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issue with LinearSolve[] when using SparseArray when size close

  • To: mathgroup at smc.vnet.net
  • Subject: [mg113622] issue with LinearSolve[] when using SparseArray when size close
  • From: "Nasser M. Abbasi" <nma at 12000.org>
  • Date: Fri, 5 Nov 2010 05:11:00 -0500 (EST)
  • Reply-to: nma at 12000.org

There is some serious problem I am getting when trying to use 
Mathematica LinearSolve[] to solve  Au=f when using SparseArray when the 
size of non-zero elements gets close to 1 GB.

The non-zero elements in A that I am building is of the order n^3.

Therefore, for n=255, memory needed should be around 130 Mega Bytes 
(255^2*8) (using 8 bytes for each value, I am using numerical everything).

For n=511, memory used should be a little over 1 Giga Bytes.

I have 8 GB Ram. Windows 7, new Intel CPU.

When I run the solver for n=31 or 63 or 127 or 255, it all works, and 
Mathematica is fast solving Au=f, takes few seconds, and no problem. 
These are the CPU times reported by Mathemtica Timing command for the 
LinearSolve[] call

n=63, cpu=0.03
n=127, cpu=0.078
n=255, cpu=0.437
n=511, been running for many hrs, memory problem.

When I changed to n=511, I see memory of the Mathematica Kernel going up 
to almost 100% of the PC memory, using almost 7 GB, and I have waited 
for 9 hrs, and Mathematica is still not done.

There seems, on the face of it, something really wrong here. It seems 
SparseArray behavior or how LinearSolve[] uses it, does not scale well 
at all? or is this a windows OS issue? if it is a bug in my code, but it 
works so well for all the other n values?

I am posting the code, it is really small code, to see if someone can 
please try it on their PC and see if they get the same behavior.

------------ code -----------------
$MinPrecision=$MachinePrecision;$MaxPrecision=$MachinePrecision;
Share[];

(* make sparse A*)
makeA[n_?(IntegerQ[#]&&Positive[#]&)]:=Module[{r,off,block},
r=Table[-4,{i,n}];
off=Table[1,{i,n-1}];

block =DiagonalMatrix[r,0]+DiagonalMatrix[off,1]+DiagonalMatrix[off,-1];

SparseArray[{Band[{1,1}]->ConstantArray[block,{n}],Band[{1,n+1}]->1,Band[{n+1,1}]->1}]
];

(* f(x,y) *)
force[i_?(IntegerQ[#]&),j_?(IntegerQ[#]&),h_?(NumericQ[#]&&Positive[#] 
&)]:=Module[{x=i*h,y=j*h},
N@Exp[-(x-0.25)^2-(y-0.6)^2]
];

(*n=127; h=2^-7;*)    (*these values are OK *)
(*n=255; h=2^-8;*)    (*these values are OK *)
n=511; h=2^-9;  (*these cause problem *)

A=N[makeA[n]];

(* fill in f vector, in correct order for problem*)
f=Table[0,{i,n^2}];
For[j=1,j<=n,j++,
For[i=1,i<=n,i++,
f[[j+n*(i-1)]]=force[i,j,h]
]
];

Print["before solver, MemoryInUse[]=",MemoryInUse[]];
{cpu,sol}=Timing[LinearSolve[A,f]];
Print["after solver, MemoryInUse[]=",MemoryInUse[]];
Print["after solver, cpu=",cpu];

---------------- end code -----------------

thanks
--Nasser


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