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Sparse Matrices

This example shows how reordering the rows and columns of a sparse matrix can influence the speed and storage requirements of a matrix operation.

Visualizing a Sparse Matrix

A SPY plot shows the nonzero elements in a matrix.

This spy plot shows a SPARSE symmetric positive definite matrix derived from a portion of the Harwell-Boeing test matrix "west0479", a matrix describing connections in a model of a diffraction column in a chemical plant.

load west0479.mat
A = west0479;
S = A * A' + speye(size(A));
pct = 100 / numel(A);

figure
spy(S)
title('A Sparse Symmetric Matrix')
nz = nnz(S);
xlabel(sprintf('nonzeros = %d (%.3f%%)',nz,nz*pct));

Computing the Cholesky Factor

Now we compute the Cholesky factor L, where S = L*L'. Notice that L contains MANY more nonzero elements than the unfactored S, because the computation of the Cholesky factorization creates "fill-in" nonzeros. This slows down the algorithm and increases storage cost.

tic
L = chol(S,'lower');
t(1) = toc;
spy(L), title('Cholesky decomposition of S')
nc(1) = nnz(L);
xlabel(sprintf('nonzeros = %d (%.2f%%)   time = %.2f sec',nc(1),nc(1)*pct,t(1)));

Reordering to Speed Up the Calculation

By reordering the rows and columns of a matrix, it may be possible to reduce the amount of fill-in created by factorization, thereby reducing time and storage cost.

We will now try three different orderings supported by MATLAB®.

  • reverse Cuthill-McKee

  • column count

  • minimum degree

Using the Reverse Cuthill-McKee

The SYMRCM command uses the reverse Cuthill-McKee reordering algorithm to move all nonzero elements closer to the diagonal, reducing the "bandwidth" of the original matrix.

p = symrcm(S);
spy(S(p,p))
title('S(p,p) after Cuthill-McKee ordering')
nz = nnz(S);
xlabel(sprintf('nonzeros = %d (%.3f%%)',nz,nz*pct));

The fill-in produced by Cholesky factorization is confined to the band, so that factorization of the reordered matrix takes less time and less storage.

tic
L = chol(S(p,p),'lower');
t(2) = toc;
spy(L)
title('chol(S(p,p)) after Cuthill-McKee ordering')
nc(2) = nnz(L);
xlabel(sprintf('nonzeros = %d (%.2f%%)   time = %.2f sec', nc(2),nc(2)*pct,t(2)));

Using Column Count

The COLPERM command uses the column count reordering algorithm to move rows and columns with higher nonzero count towards the end of the matrix.

q = colperm(S);
spy(S(q,q)), title('S(q,q) after column count ordering')
nz = nnz(S);
xlabel(sprintf('nonzeros = %d (%.3f%%)',nz,nz*pct));

For this example, the column count ordering happens to reduce the time and storage for Cholesky factorization, but this behavior cannot be expected in general.

tic
L = chol(S(q,q),'lower');
t(3) = toc;

spy(L)
title('chol(S(q,q)) after column count ordering')
nc(3) = nnz(L);
xlabel(sprintf('nonzeros = %d (%.2f%%)   time = %.2f sec',nc(3),nc(3)*pct,t(3)));

Using Minimum Degree

The SYMAMD command uses the approximate minimum degree algorithm (a powerful graph-theoretic technique) to produce large blocks of zeros in the matrix.

r = symamd(S);
spy(S(r,r)), title('S(r,r) after minimum degree ordering')
nz = nnz(S);
xlabel(sprintf('nonzeros = %d (%.3f%%)',nz,nz*pct));

The blocks of zeros produced by the minimum degree algorithm are preserved during the Cholesky factorization. This can significantly reduce time and storage costs.

tic
L = chol(S(r,r),'lower');
t(4) = toc;

spy(L)
title('chol(S(r,r)) after minimum degree ordering')
nc(4) = nnz(L);
xlabel(sprintf('nonzeros = %d (%.2f%%)   time = %.2f sec',nc(4),nc(4)*pct,t(4)));

Summarizing the Results

labels = {'original','Cuthill-McKee','column count','min degree'};

ax = subplot(2,1,1);
bar(nc*pct)
title('Nonzeros after Cholesky factorization')
ylabel('Percent');
ax.XTickLabel = labels;

ax = subplot(2,1,2);
bar(t)
title('Time to complete Cholesky factorization')
ylabel('Seconds');
ax.XTickLabel = labels;

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