Re: Re: Complexity explosion in linear solve

*To*: mathgroup at smc.vnet.net*Subject*: [mg80303] Re: [mg80264] Re: Complexity explosion in linear solve*From*: danl at wolfram.com*Date*: Fri, 17 Aug 2007 01:46:01 -0400 (EDT)*References*: <f9s3f8$a8r$1@smc.vnet.net><f9udg9$ajv$1@smc.vnet.net>

> On Aug 15, 2:28 am, Jens-Peer Kuska <ku... at informatik.uni-leipzig.de> > wrote: >> 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 >> >> car... at colorado.edu wrote: >> > 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. > > Hi Jens, > > I know *that* - I teach numerical analysis, among other stuff. > Also I teach the kids to avoid Cramer's rule for systems > of order >3. Then can you explain me why > > MyLinearSolve[A_,b_]:=Module[{i,j,Ab,detA,detAb,n=Length[A],x}, > detA=Simplify[Det[A]]; x=Table[0,{n}]; > For [i=1,i<=n,i++, Ab=Transpose[A]; > Ab[[i]]=b; detAb=Simplify[Det[Ab]]; > x[[i]]=Simplify[detAb/detA] ]; > ClearAll[Ab]; > Return[x]]; > > beats LinearSolve for a symbolic system of order 13? > Here are my times on a dual G5 Mac: > > LinearSolve 235 sec unsimplified x > MyLinraSolve 6 sec simplified x > > For the simplification of the LinearSolve solution x > I estimate 10^8 years. > Hard to say much without seeing the specific input you use. I'll venture a guess below. As for Cramer's rule...for symbolic systems it is often a "good" method but requires huge memory to do efficiently (one must memo-ize smaller cofactors in order to do the computations recursively). For sufficiently large dimension-- 12 and up if I recall correctly-- it is not used as a default method in LinearSolve et al. In any case it won't work for nonsquare or rank deficient matrices. Also for LinearSolve the built in cofactor method (Method ->CofactorExpansion) is not terribly efficient. Instead of using Cramer's rule is in effect inverting the matrix via cofactors and then multiplying the rhs by that inverse. This is probably a weakness in the implementation. I'll show what happens with what I think is the matrix and rhs you actually use (this is something you should have tried by now). I use mat as your A above. b = q/2*{0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}; rng = Complement[Range[16], {1, 2, 13}]; mat2 = mat[[rng, rng]]; b2 = b[[rng]]; In[28]:= Timing[ sol1 = LinearSolve[mat2, b2, Method -> "OneStepRowReduction"]] Out[28]= {0.271, {q/9, 0, (4 q)/3, 0, q/9, 0, (4 q)/3, 0, q/9, 0, 0, q/9, 0}} Your method also works well for this example. In[29]:= Timing[sol2 = myLinearSolve[mat2, b2]] Out[29]= {3.104, {q/9, 0, (4 q)/3, 0, q/9, 0, (4 q)/3, 0, q/9, 0, 0, q/9, 0}} In[36]:= Timing[sol3 = myLinearSolve[mat2, b2]] Out[36]= {1.882, {q/9, 0, (4 q)/3, 0, q/9, 0, (4 q)/3, 0, q/9, 0, 0, q/9, 0}} I'd define it a bit differently, though. myLinearSolve[A_, b_] := Module[{i, detA = Simplify[Det[A]], atran = Transpose[A]}, Table[Simplify[Det[ReplacePart[atran, i -> b]]/detA], {i,Length[A]}]] Daniel Lichtblau Wolfram Research