hello folks; I really wonder what "projected" gauss seidel is? Of course, I fully understand the canonical Gauss Seidel, but what "projected" means? I have no clue, anybody who can tell about it? Thank you.

While Gauss-Seidel is an iterative method for solving systems of linear equations, the projected Gauss-Seidel (PGS) solves linear complementary problems (LCP) which you typically find in rigid body simulation to handle contacts and limits. The LCP is defined as:

w = A*x + b, x >= 0, w >= 0, x_i * w_i = 0


You can look for Murty's webbook to learn about LCPs and I think it deals with iterative methods to solve these kind of problems, though not necessarily with PGS. I am relatively sure that the book by Cottle & Dantzig deals with it. The later book is only necessary if you professionally write physic engines...

of course, I will rewrite this part of code, because I will use SSE to parallel it.

By the way, I am interested in ODE as you mentioned above.
And I want to parallel the sphere vs sphere and sphere vs plane.

The PGS is used for contact handling, not for finding the contacts. Where did I mention the ODE? If you refer to their quickstep solver this is indeed an (optimized) PGS solver.

BTW: If you plan to code for the PC forget about SIMD.