Graphical Representation of Sparse Matrices

View Finite Element Mesh

First, scale x and y by $2^{-32}$ to bring them into the range $\left[0,1\right]$. 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')

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')

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')

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')

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')