Creating Sparse Finite-Element Matrices in MATLAB » Loren on the Art of MATLAB

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http://bl ogs.m athw or ks.com /lor en/2007/03/01/cr eati ng- spar se- fi ni te- element- matr ices- in- matl ab/

Posted by Loren Shure, March 1, 2007

I'm pleased to introduce Tim Davis as this week's guest blogger. Tim is a professor at the University of Florida, and is the author or co-author of

of our sparse matrix functions (lu, chol, much of sparse backslash, ordering methods such as amd and colamd, and other functions such as etre

and symbfact). He is also the author of a recent book, Direct Methods for Sparse Linear Systems, published by SIAM, where more details of MA

sparse matrices are discussed ( http://www.cise.ufl.edu/~davis ).

Contents

MATLAB is Slow? Only If You Use It Incorrectly

How Not to Create a Finite-Element Matrix

What's Wrong with It?

How MATLAB Stores Sparse Matrices

A Better Way to Create a Fin ite-Elemen t Matrix

Moral: Do Not Abuse A(i,j)=... for Sparse A Use sparse Instead

Try a Faster Sparse Function

When Fast is Not Fast Enough...

MATLAB is Slow? Only If You Use It Incorrectly

From time to time, I hear comments such as "MATLAB is slow for large finite-element problems." When I look at the details, the problem is typic

due to an overuse of A(i,j)= ... when creating the sparse matrix (assembling the finite elements). This can be seen in typical user's code,

MATLAB code in books on the topic, and even in MATLAB itself. The problem is widespread. MATLAB can be very fast for finite-element proble

but not if it's used incorrectly.

How Not to Create a Finite-Element Matrix

A goo d e xample of wh at not to do can be found in the wathen.mfunction, in MATLAB. A=gallery('wathen',200,200)takes a huge amoun

time a very minor modification cuts the run time drastically. I'm not intending to single out this one function for critique this is a very common iss

that I see over and over again. This function is built into MATLAB, which makes it an accessible example. It was first written when MATLAB did

support sparse matrices, and was modified only slightly to exploit sparsity. It was never meant for generating large sparse finite-element matric

You can see the entire function with the command:

type private/wathen. m

Below is an excerpt of the relevant parts of wathen.m. The function wathen1.mcreates a finite-element matrix of an nx-by-ny mesh. Each i lo

creates a single 8-by-8 finite-element matrix, and adds it into A.

type wathen1

tic

A = wathen1 (200,200)

toc
http://www.cise.ufl.edu/~davishttp://ec-securehost.com/SIAM/FA02.html


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function A = wathen1 (nx,ny)

rand('state',0)

e1 = [6 -6 2 -8-6 32 -6 202 -6 6 -6-8 20 -6 32]

e2 = [3 -8 2 -6-8 16 -8 202 -8 3 -8-6 20 -8 16]

e = [e1 e2 e2' e1]/45

n = 3*nx*ny+2*nx+2*ny+1

A = sparse(n,n)

RHO = 100*rand(nx,ny)

nn = zeros(8,1)

for j=1:ny

for i=1:nx

nn(1) = 3*j*nx+2*i+2*j +1

nn(2) = nn(1)-1

nn(3) = nn(2)-1

nn(4) = (3*j-1)*nx+2*j +i-1

nn(5) = 3*(j-1)*nx+2*i +2*j-3

nn(6) = nn(5)+1

nn(7) = nn(6)+1

nn(8) = nn(4)+1 em = e*RHO(i,j)

for krow=1:8

for kcol=1:8

A(nn(krow),nn( kcol)) = A(nn(krow),nn( kcol))+em(krow,k col)

end

end

end

end

Elapsed time is 305.832709 seconds.

What's Wrong with It?

The above code is fine for generating a modest sized matrix, but the A(i,j) = ... statement is quite slow whenAis large and sparse, partic

when iandj are also scalars. The inner two for-loops can be vectorized so thatiandjare vectors of length 8. Each A(i,j) = ...stateme

assembling an entire finite-element matrix into A. However, this leads to very minimal improvement in run time.

type wathen1b

tic

A1b = wathen1b (200,200)

toc


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function A = wathen1b (nx,ny)

rand('state',0)

e1 = [6 -6 2 -8-6 32 -6 202 -6 6 -6-8 20 -6 32]

e2 = [3 -8 2 -6-8 16 -8 202 -8 3 -8-6 20 -8 16]

e = [e1 e2 e2' e1]/45

n = 3*nx*ny+2*nx+2*ny+1

A = sparse(n,n)


nn = zeros(8,1)

for j=1:ny

for i=1:nx

nn(1) = 3*j*nx+2*i+2*j +1

nn(2) = nn(1)-1

nn(3) = nn(2)-1

nn(4) = (3*j-1)*nx+2*j +i-1

nn(5) = 3*(j-1)*nx+2*i +2*j-3

nn(6) = nn(5)+1

nn(7) = nn(6)+1

nn(8) = nn(4)+1 em = e*RHO(i,j)

A (nn,nn) = A (nn,nn) + em

end

end


disp (norm (A-A1b,1))

0

How MATLAB Stores Sparse Matrices

To understand why the above examples are so slow, you need to understand how MATLAB stores its sparse matrices. An n-by-n MATLAB spars

matrix is stored as three arrays I'll call them p, i, and x. These three arrays are not directly accessible from M, but they can be accessed by a

mexFunction. The nonzero entries in column jare stored in i(p(j):p(j+1)-1)andx(p(j):p(j+1)-1), where xholds the numerical value

iholds the corresponding row indices. Below is a very small example. First, I create a fullmatrix and convert it into a sparse one. This is only s

you can easily see the matrix Cand how it's stored in sparse form. You should never create a sparse matrix this way, except for tiny examples.

C = [

4.5 0.0 3.2 0.0

3.1 2.9 0.0 0.9

0.0 1.7 3.0 0.0

3.5 0.4 0.0 1.0 ]

C = sparse (C)


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C =

(1,1) 4.5000

(2,1) 3.1000

(4,1) 3.5000

(2,2) 2.9000

(3,2) 1.7000

(4,2) 0.4000

(1,3) 3.2000

(3,3) 3.0000

(2,4) 0.9000

(4,4) 1.0000

Notice that the nonzero entries in Care stored in column order, with sorted row indices. The internal p, i, and xarrays can be reconstructed as

follows. The find(C)statement returns a list of "triplets," where the kth triplet is i(k), j(k), and x(k). This specifies that C(i(k),j(k))is eq

x(k). Next, find(diff(...))constructs the column pointer array p(this only works if there are no all-zero columns in the matrix).

[i j x] = find (C)

n = size(C,2)

p = find (diff ([0 j n+1]))

for col = 1:n

fprintf ('column %d:\n k row index value\n', col)

disp ([(p(col):p(col+ 1)-1)' i(p(col):p(col+1 )-1) x(p(col):p(col+1 )-1)])

end

column 1:

k row index value

1.0000 1.0000 4.5000

2.0000 2.0000 3.1000

3.0000 4.0000 3.5000

column 2:

k row index value

4.0000 2.0000 2.9000

5.0000 3.0000 1.7000

6.0000 4.0000 0.4000

column 3:

k row index value

7.0000 1.0000 3.2000

8.0000 3.0000 3.0000

column 4:

k row index value

9.0000 2.0000 0.9000

10.0000 4.0000 1.0000

Now consider what happens when one entry is added to C:


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C(3,1) = 42

[i j x] = find (C)

n = size(C,2)

p = find (diff ([0 j n+1]))

for col = 1:n

fprintf ('column %d:\n k row index value\n', col)

disp ([(p(col):p(col+ 1)-1)' i(p(col):p(col+1 )-1) x(p(col):p(col+1 )-1)])

end

column 1:

k row index value

1.0000 1.0000 4.5000

2.0000 2.0000 3.1000

3.0000 3.0000 42.0000

4.0000 4.0000 3.5000

column 2:

k row index value

5.0000 2.0000 2.9000

6.0000 3.0000 1.7000

7.0000 4.0000 0.4000

column 3:

k row index value

8.0000 1.0000 3.2000

9.0000 3.0000 3.0000

column 4:

k row index value

10.0000 2.0000 0.9000

11.0000 4.0000 1.0000

and you can see that nearly every entry in Chas been moved down by one in the iandxarrays. In general, the single statement C(3,1)=42t

time proportional to the number of entries in matrix. Thus, looping nnz(A)times over the statementA(i,j)=A(i,j)+... takes time proportio

nnz(A)^2.

A Better Way to Create a Finite-Element Matrix

The version below is only slightly different. It could be improved, but I left it nearly the same to illustrate how simple it is to write fast MATLAB co

solve this problem, via a minor tweak. The idea is to create a list of triplets, and let MATLAB convert them into a sparse matrix all at once. If there

duplicates (which a finite-element matrix always has) the duplicates are summed, which is exactly what you want when assembling a finite-ele

matrix. In MATLAB 7.3 (R2006b), sparseuses quicksort, which takes nnz(A)*log(nnz(A))time. This is slower than it could be (a linear-time

bucket sort can be used, taking essentially nnz(A)time). However, it's still much faster thannnz(A)^2. For this matrix, nnz(A)is about 1.9 mi

type wathen2.m

tic

A2 = wathen2 (200,200)

toc


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function A = wathen2 (nx,ny)

rand('state',0)

e1 = [6 -6 2 -8-6 32 -6 202 -6 6 -6-8 20 -6 32]

e2 = [3 -8 2 -6-8 16 -8 202 -8 3 -8-6 20 -8 16]

e = [e1 e2 e2' e1]/45

n = 3*nx*ny+2*nx+2*ny+1

ntriplets = nx*ny*64

I = zeros (ntriplets, 1)

J = zeros (ntriplets, 1)

X = zeros (ntriplets, 1)

ntriplets = 0


nn = zeros(8,1)

for j=1:ny

for i=1:nx

nn(1) = 3*j*nx+2*i+2*j +1

nn(2) = nn(1)-1

nn(3) = nn(2)-1

nn(4) = (3*j-1)*nx+2*j +i-1 nn(5) = 3*(j-1)*nx+2*i +2*j-3

nn(6) = nn(5)+1

nn(7) = nn(6)+1

nn(8) = nn(4)+1

em = e*RHO(i,j)

for krow=1:8

for kcol=1:8

ntriplets = ntriplets + 1

I (ntriplets) = nn (krow)

J (ntriplets) = nn (kcol)

X (ntriplets) = em (krow,kcol)

end end

end

end

A = sparse (I,J,X,n,n)


disp (norm (A-A2,1))

1.4211e-014

If you do not know how many entries your matrix will have, you may not be able to preallocate the I, J, and Xarrays, as done in wathen2.m. In

case, start them at a reasonable size (anything larger than zero will do) and add this code to the innermost loop, just after ntripletsis

incremented:


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len = length (X)

if (ntriplets > len)

I (2*len) = 0

J (2*len) = 0

X (2*len) = 0

end

and when done, use sparse(I(1:ntriplets),J(1:ntriplets),X(1:ntriplets),n,n).

Moral: Do Not Abuse A(i,j)=... for Sparse A Use sparse Instead

Avoid statements such as

C(4,2) = C(4,2) + 42

in a loop. Create a list of triplets (i,j,x) and use sparse instead. This advice holds for any sparse matrix, not just finite-element ones.

Try a Faster Sparse Function

CHOLMOD includes a sparse2mexFunction which is a replacement forsparse. It uses a linear-time bucket sort. The MATLAB 7.3 (R2006b)

sparseaccounts for about 3/4ths the total run time of wathen2.m. For this matrix sparse2in CHOLMOD is about 10 times faster than the MAT

sparse. CHOLMOD can be found in the SuiteSparse package, in MATLAB Central.

If you would like to see a short and concise C mexFunction implementation of the method used by sparse2, take a look at CSparse, which wa

written for a concise textbook-style presentation. The cs_sparse mexFunction usescs_compress.c, to convert the triplets to a compressed c

form of A', cs_dupl.cto sum up duplicate entries, and then cs_transpose.cto transpose the result (which also sorts the columns).

When Fast is Not Fast Enough...

Even with this dramatic improvement in constructing the matrix A, MATLAB could still use additional features for faster construction of sparse fin

element matrices. Constructing the matrix should be much faster than x=A\b, since cholis doing about 700 times more work as sparsefor th

matrix (1.3 billion flops, vs 1.9 million nonzeros in A). The run times are not that different, however:

tic

A = wathen2 (200,200)

toc

b = rand (size(A,1),1)

tic

x=A\b

toc



Get the MATLAB

Published with MATL
http://grabcode_21d29e7e6e0e4c2e827c670f6ecf434e%28%29/http://www.mathworks.com/access/helpdesk/help/techdoc/ref/chol.htmlhttp://www.cise.ufl.edu/research/sparse/CSparse/CSparse/Source/cs_transpose.chttp://www.cise.ufl.edu/research/sparse/CSparse/CSparse/Source/cs_dupl.chttp://www.cise.ufl.edu/research/sparse/CSparse/CSparse/Source/cs_compress.chttp://www.cise.ufl.edu/research/sparse/CSparse/CSparse/MATLAB/CSparse/cs_sparse_mex.chttp://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=11740http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=12233&objectType=FILEhttp://www.mathworks.com/access/helpdesk/help/techdoc/ref/sparse.html


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