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Graphical Representation of Sparse Matrices

This example shows the finite element mesh for a NASA airfoil, including two trailing flaps. More information about the history of airfoils is available at NACA Airfoils (nasa.gov).

The data is stored in the file airfoil.mat. The data consists of 4253 pairs of (x,y) coordinates of the mesh points. It also contains an array of 12,289 pairs of indices, (i,j), specifying connections between the mesh points.

Load the data file into the workspace.

load airfoil

View Finite Element Mesh

First, scale x and y by 2-32 to bring them into the range [0,1]. Then form a sparse adjacency matrix from the (i,j) connections and make it positive definite. Finally, plot the adjacency matrix using (x,y) as the coordinates for the vertices (mesh points).

% Scaling x and y
x = pow2(x,-32); 
y = pow2(y,-32);

% Forming the sparse adjacency matrix and making it positive definite
n = max(max(i),max(j));
A = sparse(i,j,-1,n,n);
A = A + A';
d = abs(sum(A)) + 1;
A = A + diag(sparse(d));

% Plotting the finite element mesh
gplot(A,[x y])
title('Airfoil Cross-Section')

Figure contains an axes object. The axes object with title Airfoil Cross-Section contains an object of type line.

Visualize Sparsity Pattern

You can use spy to visualize the nonzero elements in a matrix, so it is a particularly useful function to see the sparsity pattern in sparse matrices. spy(A) plots the sparsity pattern of the matrix A.

spy(A)
title('Airfoil Adjacency Matrix')

Figure contains an axes object. The axes object with title Airfoil Adjacency Matrix contains an object of type line.

Symmetric Reordering - Reverse Cuthill-McKee

symrcm uses the Reverse Cuthill-McKee technique for reordering the adjacency matrix. r = symrcm(A) returns a permutation vector r such that A(r,r) tends to have its diagonal elements closer to the diagonal than A. This form is a good preordering for LU or Cholesky factorization of matrices that come from "long, skinny" problems. It works for both symmetric and nonsymmetric matrices.

r = symrcm(A);
spy(A(r,r))
title('Reverse Cuthill-McKee')

Figure contains an axes object. The axes object with title Reverse Cuthill-McKee contains an object of type line.

Symmetric Reordering - Column Permutations

Use j = COLPERM(A) to return a permutation vector that reorders the columns of the sparse matrix A in nondecreasing order of nonzero count. This form is sometimes useful as a preordering for LU factorization, as in lu(A(:,j)).

j = colperm(A);
spy(A(j,j))
title('Column Count Reordering')

Figure contains an axes object. The axes object with title Column Count Reordering contains an object of type line.

Symmetric Reordering - Symmetric Approximate Minimum Degree

symamd gives a symmetric approximate minimum degree permutation. For a symmetric positive definite matrix A, the command p = symamd(S) returns the permutation vector p such that S(p,p) tends to have a sparser Cholesky factor than S. Sometimes symamd works well for symmetric indefinite matrices too.

m = symamd(A);
spy(A(m,m))
title('Approximate Minimum Degree')

Figure contains an axes object. The axes object with title Approximate Minimum Degree contains an object of type line.

See Also

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