MATLAB Mathematics - SERC - Index of
MATLAB Mathematics - SERC - Index of MATLAB Mathematics - SERC - Index of
6 Sparse Matrices simply sparse(m,n). The sparse analogs of rand and eye are sprand and speye, respectively. There is no sparse analog for the function ones. • Unary functions that accept a matrix and return a matrix or vector preserve the storage class of the operand. If S is a sparse matrix, then chol(S) is also a sparse matrix, and diag(S) is a sparse vector. Columnwise functions such as max and sum also return sparse vectors, even though these vectors may be entirely nonzero. Important exceptions to this rule are the sparse and full functions. • Binary operators yield sparse results if both operands are sparse, and full results if both are full. For mixed operands, the result is full unless the operation preserves sparsity. If S is sparse and F is full, then S+F, S*F, and F\S are full, while S.*F and S&F are sparse. In some cases, the result might be sparse even though the matrix has few zero elements. • Matrix concatenation using either the cat function or square brackets produces sparse results for mixed operands. • Submatrix indexing on the right side of an assignment preserves the storage format of the operand unless the result is a scalar. T = S(i,j) produces a sparse result if S is sparse and either i or j is a vector. It produces a full scalar if both i and j are scalars. Submatrix indexing on the left, as in T(i,j) = S, does not change the storage format of the matrix on the left. • Multiplication and division are performed on only the nonzero elements of sparse matrices. Dividing a sparse matrix by zero returns a sparse matrix with Inf at each nonzero location. Because the zero-valued elements are not operated on, these elements are not returned as NaN. Similarly, multiplying a sparse matrix by Inf or NaN returns Inf or NaN, respectively, for the nonzero elements, but does not fill in NaN for the zero-valued elements. Permutation and Reordering A permutation of the rows and columns of a sparse matrix S can be represented in two ways: • A permutation matrix P acts on the rows of S as P*S or on the columns as S*P'. • A permutation vector p, which is a full vector containing a permutation of 1:n, acts on the rows of S as S(p,:), or on the columns as S(:,p). 6-26
Sparse Matrix Operations For example, the statements p = [1 3 4 2 5] I = eye(5,5); P = I(p,:); e = ones(4,1); S = diag(11:11:55) + diag(e,1) + diag(e,-1) produce p = P = S = 1 3 4 2 5 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 11 1 0 0 0 1 22 1 0 0 0 1 33 1 0 0 0 1 44 1 0 0 0 1 55 You can now try some permutations using the permutation vector p and the permutation matrix P. For example, the statements S(p,:) and P*S produce ans = 11 1 0 0 0 0 1 33 1 0 0 0 1 44 1 1 22 1 0 0 0 0 0 1 55 6-27
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Sparse Matrix Operations<br />
For example, the statements<br />
p = [1 3 4 2 5]<br />
I = eye(5,5);<br />
P = I(p,:);<br />
e = ones(4,1);<br />
S = diag(11:11:55) + diag(e,1) + diag(e,-1)<br />
produce<br />
p =<br />
P =<br />
S =<br />
1 3 4 2 5<br />
1 0 0 0 0<br />
0 0 1 0 0<br />
0 0 0 1 0<br />
0 1 0 0 0<br />
0 0 0 0 1<br />
11 1 0 0 0<br />
1 22 1 0 0<br />
0 1 33 1 0<br />
0 0 1 44 1<br />
0 0 0 1 55<br />
You can now try some permutations using the permutation vector p and the<br />
permutation matrix P. For example, the statements S(p,:) and P*S produce<br />
ans =<br />
11 1 0 0 0<br />
0 1 33 1 0<br />
0 0 1 44 1<br />
1 22 1 0 0<br />
0 0 0 1 55<br />
6-27
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