SV: [ODE] Iterative solution

[Russ Smith]

> solving iteratively an LCP problem is not the same as solving a linear
> system or i am missing out something?
> linear systems means that we have only equality constraints and no
> contacts(inequality constraints) are involved.

'iterative' is one of those overused words that causes confusion. in one
sense all algorithms that invalve loops are iterative (but that is a
useless classification). here's the terminology i use:

* a 'direct' solution of the matrix problem A*x=b has a fixed number of
  predetermined steps. you know in advance how long it will take.

* an 'iterative' solution of the matrix problem A*x=b involves a varying
  number of iterations, each of which is much simpler than the direct
  solution. you may not know in advance how many steps will be needed to
  acheve a given accurary (you may still use a fixed number of iterations
  anyway).

* an LCP problem is [ A*x=b+w, x>=0, y>=0 ]. all algorithms i know of to
  solve LCP are iterative in the sense that they have this structure:
      loop:
        do something
        have we got the solution yet? exit if so.
      endloop
  i.e. their running time is not predetermined. at each iteration some of
  these algorithms (e.g. dantzig, principal pivoting) may perform a direct
  solution of a sub-problem of A*x=b+w. i call these 'direct LCP' methods.
  other algorithms (e.g. SOR) converge on the matrix solution at the same
  time as they converge on the active index set. i call these 'iterative
  LCP' methods.

russ.

--
Russ Smith
http://www.q12.org/