which Solvers are efficient?
May explain that "TDMA algorithm" is efficient yet or not?
which new solvers are there for solving equations system? 
You should make your question more precise. Which algorithm is efficient often depends on the structure and properties of your system of equations. Why TDMA is efficient is more or less obvious... For tridiagonal matrices, the Gaussian elimination can be done in O(n) instead of O(n^3) due to the tridiagonal pattern. Unfortunately, tridiagonal matrices are not very common in real applications. Typically, only simple onedimensional problems can be formulated as a tridiagonal system.

Well for the case of TDMA, it's actually just a backward substitution. So I think for the case of tridiagonal matrix TDMA is most efficient solver.

TDMA can be used only when the matrix have five diagonals
if you obtaned the matrix on a unstructed grid system, krylov iteration method is strongly sugguested. Regards 
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