periodic, are proposed. In this connection, computation of periods
of the both first and second kinds is reconsidered. An analytical approach to
constructing the Reimann surface of a plane algebraic curves of low gonalities
is used. The approach is based on explicit radical solutions to quadratic,
cubic, and quartic equations, which serve for hyperelliptic, trigonal, and
tetragonal curves, respectively. The proposed analytical construction of the
Riemann surface gives a full control over computation of the Abel image of any
point on a curve. Therefore, computation of $\wp$-functions on given divisors
can be done directly. An alternative computation with the help of the Jacobi
inversion problem is used for verification. Hyperelliptic and trigonal curves
are considered in detail, and illustrated by examples.
