Re: Complexity explosion in linear solve

• To: mathgroup at smc.vnet.net
• Subject: [mg80229] Re: Complexity explosion in linear solve
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Wed, 15 Aug 2007 04:18:19 -0400 (EDT)
• Organization: Uni Leipzig
• References: <f9s3f8\$a8r\$1@smc.vnet.net>

Hi,

what do you expect ? matrix inversion is an N^3 process
when N is the dimension of the matrix. This mean for numerical
data, that you have to do N^3 operations to get the inverse.
For symbolic data a single operation gives not a single number
it simply add another leaf to your expression because 1+2 is 3
(leaf count 1) but a+b is a+b (leaf count 2) so every of your N^3
operations add a new leaf to your expression and you end up (in the
worst case) with N^3 more leafs in your expression ..

Regards
Jens

> LinearSolve (and cousins Inverse, LUDecomposition etc) works
> efficiently with
> numerical float data.  But it is prone to complexity explosion in
> solving symbolic
> systems of moderate size.  Here is an example from an actual
> application.
>
> Task: solve A x = b for x where A is 16 x 16 and b is 16 x 1:
>
> A={{7/12,1/24,-1/2,0,-1/4,-5/24,-1/2,0,
> 0,-1/24,-1/4,0,-1/3,5/24,-1/4,0},{1/24,91/144,-2/3,0,
> 5/24,-3/16,-1/3,0,-1/24,0,-1/3,0,-5/24,-4/9,-2/3,
> 0},{-1/2,-2/3,(-9+(1378*C11*hh)/225)/2,(-4*C13*hh)/15,
> 1/2,-1/3,(-3-2*C11*hh)/2,4*C13*hh,1/4,
> 1/3,(-3-(32*C11*hh)/25)/2,0,-1/4,
> 2/3,(-3-(128*C11*hh)/45)/2,(-64*C13*hh)/15},{0,
> 0,(-4*C13*hh)/15,(224*C22*hh)/5,0,0,-4*C13*hh,16*C23*hh,0,0,0,
> 16*C23*hh,0,0,(64*C13*hh)/15,(64*C23*hh)/5},{-1/4,5/24,1/2,0,
> 7/12,-1/24,1/2,0,-1/3,-5/24,1/4,0,0,1/24,1/4,
> 0},{-5/24,-3/16,-1/3,0,-1/24,91/144,-2/3,0,5/24,-4/9,-2/3,
> 0,1/24,0,-1/3,0},{-1/2,-1/3,(-3-2*C11*hh)/2,-4*C13*hh,
> 1/2,-2/3,(-9+(314*C11*hh)/45)/2,(4*C13*hh)/15,1/4,
> 2/3,(-3-(128*C11*hh)/45)/2,(64*C13*hh)/15,-1/4,
> 1/3,(-3-(32*C11*hh)/15)/2,0},{0,0,4*C13*hh,16*C23*hh,0,
> 0,(4*C13*hh)/15,(832*C22*hh)/15,0,
> 0,(-64*C13*hh)/15,(64*C23*hh)/5,0,0,0,(80*C23*hh)/3},{0,-1/24,
> 1/4,0,-1/3,5/24,1/4,0,7/12,1/24,1/2,0,-1/4,-5/24,1/2,
> 0},{-1/24,0,1/3,0,-5/24,-4/9,2/3,0,1/24,91/144,2/3,0,
> 5/24,-3/16,1/3,0},{-1/4,-1/3,(-3-(32*C11*hh)/25)/2,0,
> 1/4,-2/3,(-3-(128*C11*hh)/45)/2,(-64*C13*hh)/15,1/2,
> 2/3,(-9+(1378*C11*hh)/225)/2,(-4*C13*hh)/15,-1/2,
> 1/3,(-3-2*C11*hh)/2,4*C13*hh},{0,0,0,16*C23*hh,0,
> 0,(64*C13*hh)/15,(64*C23*hh)/5,0,0,(-4*C13*hh)/15,(224*C22*hh)/5,
> 0,0,-4*C13*hh,16*C23*hh},{-1/3,-5/24,-1/4,0,0,1/24,-1/4,
> 0,-1/4,5/24,-1/2,0,7/12,-1/24,-1/2,0},{5/24,-4/9,2/3,0,
> 1/24,0,1/3,0,-5/24,-3/16,1/3,0,-1/24,91/144,2/3,
> 0},{-1/4,-2/3,(-3-(128*C11*hh)/45)/2,(64*C13*hh)/15,
> 1/4,-1/3,(-3-(32*C11*hh)/15)/2,0,1/2,
> 1/3,(-3-2*C11*hh)/2,-4*C13*hh,-1/2,
> 2/3,(-9+(314*C11*hh)/45)/2,(4*C13*hh)/15},{0,
> 0,(-64*C13*hh)/15,(64*C23*hh)/5,0,0,0,(80*C23*hh)/3,0,0,
> 4*C13*hh,16*C23*hh,0,0,(4*C13*hh)/15,(832*C22*hh)/15}}
>
> b={0,0,0,0,q*b/2,0,0,0,q*b/2,0,0,0,0,0,0,0}
>
> in which all symbolic variables are atoms. Matrix A has rank 13.
> Three BCs: x1=x2=x13=0 are imposed by removing  equations 1, 2 and
> 13,
> which gives the reduced 13-system Ahat xhat = bhat. Matrix Ahat is
> symmetric but indefinite, so Cholesky is not an option. Instead
> LinearSolve is used
>
>  xhat=LinearSolve[Ahat,bhat]
>
> Solving takes about 3 mins on a dual G5 running 5.2.  LeafCounts of
> the xhat entries reach hundreds of millions:
>
> {325197436,235675446,292306655,256982512,146153454,73076907,35324210,
> 18877804,9441784,4745440,2429139,1354800,903023}
>
> Obviously Simplify would take a long, long time so I didnt attempt it.
> Another solution method, however, gave this solution in about 10 sec:
>
> xhat={q/3, 0, 4*q, 0, q/3, 0, 4*q, 0, q/3, 0, 0, q/3, 0}
>
> My question is: is there a way to tell LinearSolve to Simplify as it
> goes
> along? That would preclude or at least alleviate the leafcount
> explosion.
>
>

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