I am working on some ill-conditioned large sparse linear system of equations. I want to use double-double arithmetic or quad-double arithmetic to solve them. I know that there is a package named MPACK developed by Nakata, Maho, which can perform numerical linear algebraic computations under quad-double arithmetic. However, it is designed for dense matrix, not sparse matrix. Do you know whether there is any quad-double arithmetic sparse matrix package?
As of version 3.2, PETSc supports sparse quad-precision computations on gcc/gfortran 4.6 and newer.
You'll need a quad-precision BLAS and LAPACK, which PETSc can provide to you (along with quad support) with the following (partial) configure command:
./configure --with-precision=__float128 --download-f2cblaslapack
See the FAQ for a little more information.
Also, I agree with nOOb, if at all possible, try to regularize the system before switching out to quad-precision.
Ill-conditioned systems are better solved by regularisation than by increasing the numerical precision. Search for "regularisation ill-posed" for the gory details.
I might give the Trilinos library a try. They have templated sparse matrix libraries under Tpetra (which is supposed to replace Epetra, their original sparse matrix library). You can template double, complex, quad, etc, and they have possibly the largest selection of solvers (both direct and iterative) next to PETSc.
Edit: after reading the comments, the immediate usefulness of Tpetra seems to be a bit questionable w.r.t quad precision...
Multiprecision computing toolbox for MATLAB has support for sparse matrices and specifically optimized for computations with quadruple precision.
Here is timing details for quad-precision sparse solvers: Direct Solvers for Sparse Matrices
(I am the author of the toolbox).