Probably among the LAPACK solvers, you'd need to look at:

DGESV (for general matrices), or

DSYSV, DSYSV_ROOK, DSYSVX (for symmetric matrices).

All the above are for double precision numbers - so the names begin with "D".

The speed of LAPACK routines highly depends on the underlying BLAS routines performance. If it is reference BLAS you are using, don't expect too good speed. A better option might be to use a tuned version of BLAS (e.g. ATLAS), of Intel MKL. Probably Intel MKL will be the simplest workaround.

Furthermore, when solving linear systems arising from continuous medium problems of large size (as in your case: you mention the solution time is not satisfactory) - it's better to rely on sparse matrices instead of dense matrices you currently use. There exist libraries which provide both direct and iterative solvers for sparse systems.